1   Rules

1. You may use any books, notes, and internet sites.
2. You may use calculators and SW like Matlab or Mathematica.
3. You may not ask anyone for help.
4. You may not communicate with anyone else about the course or the exam.
5. You may not accept help from other people. E.g., if someone offers to give you help w/o your asking, you may not accept.
6. You have 24 hours.
7. That is, your answers must be in gradescope by 4pm (1600) EDT Tues.
8. Email me with any questions. Do not wait until just before the due time.
9. Write your answers on blank sheets of paper and scan them, or use a notepad or app to write them directly into a file. Upload it to gradescope.
10. You may mark any 10 points as FREE and get the points.
11. Print your name and rcsid at the top.

2   Questions

1. Consider this pdf:

   $f_{X,Y} (x,y) = c (x^2 + 2 x y + y ^2) $ for $ 0\le x,y \le 1 $, 0 otherwise.

   1. (5 points) What must $c$ be?
   2. (5) What is F(x,y)?
   3. (5) What is the marginal $f_X(x)$?
   4. (5) What is the marginal $f_Y(y)$?
   5. (5) Are X and Y independent? Justify your answer.
   6. (30) What are $E[X], E[X^2], VAR[X], E[Y], E[Y^2], VAR[Y]$ ?
   7. (15) What are $E[XY], COV[X,Y], \rho_{X,Y}$ ?

2. (5) This question is about how late a student can sleep in before class. He can take a free bus, if he gets up in time. Otherwise, he must take a $10 Uber.

   The bus arrival time is not predictable but is uniform in [9:00, 9:20]. What's the latest time that the student can arrive at the bus stop and have his expected cost be no more than $5?

3. (5) X is a random variable (r.v.) that is U[0,1], i.e., uniform [0,1]. Y is a r.v. that is U[0,X]. What is $f_Y(y)$ ?

4. (5) X is an r.v. U[0,y] but we don't know y. We observe one sample $x_1$. What is maximum likelihood for y?

5. This is a noisy transmission question. X is the transmitted signal. It is 0 or 1. P[X=0] = 2/3. N is the noise. It is Gaussian with mean 0 and sigma 1.

   Y = X + N

   1. (5) Compute P[X=0|Y].
   2. (5) Compute $g_{MAP}(Y)$.

6. Let X be a Gaussian r.v. with mean 5 and sigma 10. Let Y be an independent exponential r.v. with lambda 3. Let Z be an independent continuous uniform r.v. in the interval [-1,1].

   1. (5) Compute E[X+Y+Z].
   2. (5) Compute VAR[X+Y+Z].

7. (5) We have a Gaussian r.v. with unknown mean $\mu$ and known $\sigma = 100$. We take a sample of 100 observations. The mean of that sample is 100. Compute $a$ such that with probability .68, $100-a \le \mu \le 100+a$.

8. (5) You're testing whether a new drug works. You will give 100 sick patients the drug and another 100 a placebo. The random variable X will be the number of days until their temperature drops to normal. You don't know in advance what $\sigma_X$ is. The question is whether E[X] over the patients with the drug is significantly different from E[X] over the patients with the placebo.

   What's the best statistical test to use?

9. You're tossing 1000 paper airplanes off the roof of the JEC onto the field, trying to hit a 1m square target. The airplanes are independent. The probability of any particular airplane hitting the target is 0.1%. The random variable X is the number of airplanes hitting the target.

   1. (5) What's the best probability distribution for X?

   2. (5) Name another distribution that would work if you computed with very large numbers.

   3. (5) Name another distribution that does not work in this case, but would work if the probability of any particular airplane hitting the target is 10%

      Historical note: for many years, GM week had an egg toss. Students designed a protective packaging for an egg and tossed it off the JEC onto the brick patio. Points were given for the egg surviving and landing near the target.

   4. You want to test a suspect die by tossing it 100 times. The number of times that each face from 1 to 6 shows is this: 12, 20, 15, 18, 15, 20.

      1. (5) What's the appropriate distribution?
      2. (5) If the die is fair, what's the probability that the observed distribution could be that far from the actual probability?

Total: 140 points.

1   My opinion of Matlab

(copied from class 9).

1. Advantages
   1. Excellent quality numerical routines.
   2. Free at RPI.
   3. Many toolkits available.
   4. Uses parallel computers and GPUs.
   5. Interactive - you type commands and immediately see results.
   6. No need to compile programs.
2. Disadvantages
   1. Very expensive outside RPI.
   2. Once you start using Matlab, you can't easily move away when their prices rise.
   3. You must force your data structures to look like arrays.
   4. Long programs must still be developed offline.
   5. Hard to write in Matlab's style.
   6. Programs are hard to read.
3. Alternatives
   1. Free clones like Octave are not very good
   2. The excellent math routines in Matlab are also available free in C++ librarues
   3. With C++ libraries using template metaprogramming, your code looks like Matlab.
   4. They compile slowly.
   5. Error messages are inscrutable.
   6. Executables run very quickly.

2   Homework grading rules

1. Each homework grade will be normalized to be out of 100.
2. Then the lowest homework grade will be dropped.
3. There will be only 9 homeworks.

3   Useful tables in book

1. TABLE 3.1 Discrete random variables is page 115.
2. TABLE 4.1 Continuous random variables is page 164.

4   Worked out problems from book

1. 5.30, page 263
2. 5.33, p 267

5   Review questions

1. What is $$\int_{-\infty}^\infty e^{\big(-\frac{x^2}{2}\big)} dx$$?

   1. 1/2
   2. 1
   3. $2\pi$
   4. $\sqrt{2\pi}$
   5. $1/\sqrt{2\pi}$

2. What is the largest possible value for a correlation coefficient?

   1. 1/2
   2. 1
   3. $2\pi$
   4. $\sqrt{2\pi}$
   5. $1/\sqrt{2\pi}$

3. The most reasonable probability distribution for the number of defects on an integrated circuit caused by dust particles, cosmic rays, etc, is

   1. Exponential
   2. Poisson
   3. Normal
   4. Uniform
   5. Binomial

4. The most reasonable probability distribution for the time until the next request hits your web server is:

   1. Exponential
   2. Poisson
   3. Normal
   4. Uniform
   5. Binomial

5. If you add two independent normal random variables, each with variance 10, what is the variance of the sum?

   1. 1
   2. $\sqrt2$
   3. 10
   4. $10\sqrt2$
   5. 20

6. X and Y are two uniform r.v. on the interval [0,1]. X and Y are independent. Z=X+Y. What is E[Z]?

   1. 0
   2. 1/2
   3. 2/3

7. Now let W=max(X,Y). What is E[W]?

   1. 0
   2. 1/2
   3. 2/3

8. Experiment: toss two fair coins, one after the other. Observe two random variables:

   1. X is the number of heads.
   2. Y is the toss when the first head occurred, with 0 meaning both coins were tails.

   What is P[X=1]?

   1. 0
   2. 1/4
   3. 1/2
   4. 3/4
   5. 1

9. What is P[Y=1]?

   1. 0
   2. 1/4
   3. 1/2
   4. 3/4
   5. 1

10. What is P[Y=1 & X=1]?

    1. 0
    2. 1/4
    3. 1/2
    4. 3/4
    5. 1

11. What is P[Y=1|X=1]?

    1. 0
    2. 1/4
    3. 1/2
    4. 3/4
    5. 1

12. What is P[X=1|Y=1]?

    1. 0
    2. 1/4
    3. 1/2
    4. 3/4
    5. 1

13. These next few questions concern math SAT scores, which we assume have a mean of 500 and standard deviation of 100.

    What is the probability that one particular score is between 400 and 600?

    1. .34
    2. .68
    3. .96
    4. .98
    5. .9974

14. I take a random sample of 4 students, and compute the mean of their 4 scores. What is the probability that that mean is between 400 and 600?

    1. .34
    2. .68
    3. .96
    4. .98
    5. .9974

15. I take a random sample of 9 students, and compute the mean of their 9 scores. What is the probability that that mean is between 400 and 600?

    1. .34
    2. .68
    3. .96
    4. .98
    5. .9974

16. What is the standard deviation of the 4-student sample?

    1. 25
    2. 33
    3. 50
    4. 100
    5. 200

17. What is the standard deviation of the 9-student sample?

    1. 25
    2. 33
    3. 50
    4. 100
    5. 200

1   Final exam

(Most of these points have been already announced.)

1. Mon April 27.
2. 24 hours to do.
3. I intend to make it available on 4pm RPI time, and allow until 4pm April 28.
4. Open book, open notes.
5. You will need an internet connection.
6. It will probably use gradescope.
7. This exam is optional. It will replace exam 1 if higher.
8. Course weights:
   1. exam: 40%.
   2. all homeworks: 60%.
9. Letter grades will be eased by one category. E.g., 90% and above earns an A.
10. You also have the right, after seeing your grade, to convert it to a pass/no credit.

2   Daily comic

The Born Loser by Art and Chip Sansom for April 19, 2020 is relevant to one of my statistics paradoxes from a few classes ago.

3   Homework 9

is online, due in a week.

4   Matlab tutorial

I was surprised that some students didn't know it, so I asked a few other profs. They tell me that you should have already seen Matlab in ECSE 1010 and 2010. Nevertheless, I'll do a tutorial. Matlab is a very useful package for engineers. As RPI students, you can install it on your own computers. If you have any free time, you might check other free SW that RPI has, play with it, and put it on your resume.

Some functions:

last 2 args usually are size of output matrix.   1 arg -> square.

randi(10,1,5) - 5 random ints in [1,10]
randn(1,5) - 5 random normals
random('unif',0,1,1,10)  - 10 random uniform [0,1]
rand(1,10) -  10 random uniform [0,1]


last 2 args are not size of output matrix:

normcdf([1 2 3], 0,1) - normal (0,1) cdf at 1, 2, 3
normpdf([1 2 3], 0,1) - normal (0,1) pdf at 1, 2, 3
randperm(6)
randsample(10,3) - random sample w/o replacement of 3 from 10.
randsample(10,3,true) - " w replacement

a=[1 2;3 4]
ai=inv(a)
a*ai            - matrix mult
a.*a            - element by element mult
a+a             - element by element add; mult was a special case
eye(3)          - identity
eig(a)          - eigenvalues
sin(a)          - usual functions, applied to each element
sin(a).*sin(a)+cos(a).*cos(a)   - identity
x = 0: .1 : 10   - sequence
y = sin(x)
plot(x,y)    - many options for prettifying it

2D plotting is messier:

[X,Y] = meshgrid(1:0.5:10,1:20);

  X is the list of x's repeated row after row
  Y is the list of y's repeated col after col

Z = sin(X) + cos(Y);  - computed element by element
surfc(X,Y,Z)

For the homework question, this is useful: https://www.mathworks.com/help/stats/multivariate-normal-distribution.html

e.g.:

xr = 1:5   - list of x's
yr = xr
[xm,ym]=meshgrid(xr,yr)  - mesh of (x,y)
p = [xm(:) ym(:)]  - list of pairs
mu = [0 0]
cov = [1 1.6; 1.6 4]
y = mvnpdf(p,mu,cov);  - multivar normal pdf
yr = reshape(y,length(xm),length(ym));
surfc(xm,ym,yr)

5   Statistics

1. How many students have had statistics like this in highschool?
2. 15-1 Why Non Parametric Statistics? (6.52) https://www.youtube.com/watch?v=xA0QcbNxENs
3. Regression: Crash Course Statistics #32 (12:40) https://www.youtube.com/watch?v=WWqE7YHR4Jc

Questions

1. 6.32, p 352 (2 parts: mean, covariance).
2. 6.68 (c), p 355. Use pmf (i).
3. 6.92 (a-d), p 358.
4. 8.1 (a-e) p 471. You decide how to generate the random samples, perhaps with Matlab or Mathematica.
5. 8.2 (a-e).
6. 8.49 (a-b), p 478.
7. 8.101, p 486. 10 points.

1   Prof Radke lecture 22

His notes

2   Final exam

1. Optional; will replace exam 1 if higher.
2. Open book, open notes.
3. 24 hours to do.
4. You will need an internet connection.
5. Submit to gradescope.

3   Worked out book examples

1. Section 7.2, p365.

   Note: Chebyshev inequality, Eqn 4.76, p 182. says

   \(P[ | X-m | \ge a ] \le \sigma^2/ a^2\)

   Eqn 7.20, p 366 is useful:

   \(P[ | M_n-\mu | < \epsilon ] \ge 1 - \sigma^2 / (n \epsilon^2)\)

2. Example 7.10, 7.11, 7.12.

Questions

All questions are from the text.

Each part of a question is worth 5 points.

1. 5.103, page 298.
2. 5.107(a), page 299.
3. 5.114(e), page 299.
4. 5.131(a-c), page 301. This is a chance to learn the gamma distribution.
5. 6.7(a-c), page 349.
6. 6.14(a-d), page 350.
7. 6.23 (a-c), p 351.

Questions

All questions are from the text.

Each part of a question is worth 5 points.

1. 5.3 (a-c), page 288.
2. 5.11 (a-b). Do only distribution (i).
3. 5.17 (a-c).
4. 5.25 (a-c), page 291.
5. 5.41 (a-b).
6. 5.56 (a-c), page 294.

Total: 80 points.

1   Homework 8

is online, due next Mon.

2   To watch before next class

Radke's Engineering Probability Lecture 22: Testing the fit of a distribution; generating random samples.

3   Statistics

Now, we learn statistics. That means, determining parameters of a population by sampling it. In probability, we already know that parameters, and calculate things from them.

We'll start with Leon-Garcia Chapter 8, and add stuff to it.

This course module fits with RPI's goal of a data dexterity requirement for undergrads. Pres Jackson mentioned this at the 2019 spring town meet; see https://president.rpi.edu/speeches/2019/remarks-spring-town-meeting .

Disclosure: Prof Dave Mendonca and I are co-chairs of the SoE Data Dexterity Task Force, working out details of this.

1   Today

Probability Lecture 20: Engineering Probability Lecture 20: MAP, ML, and MMSE estimation.

Radke's handwritten notes.

2   To watch before next class

Radke's Engineering Probability Lecture 21: Hypothesis testing.

3   Exercises

1. 6.1 page 348.
2. 6.3 (a) and (d)
3. 6.23 (a,c) p 351.

1   Radke's lectures

1. Today 19, The Central Limit Theorem.
2. To watch for Thurs: Probability Lecture 20: Engineering Probability Lecture 20: MAP, ML, and MMSE estimation.

2   Material from text

1   Exam 2

is replaced by a normal class.

2   Today

Radke's Engineering Probability Lecture 18: Sums of random variables and laws of large numbers.

Notes .

3   Mathematica

Today I will also (try to) demonstrate using Mathematica for the sum of 2 and 4 uniform random variables.

The pdf is a conditional, which is messy to work with by hand.

4   Homework

I will put a new homework online, due in a week.

5   Presentation ideas wanted

If another of your profs is using a remote teaching technique that you like, tell me. I might try it.

6   Watching lecture videos

Following the suggestion of some ECSE people, I've been uploading videos from Webex to Mediasite. I'm going to try just posting the Webex link for you. It will save me a step, and solve the impending problem of Mediasite running out of space soon. Tell me how you like this.

7   To watch before next class

Radke's Engineering Probability Lecture 19: The Central Limit Theorem.

8   Min, max of 2 r.v.

1. Example 5.43, page 274.

9   Chapter 6: Vector random variables, page 303-

1. Skip the starred sections.
2. Examples:
   1. arrivals in a multiport switch,
   2. audio signal at different times.
3. pmf, cdf, marginal pmf and cdf are obvious.
4. conditional pmf has a nice chaining rule.
5. For continuous random variables, the pdf, cdf, conditional pdf etc are all obvious.
6. Independence is obvious.
7. Work out example 6.5, page 306. The input ports are a distraction. This problem reduces to a multinomial probability where N is itself a random variable.

1   Ensure that you're in piazza

I use this for quick announcements that are not so much of permanent interest, and for things that don't necessarily need to be on the global internet forever.

I think I added everyone who wasn't already in it, but for complicated reasons might have missed someone.

So, if you're not in it, please add yourself or email me to add you.

I'll send a test message. If you don't get it, check your spam filter. RPI's spam filter recently blocked my own messages to piazza being forwarded to myself.

2   Exams

1. We had exam 1.
2. Based on your feedback, I canceled exam 2.
3. There will be an optional final exam (formerly called exam 3).
   1. It will be a take-home exam...
   2. probably on the day of the last class. (Or, do you prefer the official final exam date)?
4. The grading formula will use the max of exam 1 and the final exam.

3   Radke's lecture notes

that he was writing during the videos are online .

4   Today

You should have watched Radke's Engineering Probability Lecture 17: Conditional expectations.

5   To watch before next class on Thurs 2020-04-06

Radke's Engineering Probability Lecture 18: Sums of random variables and laws of large numbers.

1   Webex guidelines

1. Use chat to ask me questions.
2. Please mute audio. Be ready to unmute when you talk
3. Turn on video (unless the connection is limited or you prefer not to)
4. This session is being recorded. If you don’t want to be recorded: a. Stay in the session but turn off video and don’t speak/chat – this still may show your name, or, #. Leave the session, and watch the recording later.

2   Mediatesite

The videos of our classes will be in my Mediatesite channel ECSE-2500 Engineering Probability.

Remember that the classes will not be formal lectures, but will be optional discussions and chances for you to ask questions. This is to accommodate students in different time zones.

3   No exam 2

just lots of small homeworks.

4   To watch before next class

Radke's Engineering Probability Lecture 17: Conditional expectations.

1   Revised course format because of the current situation

1. No physical lectures.

2. I will use Webex.

   Connect via W Randolph Franklin's Personal Room.

3. For most classes after today, I'll assign one of Prof Rich Radke's lectures to watch before class.

4. Class time will be used for enrichment material, as discussion time for you to ask questions, and for me to work out textbook exercises.

5. I will add a small homework after each class, due in a few days.

6. I will continue to distribute material with this class blog.

7. Continue to use piazza for questions and written discussions.

8. Continue to submit work with gradescope.

9. I do not plan to use LMS.

10. Your feedback is welcome. This is an unprecedented situation. We profs have been officially told to be humane. So, no more 57 question homeworks. :-)

2   Normal distribution table

For your convenience. I computed it with Matlab.:

x          f(x)      F(x)      Q(x)
-3.0000    0.0044    0.0013    0.9987
-2.9000    0.0060    0.0019    0.9981
-2.8000    0.0079    0.0026    0.9974
-2.7000    0.0104    0.0035    0.9965
-2.6000    0.0136    0.0047    0.9953
-2.5000    0.0175    0.0062    0.9938
-2.4000    0.0224    0.0082    0.9918
-2.3000    0.0283    0.0107    0.9893
-2.2000    0.0355    0.0139    0.9861
-2.1000    0.0440    0.0179    0.9821
-2.0000    0.0540    0.0228    0.9772
-1.9000    0.0656    0.0287    0.9713
-1.8000    0.0790    0.0359    0.9641
-1.7000    0.0940    0.0446    0.9554
-1.6000    0.1109    0.0548    0.9452
-1.5000    0.1295    0.0668    0.9332
-1.4000    0.1497    0.0808    0.9192
-1.3000    0.1714    0.0968    0.9032
-1.2000    0.1942    0.1151    0.8849
-1.1000    0.2179    0.1357    0.8643
-1.0000    0.2420    0.1587    0.8413
-0.9000    0.2661    0.1841    0.8159
-0.8000    0.2897    0.2119    0.7881
-0.7000    0.3123    0.2420    0.7580
-0.6000    0.3332    0.2743    0.7257
-0.5000    0.3521    0.3085    0.6915
-0.4000    0.3683    0.3446    0.6554
-0.3000    0.3814    0.3821    0.6179
-0.2000    0.3910    0.4207    0.5793
-0.1000    0.3970    0.4602    0.5398
      0    0.3989    0.5000    0.5000
 0.1000    0.3970    0.5398    0.4602
 0.2000    0.3910    0.5793    0.4207
 0.3000    0.3814    0.6179    0.3821
 0.4000    0.3683    0.6554    0.3446
 0.5000    0.3521    0.6915    0.3085
 0.6000    0.3332    0.7257    0.2743
 0.7000    0.3123    0.7580    0.2420
 0.8000    0.2897    0.7881    0.2119
 0.9000    0.2661    0.8159    0.1841
 1.0000    0.2420    0.8413    0.1587
 1.1000    0.2179    0.8643    0.1357
 1.2000    0.1942    0.8849    0.1151
 1.3000    0.1714    0.9032    0.0968
 1.4000    0.1497    0.9192    0.0808
 1.5000    0.1295    0.9332    0.0668
 1.6000    0.1109    0.9452    0.0548
 1.7000    0.0940    0.9554    0.0446
 1.8000    0.0790    0.9641    0.0359
 1.9000    0.0656    0.9713    0.0287
 2.0000    0.0540    0.9772    0.0228
 2.1000    0.0440    0.9821    0.0179
 2.2000    0.0355    0.9861    0.0139
 2.3000    0.0283    0.9893    0.0107
 2.4000    0.0224    0.9918    0.0082
 2.5000    0.0175    0.9938    0.0062
 2.6000    0.0136    0.9953    0.0047
 2.7000    0.0104    0.9965    0.0035
 2.8000    0.0079    0.9974    0.0026
 2.9000    0.0060    0.9981    0.0019
 3.0000    0.0044    0.9987    0.0013

x is often called z.

More info: https://en.wikipedia.org/wiki/Standard_normal_table

3   The large effect of a small bias

This is enrichment material. It is not in the text, and will not be on the exam. However, it might be in a future homework.

Consider tossing $n=10^6$ fair coins.

1. P[more heads than tails] = 0.5

2. Now assume that each coin has chance of being heads $p=0.5005$.

   What's P[more heads than tails]?

   1. Approx with a Gaussian. $\mu=500500, \sigma=500$.
   2. Let X be the r.v. for the number of heads.
   3. P[X>500000] = Q(-1) = .84
   4. I.e., increasing the probability of winning 1 toss by 1 part in 1000, increased the probability of winning 1,000,000 tosses from 50% to 84%.

3. Now assume that 999,000 of the coins are fair, but 1,000 will always be heads.

   What's P[more heads than tails]?

   1. Let X = number of heads in 999,000 tosses.
   2. We want P[X>499,000].
   1. Approx with a Gaussian. $\mu=499,500, \sigma=500$.
   2. P[X>499,000] = Q(-1) = .84 as before.
   3. I.e., fixing 0.1% of the coins increased the probability of winning 1,000,000 tosses from 50% to 84%.

4   Section 5.7 Conditional probability ctd

I'll do these sections only if there's time and interest.

1. Example 5.35 Maximum A Posteriori Receiver on page 268.
2. Example 5.37, page 270.
3. Remember equations 5.49 a,b for total probability on page 269-70 for conditional expectation of Y given X.

5   Section 5.8 page 271: Functions of two random variables, ctd

1. Example 5.39 Sum of Two Random Variables, page 271.

2. Example 5.40 Sum of Nonindependent Gaussian Random Variables, page 272.

   I'll do an easier case of independent N(0,1) r.v. The sum will be N(0, $\sqrt{2}$ ).

3. Example 5.44, page 275. Tranform two independent Gaussian r.v from

   (X,Y) to (R, $\theta$).

6   Section 5.9, page 278: pairs of jointly Gaussian r.v.

1. I will simplify formula 5.61a by assuming that $\mu=0, \sigma=1$.

   $$f_{XY}(x,y)= \frac{1}{2\pi \sqrt{1-\rho^2}} e^{ \frac{-\left( x^2-2\rho x y + y^2\right)}{2(1-\rho^2)} } $$ .

2. The r.v. are probably dependent. $\rho$} says how much.

3. The formula degenerates if $|\rho|=1$ since the numerator and denominator are both zero. However the pdf is still valid. You could make the formula valid with l'Hopital's rule.

4. The lines of equal probability density are ellipses.

5. The marginal pdf is a 1 variable Gaussian.

6. Example 5.47, page 282: Estimation of signal in noise

   1. This is our perennial example of signal and noise. However, here the signal is not just $\pm1$ but is normal. Our job is to find the ''most likely'' input signal for a given output.

7. Important concept in the noisy channel example (with X and N both being Gaussian): The most likely value of X given Y is not Y but is somewhat smaller, depending on the relative sizes of \(\sigma_X\) and \(\sigma_N\). This is true in spite of \(\mu_N=0\). It would be really useful for you to understand this intuitively. Here's one way:

   If you don't know Y, then the most likely value of X is 0. Knowing Y gives you more information, which you combine with your initial info (that X is \(N(0,\sigma_X)\) to get a new estimate for the most likely X. The smaller the noise, the more valuable is Y. If the noise is very small, then the mostly likely X is close to Y. If the noise is very large (on average) then the most likely X is still close to 0.

7   Tutorial on probability density - 2 variables

In class 15, I tried to motivate the effect of changing one variable on probability density. Here's a try at motivating changing 2 variables.

1. We're throwing darts uniformly at a one foot square dartboard.
2. We observe 2 random variables, X, Y, where the dart hits (in Cartesian coordinates).
3. $$f_{X,Y}(x,y) = \begin{cases} 1& \text{if}\,\, 0\le x\le1 \cap 0\le y\le1\\ 0&\text{otherwise} \end{cases}$$
4. $$P[.5\le x\le .6 \cap .8\le y\le.9] = \int_{.5}^{.6}\int_{.8}^{.9} f_{XY}(x,y) dx \, dy = 0.01 $$
5. Transform to centimeters: $$\begin{bmatrix}V\\W\end{bmatrix} = \begin{pmatrix}30&0\\0&30\end{pmatrix} \begin{bmatrix}X\\Y\end{bmatrix}$$
6. $$f_{V,W}(v,w) = \begin{cases} 1/900& \text{if } 0\le v\le30 \cap 0\le w\le30\\ 0&\text{otherwise} \end{cases}$$
7. $$P[15\le v\le 18 \cap 24\le w\le27] = \\ \int_{15}^{18}\int_{24}^{27} f_{VW}(v,w)\, dv\, dw = \frac{ (18-15)(27-24) }{900} = 0.01$$
8. See Section 5.8.3 on page 286.
9. Next: We've seen 1 r.v., we've seen 2 r.v. Now we'll see several r.v.

8   To watch before next class

Radke's Engineering Probability Lecture 16: Conditional PDFs; Bayesian and maximum likelihood estimation.

9   Xkcd comic

Conditional Risk

1   Review of normal (Gaussian) distribution

1. Review of the normal distribution. If $\mu=0, \sigma=1$ (to keep it simple), then: $$f_N(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} $$
2. Show that $\int_{-\infty}^{\infty} f(x) dx =1$. This is example 4.21 on page 168.
3. Review: Consider a normal r.v. with $\mu=500, \sigma=100$. What is the probability of being in the interval [400,600]? Page 169 might be useful.
   1. .02
   2. .16
   3. .48
   4. .68
   5. .84
4. Repeat that question for the interval [500,700].
5. Repeat that question for the interval [0,300].

2   Varieties of Gaussian functions

1. Book page 167: $\Phi(x)$ is the CDF of the Gaussian.
2. Book page 168 and table on page 169: $Q(x) = 1 - \Phi(x)$.
3. Mathematica (and other SW packages): Erf[x] is integral of pdf from 0 to x $Erf(x) = Q(x)-.5$ .
4. Erfx(x) = 1-Erf(x).

(The nice thing about standards is that there are so many of them.)

3   Mathematica on Gaussians

1. NormalDistribution[m,s] is the abstract pdf.
2. get functions of it thus:
   1. PDF[NormalDistribution[m,s][x]]
   2. CDF ...
   3. Mean, Variance, Median ..
3. MultinormalDistribution[{mu1, mu2}, {{sigma11, sigma12}, {sigma12, sigma22}}] (details later).

4   Chapter 5, Two Random Variables

1. See intro I did in last class.
2. Today's reading: Chapter 5, page 233-242.
3. Review: An outcome is a result of a random experiment. It need not be a number. They are selected from the sample space. A random variable is a function mapping an outcome to a real number. An event is an interesting set of outcomes.
4. Example 5.3 on page 235. There's no calculation here, but this topic is used for several future problems.
5. Example 5.5 on page 238.
6. Example 5.6 on page 240. Easy, look at it yourself.
7. Example 5.7 on page 241. Easy, look at it yourself.
8. Example 5.8 on page 242. Easy, look at it yourself.
9. Example 5.9 on page 242.
10. 5.3 Joint CDF page 242.
11. Example 5.11 on page 245. What is f(x,y)?
12. Example 5.12 p 246
13. Cdf of mixed continuous - discrete random variables: section 5.3.1 on page 247. The input signal X is 1 or -1. It is perturbed by noise N that is U[-2,2] to give the output Y.. What is P[X=1|Y<=0]?
14. Example 5.14 on page 247.
15. Example 5.16 on page 252.

5   Xkcd comic

Conditional Risk

1   Markov and Chebyshev inequalities (Section 4.6, page 181)

1. Your web server averages 10 hits/second.
2. It will crash if it gets 20 hits.
3. By the Markov inequality, that has a probability at most 0.5.
4. That is way way too conservative, but it makes no assumptions about the distribution of hits.
5. For the Chebyshev inequality, assume that the variance is 10.
6. It gives the probability of crashing at under 0.1. That is tighter.
7. Assuming the distribution is Poisson with a=10, use Matlab 1-cdf('Poisson',20,10). That gives 0.0016.
8. The more we assume, the better the answer we can compute.
9. However, our assumptions had better be correct.
10. (Editorial): In the real world, and especially economics, the assumptions are, in fact, often false. However, the models still usually work (at least, we can't prove they don't work). Until they stop working, e.g., https://en.wikipedia.org/wiki/Long-Term_Capital_Management . Jamie Dimon, head of JP Morgan, has observed that the market swings more widely than is statistically reasonable.

2   Reliability (section 4.8, page 189)

1. The reliability R(t) is the probability that the item is still functioning at t. R(t) = 1-F(t).

2. What is the reliability of an exponential r.v.? ( $F(t)=1-e^{\lambda t}$ ).

3. The Mean Time to Failure (MTTF) is obvious. The equation near the top of page 190 should be

   $E[T] = \int_0^\infty \textbf{t} f(t) dt$

4. ... for an exponential r.v.?

5. The failure rate is the probability of a widget that is still alive now dying in the next second.

6. The importance of getting the fundamentals (or foundations) right:

   In the past 40 years, two major bridges in the Capital district have collapsed because of inadequate foundations. The Green Island Bridge collapsed on 3/15/77, see http://en.wikipedia.org/wiki/Green_Island_Bridge , http://cbs6albany.com/news/local/recalling-the-schoharie-bridge-collapse-30-years-later . The Thruway (I-90) bridge over Schoharie Creek collapsed on 4/5/87, killing 10 people.

   Why RPI likes the Roeblings: none of their bridges collapsed. E.g., when designing the Brooklyn Bridge, Roebling Sr knew what he didn't know. He realized that something hung on cables might sway in the wind, in a complicated way that he couldn't analyze. So he added a lot of diagonal bracing. The designers of the original Tacoma Narrows Bridge were smart enough that they didn't need this expensive margin of safety.

7. Another way to look at reliability: think of people.

   1. Your reliability R(t) is the probability that you live to age t, given that you were born alive. In the US, that's 98.7% for age 20, 96.4% for 40, 87.8% for 60.
   2. MTTF is your life expectancy at birth. In the US, that's 77.5 years.
   3. Your failure rate, r(t), is your probability of dying in the next dt, divided by dt, at different ages. E.g. for a 20-year-old, it's 0.13%/year for a male and 0.046%/year for a female http://www.ssa.gov/oact/STATS/table4c6.html . For 40-year-olds, it's 0.24% and 0.14%. For 60-year-olds, it's 1.2% and 0.7%. At 80, it's 7% and 5%. At 100, it's 37% and 32%.

8. Example 4.47, page 190. If the failure rate is constant, the distribution is exponential.

9. If several subsystems are all necessary, e.g., are in serial, then their reliabilities multiply. The result is less reliable.

   If only one of them is necessary, e.g. are in parallel, then their complementary reliabilities multiply. The result is more reliable.

   An application would be different types of RAIDs. (Redundant Array of Inexpensivexxxxxxxxxxxxx Independent Disks). In one version you stripe a file over two hard drives to get increased speed, but decreased reliability. In another version you triplicate the file over three drives to get increased reliability. (You can also do a hybrid setup.)

   (David Patterson at Berkeley invented RAID (and also RISC). He intended I to mean Inexpensive. However he said that when this was commercialized, companies said that the I meant Independent.)

10. Example 4.49 page 193, reliability of series subsystems.

11. Example 4.50 page 193, increased reliability of parallel subsystems.

3   4.9 Generating r.v

Ignore. It's surprisingly hard to do right, and has been implemented in builtin routines. Use them.

4   4.10 Entropy

Ignore since it's starred.

5   Chapter 5, Two Random Variables

1. One experiment might produce two r.v. E.g.,
   1. Shoot an arrow; it lands at (x,y).
   2. Toss two dice.
   3. Measure the height and weight of people.
   4. Measure the voltage of a signal at several times.
2. The definitions for pmf, pdf and cdf are reasonable extensions of one r.v.
3. The math is messier.
4. The two r.v. may be *dependent* and *correlated*.
5. The *correlation coefficient*, $\rho$, is a dimensionless measure of linear dependence. $-1\le\rho\le1$.
6. $\rho$ may be 0 when the variables have a nonlinear dependent relation.
7. Integrating (or summing) out one variable gives a marginal distribution.
8. We'll do some simple examples:
   1. Toss two 4-sided dice.
   2. Toss two 4-sided ''loaded'' dice. The marginal pmfs are uniform.
   3. Pick a point uniformly in a square.
   4. Pick a point uniformly in a triangle. x and y are now dependent.
9. The big example is a 2 variable normal distribution.
   1. The pdf is messier.
   2. It looks elliptical unless $\rho$=0.
10. I finished the class with a high level overview of Chapter 5, w/o any math.

1   Prof Ali Tajer

covered this class. His notes are here .

1   Tutorial on probability density

Since the meaning of probability density when you transform variables is still causing problems for some people, think of changing units from English to metric. First, with one variable, X.

1. Let X be in feet and be U[0,1].

   $$f_X(x) = \begin{cases} 1& \text{if } 0\le x\le1\\ 0&\text{otherwise} \end{cases}$$

2. $P[.5\le x\le .51] = 0.01$.

3. Now change to centimeters. The transformation is $Y=30X$.

4. $$f_Y(y) = \begin{cases} 1/30 & \text{if } 0\le y\le30\\ 0&\text{otherwise} \end{cases}$$

5. Why is 1/30 reasonable?

6. First, the pdf has to integrate to 1: $$\int_{-\infty}^\infty f_Y(y) =1$$

7. Second, $$\begin{align} & P[.5\le x\le .51] \\ &= \int_.5^.51 f_X(x) dx \\& =0.01 \\& = P[15\le y\le 15.3] \\& = \int_{15}^{15.3} f_Y(y) dy \end{align}$$

2   Mathematica demo

1. Int
2. Sum
3. Manipulate
4. Binomial etc

3   Examples

4.11, p153.

4   4.3.2 Variance

p160

5   Memoryless Exponential Distn

p 166.

6   4.4.3 Normal (Gaussian) dist

p 167.

Show that the pdf integrates to 1.

Lots of different notations:

Generally, F(x) = P(X<=x).

For normal: that is called $\Psi(x)$ .

$Q(x) = 1-\Psi(x)$ .

Example 4.22 page 169.

7   4.4.4 Gamma r.v.

1. 2 parameters
2. Has several useful special cases, e.g., chi-squared and m-Erlang.
3. The sum of m exponential r.v. has the m-Erlang dist.
4. Example 4.24 page 172.

8   Functions of a r.v.

1. Example 4.29 page 175.
2. Linear function: Example 4.31 on page 176.

9   Comic

Dilbert

Questions

What is the best discrete probability distribution in the following cases.

1. (2 points) Your car has five tires (including the spare), which may each independently be flat. The event is that not more than one tire is flat.

   1. Bernoulli
   2. binomial
   3. geometric
   4. Poisson
   5. uniform

2. (2 points) 1,000,000 widgets are made this year, of which 1,000 are bad. You buy 5 at random. The event is that not more than one widget is bad.

   1. Bernoulli
   2. binomial
   3. geometric
   4. Poisson
   5. uniform

3. (2 points) You toss a weighted coin, which lands heads 3/4 of the time.

   1. Bernoulli
   2. binomial
   3. geometric
   4. Poisson
   5. uniform

4. (2 points) You toss a fair 12-sided die.

   1. Bernoulli
   2. binomial
   3. geometric
   4. Poisson
   5. uniform

5. (2 points) You're learning to drive a car, and trying to pass the test. The event of interest is the number of times you have to take the test to pass. Assume that the tests are independent of each other and have equal probability.

   1. Bernoulli
   2. binomial
   3. geometric
   4. Poisson
   5. uniform

6. (2 points) It's Nov 17 and you're outside in a dark place looking for Leonid meteorites. The event of interest is the number of meteorites per hour that you see.

   1. Bernoulli
   2. binomial
   3. geometric
   4. Poisson
   5. uniform

7. (2 points) It's Nov 17.... The new event of interest is the number of seconds until you see the next meteorite.

   1. Bernoulli
   2. binomial
   3. geometric
   4. Poisson
   5. uniform

8. Taxi example: Sometimes there are mixed discrete and continuous r.v.

   1. Let X be the time X to get a taxi at the airport.
   2. 80% of the time a taxi is already there, so p(X=0)=.8.
   3. Otherwise we wait a uniform time from 0 to 20 minutes, so p(a<x<b)=.01(b-a), for 0<a<b<20.

   Questions:

   1. (2 points) For the taxi example, what is F(0)?
      1. 0
      2. .2
      3. .8
      4. .81
      5. 1
   2. (2 points) For the taxi example, what is F(1)?
      1. 0
      2. .8
      3. .81
      4. .9
      5. 1

9. (10 points) Problem 3.50 on page 135 of the text.

10. (15 points) 3.88 on page 139.

11. (15 points) 3.91 (p 139).

Total: 58

1   Super secret plan for creating the exam

As stolen by

1. Copy many questions from the last few exams, perhaps changing some numbers.
2. Copy some questions from the homeworks.
3. Go through the class blog, creating one question from each class.
4. Make a note of any particularly interesting question asked in class during the review session, and ....
5. The ulterior motive is that any student who studies according to the above plan will know the material.

2   Review for exam 1

2018 exam 1.

2018 solutions.

2019 exam 1.

2019 solutions.

1   Exam 1

next Thurs Feb 20.

one 2-sided crib sheet allowed.

2   Notation

How to parse \(F_X(x)\)

1. Uppercase F means that this is a cdf. Different letters may indicate different distributions.
2. The subscript X is the name of the random variable.
3. The x is an argument, i.e., an input.
4. \(F_X(x)\) returns the probability that the random variable is less or equal to the value x, i.e. prob(X<=x).

3   Matlab

1. Matlab, Mathematica, and Maple all will help you do problems too big to do by hand. Sometime I'll demo one or the other.

2. Matlab

   1. Major functions:

      cdf(dist,X,A,...)
      pdf(dist,X,A,...)
      

   2. Common cases of dist (there are many others):

      'Binomial'
      'Exponential'
      'Poisson'
      'Normal'
      'Geometric'
      'Uniform'
      'Discrete Uniform'
      

   3. Examples:

      pdf('Normal',-2:2,0,1)
      cdf('Normal',-2:2,0,1)
      
      p=0.2
      n=10
      k=0:10
      bp=pdf('Binomial',k,n,p)
      bar(k,bp)
      grid on
      
      bc=cdf('Binomial',k,n,p)
      bar(k,bc)
      grid on
      
      x=-3:.2:3
      np=pdf('Normal',x,0,1)
      plot(x,np)
      

   4. Interactive GUI to explore distributions: disttool

   5. Random numbers:

      rand(3)
      rand(1,5)
      randn(1,10)
      randn(1,10)*100+500
      randi(100,4)
      

   6. Interactive GUI to explore random numbers: randtool

   7. Plotting two things at once:

      x=-3:.2:3
      n1=pdf('Normal',x,0,1)
      n2=pdf('Normal',x,0,2)
      plot(x,n1,n2)
      plot(x,n1,x,n2)
      plot(x,n1,'--r',x,n2,'.g')
      

3. Use Matlab to compute a geometric pdf w/o using the builtin function.

4. Review. Which of the following do you prefer to use?

   1. Matlab
   2. Maple
   3. Mathematica
   4. Paper. It was good enough for Bernoulli and Gauss; it's good enough for me.
   5. Something else (please email about it me after the class).

4   Chapter 4 ctd

1. Text 4.2 p 148 pdf

2. Simple continuous r.v. examples: uniform, exponential.

3. The exponential distribution complements the Poisson distribution. The Poisson describes the number of arrivals per unit time. The exponential describes the distribution of the times between consecutive arrivals.

   The exponential is the continuous analog to the geometric. If the random variable is the integral number of seconds, use geometric. If the r.v. is the real number time, use exponential.

   Ex 4.7 p 150: exponential r.v.

4. Properties

   1. Memoryless.
   2. \(f(x) = \lambda e^{-\lambda x}\) if \(x\ge0\), 0 otherwise.
   3. Example: time for a radioactive atom to decay.

5. Skip 4.2.1 for now.

6. The most common continuous distribution is the normal distribution.

7. 4.2.2 p 152. Conditional probabilities work the same with continuous distributions as with discrete distributions.

8. p 154. Gaussian r.v.

   1. \(f(x) = \frac{1}{\sqrt{2\pi} \cdot \sigma} e^{\frac{-(x-\mu)^2}{2\sigma^2}}\)
   2. cdf often called \(\Psi(x)\)
   3. cdf complement:
      1. \(Q(x)=1-\Psi(x) = \int_x^\infty \frac{1}{\sqrt{2\pi} \cdot \sigma} e^{\frac{-(t-\mu)^2}{2\sigma^2}} dt\)
      2. E.g., if \(\mu=500, \sigma=100\),
         1. P[x>400]=0.66
         2. P[x>500]=0.5
         3. P[x>600]=0.16
         4. P[x>700]=0.02
         5. P[x>800]=0.001

9. Text 4.3 p 156 Expected value

10. Skip the other distributions (for now?).

5   Comic

Broomhilda

1   Probability in the real world - enrichment

Oct. 5, 1960: The moon tricks a radar.

Where would YOU make the tradeoff between type I and type II errors?

2   Chapter 3 ctd

1. 3.4 page 111 Conditional pmf

2. Example 3.24 Residual waiting time

   1. X, time to xmit message, is uniform in 1...L.
   2. If X is over m, what's probability that remaining time is j?
   3. \(p_X(m+j|X>m) = \frac{P[X =m+j]}{P[X>m]} = \frac{1/L}{(L-m)/L} = 1/(L-m)\)

3. \(p_X(x) = \sum p_X(x|B_i) P[B_i]\)

4. Example 3.25 p 113 device lifetimes

   1. 2 classes of devices, geometric lifetimes.
   2. Type 1, probability \(\alpha\), parameter r. Type 2 parameter s.
   3. What's pmf of the total set of devices?

5. Example 3.26, p114.

6. 3.5 p115 More important discrete r.v

7. Table 3.1: We haven't seen \(G_X(z)\) yet.

8. 3.5.1 p 117 The Bernoulli Random Variable

   We'll do mean and variance.

9. Example 3.28 p119 Variance of a Binomial Random Variable

10. Example 3.29 Redundant Systems

11. 3.5.3 p119 The Geometric Random Variable

    It models the time between two consecutive occurrences in a sequence of independent random events. E.g., the length of a run of white bits in a scanned image (if the bits are independent).

12. 3.5.4 Poisson r.v.

    1. The experiment is observing how many of a large number of rare events happen in, say, 1 minute.

    2. E.g., how many cosmic particles hit your DRAM, how many people call to call center.

    3. The individual events are independent. (In the real world this might be false. If a black hole occurs, you're going to get a lot of cosmic particles. If the ATM network crashes, there will be a lot of calls.)

    4. The r.v. is the number that happen in that period.

    5. There is one parameter, \(\alpha\). Often this is called \(\lambda\).

       \begin{equation*} p(k) = \frac{\alpha^k}{k!}e^{-\alpha} \end{equation*}

    6. Mean and std dev are both \(\alpha\).

    7. In the real world, events might be dependent.

13. Example 3.32 p123 Errors in Optical Transmission

14. 3.5.5 p124 The Uniform Random Variable

3   Poisson vs Binomial vs Normal distributions

The binomial distribution is the exact formula for the probability of k successes from n trials (with replacement).

When n and k are large but p=k/n is small, then the Poisson distribution is a good approximation to the binomial. Roughly, n>10, k<5.

When n is large and p is not too small or too large, then the normal distribution, which we haven't seen yet, is an excellent approximation. Roughly, n>10 and \(|n-k|>2\ \sqrt{n}\) .

For big n, you cannot use binomial, and for really big n, cannot use Poisson. Imagine that your experiment is to measure the number of atoms decaying in this uranium ore . How would you compute \(\left(10^{23}\right)!\) ?

OTOH, for small n, you can compute binomial by hand. Poisson and normal probably require a calculator.

4   Chapter 4

1. I will try to ignore most of the theory at the start of the chapter.
2. Now we will see continuous random variables.
   1. The probability of the r.v being any exact value is infinitesimal,
   2. so we talk about the probability that it's in a range.
3. Sometimes there are mixed discrete and continuous r.v.
   1. Let X be the time X to get a taxi at the airport.
   2. 80% of the time a taxi is already there, so p(X=0)=.8.
   3. Otherwise we wait a uniform time from 0 to 20 minutes, so p(a<x<b)=.01(b-a), for 0<a<b<20.
4. Remember that for discrete r.v. we have a probability mass function (pmf).
5. For continuous r.v. we now have a probability density function (pdf), \(f_X(x)\).
6. p(a<x<a+da) = f(a)da
7. For any r.v., we have a cumulative distribution function (cdf) \(F_X(x)\).
8. The subscript is interesting only when we are using more than one cdf and need to tell them apart.
9. Definition: F(x) = P(X<=x).
10. The <= is relevant only for discrete r.v.
11. As usual Wikipedia isn't bad, and is deeper than we need here, Cumulative_distribution_function.
12. We compute means and other moments by the obvious integrals.

5   Xkcd comic

Seashell

Questions

1. (5 points) This is a followup on last week's first question, which was this:

   Assume that it is known that one person in a group of 100 committed a crime. You're in the group, so there's a prior probability of 1/100 that you are it. There is a pretty good forensic test. It makes errors (either way) only 0.1% of the time. You are given the test; the result is positive. Using this positive test, what's the probability now that you are the criminal? (Use Bayes.)

   With a lot of tests, the results are grey, and the person running them has a choice in how to interpret them: lean towards finding someone guilty (but falsely accusing an innocent person), or the other way toward finding someone innocent (but letting a guilty person go free).

   Assume that in this example, the administrator can choose the bias. However the sum of the two types of errors is constant at 0.2%. (Whether that relation is really true would depend on the test.)

   This question is to plot both the number of innocent people falsely found guilty and the number of guilty people wrongly let go, as a function of the false positive rate. Use any plot package. Both numbers of people will usually be fractional.

2. (5 pts) Do exercise 2.126, page 95.

3. (5 pts) Do exercise 2.127.

4. (5 pts) Do exercise 3.1 on page 130.

5. (5 pts) Do exercise 3.5.

6. (5 pts) Do exercise 3.13 on page 132.

7. (5 pts) Do exercise 3.15.

Total: 35 pts.

1   Piazza

Remember that we have a piazza site for posting questions and answers.

2   Probability in the real world - enrichment

2020 U.S. Presidential Nomination Market

3   Review questions

1. Imagine that the coin you toss might land on its edge (and stay there). P(head)=.5, p(tail)=.4, p(edge)=.1. You toss it 3 times. What's the probability that it lands on its head twice, and on edge once?
   1. .025
   2. .05
   3. .075
   4. .081
   5. .1
2. Now you toss the coin repeatedly until it lands on edge. What's the probability that this happens for the first time on the 3rd toss?
   1. .025
   2. .05
   3. .081
   4. .1
   5. .333
3. review: You have a coin where the probability of a head is p=2/3 What's the probability that the 1st head occurs on the 2nd toss?
   1. 1/2
   2. 1/3
   3. 2/9
   4. 5/9
   5. 4/9

4   Wikipedia

Wikipedia's articles on technical subjects can be excellent. In fact, they often have more detail than you want. Here are some that are relevant to this course. Read at least the first few paragraphs.

1. https://en.wikipedia.org/wiki/Outcome_(probability)
2. https://en.wikipedia.org/wiki/Random_variable
3. https://en.wikipedia.org/wiki/Indicator_function
4. https://en.wikipedia.org/wiki/Gambler%27s_fallacy
5. https://en.wikipedia.org/wiki/Fat-tailed_distribution
6. https://en.wikipedia.org/wiki/St._Petersburg_paradox

5   Two types of testing errors

1. There's an event A, with probability P[A]=p.
2. There's a dependent event, perhaps a test or a transmission, B.
3. You know P[B|A] and P[B|A'].
4. Wikipedia:
   1. https://en.wikipedia.org/wiki/Type_I_and_type_II_errors
   2. https://en.wikipedia.org/wiki/Sensitivity_and_specificity
5. Terminology:
   1. Type I error, False negative.
   2. Type II error, false positive.
   3. Sensitivity, true positive proportion.
   4. Selectivity, true negative proportion.

6   Chapter 3 ctd

1. This chapter covers Discrete (finite or countably infinite) r.v.. This contrasts to continuous, to be covered later.
2. Discrete r.v.s we've seen so far:
   1. uniform: M events 0...M-1 with equal probs
   2. bernoulli: events: 0 w.p. q=(1-p) or 1 w.p. p
   3. binomial: # heads in n bernoulli events
   4. geometric: # trials until success, each trial has probability p.
3. 3.1.1 p107 Expected value of a function of a r.v.
   1. Z=g(X)
   2. E[Z] = E[g(x)] = \(\sum_k g(x_k) p_X(x_k)\)
4. Example 3.17 p107 square law device
5. \(E[a g(X)+b h(X)+c] = a E[g(X)] + b E[h(x)] + c\)
6. Example 3.18 Square law device continued
7. Example 3.19 Multiplexor discards packets
8. Compute mean of a binomial distribution.
9. Compute mean of a geometric distribution.
10. 3.3.1, page 107: Operations on means: sums, scaling, functions
11. review: From a deck of cards, I draw a card, look at it, put it back and reshuffle. Then I do it again. What's the probability that exactly one of the 2 cards is a heart?
    • A: 2/13
    • B: 3/16
    • C: 1/4
    • D: 3/8
    • E: 1/2
12. review: From a deck of cards, I draw a card, look at it, put it back and reshuffle. I keep repeating this. What's the probability that the 2nd card is the 1st time I see hearts?
    • A: 2/13
    • B: 3/16
    • C: 1/4
    • D: 3/8
    • E: 1/2
13. 3.3.2 page 109 Variance of an r.v.
    1. That means, how wide is its distribution?
    2. Example: compare the performance of stocks vs bonds from year to year. The expected values (means) of the returns may not be so different. (This is debated and depends, e.g., on what period you look at). However, stocks' returns have a much larger variance than bonds.
    3. \(\sigma^2_X = VAR[X] = E[(X-m_X)^2] = \sum (x-m_x)^2 p_X(x)\)
    4. standard deviation \(\sigma_X = \sqrt{VAR[X]}\)
    5. \(VAR[X] = E[X^2] - m_X^2\)
    6. 2nd moment: \(E[X^2]\)
    7. also 3rd, 4th... moments, like a Taylor series for probability
    8. shifting the distribution: VAR[X+c] = VAR[X]
    9. scaling: \(VAR[cX] = c^2 VAR[X]\)
14. Derive variance for Bernoulli.
15. Example 3.20 3 coin tosses
    1. general rule for binomial: VAR[X]=npq
    2. Derive it.
    3. Note that it sums since the events are independent.
    4. Note that variance/mean shrinks as n grows.
16. review: The experiment is drawing a card from a deck, seeing if it's hearts, putting it back, shuffling, and repeating for a total of 100 times. The random variable is the total # of hearts seen, from 0 to 100. What's the mean of this r.v.?
    • A: 1/4
    • B: 25
    • C: 1/2
    • D: 50
    • E: 1
17. The experiment is drawing a card from a deck, seeing if it's hearts, putting it back, shuffling, and repeating for a total of 100 times. The random variable is the # of hearts seen, from 0 to 100. What's the variance of this r.v.?
    • A: 3/16
    • B: 1
    • C: 25/4
    • D: 75/4
    • E: 100

7   Chapter 3 ctd

1. Geometric distribution: review mean and variance.

2. Suppose that you have just sold your internet startup for $10M. You have retired and now you are trying to climb Mt Everest. You intend to keep trying until you make it. Assume that:

   1. Each attempt has a 1/3 chance of success.
   2. The attempts are independent; failure on one does not affect future attempts.
   3. Each attempt costs $70K.

   Review: What is your expected cost of a successful climb?

   1. $70K.
   2. $140K.
   3. $210K.
   4. $280K.
   5. $700K.

8   Xkcd comic

Frequentists vs. Bayesians

1   Probability in the real world - enrichment

The Million Random Digit book below.

2   Chapter 2 ctd

1. Multinomial probability law

   1. There are M different possible outcomes from an experiment, e.g., faces of a die showing.

   2. Probability of particular outcome: \(p_i\)

   3. Now run the experiment n times.

   4. Probability that i-th outcome occurred \(k_i\) times, \(\sum_{i=1}^M k_i = n\)

      \begin{equation*} P[(k_1,k_2,...,k_M)] = \frac{n!}{k_1! k_2! ... k_M!} p_1^{k_1} p_2^{k_2}...p_M^{k_M} \end{equation*}

2. Example 2.41 p63 dartboard.

3. Example 2.42 p63 random phone numbers.

4. 2.7 Computer generation of random numbers

   1. Skip this section, except for following points.
   2. Executive summary: it's surprisingly hard to generate good random numbers. Commercial SW has been known to get this wrong. By now, they've gotten it right (I hope), so just call a subroutine.
   3. Arizona lottery got it wrong in 1998.
   4. Even random electronic noise is hard to use properly. The best selling 1955 book A Million Random Digits with 100,000 Normal Deviates had trouble generating random numbers this way. Asymmetries crept into their circuits perhaps because of component drift. For a laugh, read the reviews.
   5. Pseudo-random number generator: The subroutine returns numbers according to some algorithm (e.g., it doesn't use cosmic rays), but for your purposes, they're random.
   6. Computer random number routines usually return the same sequence of number each time you run your program, so you can reproduce your results.
   7. You can override this by seeding the generator with a genuine random number from linux /dev/random.

5. 2.8 and 2.9 p70 Fine points: Skip.

6. Review Bayes theorem, since it is important. Here is a fictitious (because none of these probilities have any justification) SETI example.

   1. A priori probability of extraterrestrial life = P[L] = \(10^{-8}\).
   2. For ease of typing, let L' be the complement of L.
   3. Run a SETI experiment. R (for Radio) is the event that it has a positive result.
   4. P[R|L] = \(10^{-5}\), P[R|L'] = \(10^{-10}\).
   5. What is P[L|R] ?

7. Some specific probability laws

   1. In all of these, successive events are independent of each other.
   2. A Bernoulli trial is one toss of a coin where p is probability of head.
   3. We saw binomial and multinomial probilities in class 4.
   4. The binomial law gives the probability of exactly k heads in n tosses of an unfair coin.
   5. The multinomial law gives the probability of exactly ki occurrances of the i-th face in n tosses of a die.

3   Review questions

1. You have a coin where the probability of a head is p=2/3 If you toss it twice, what's the probability that you will see one head and one tail?
   1. 1/2
   2. 1/3
   3. 2/9
   4. 5/9
   5. 4/9
2. Imagine that the coin you toss might land on its edge (and stay there). P(head)=.5, p(tail)=.4, p(edge)=.1. You toss it 3 times. What's the probability that it lands on its head twice, and on edge once?
   1. .025
   2. .05
   3. .081
   4. .1
   5. .333
3. Now you toss the coin repeatedly until it lands on edge. What's the probability that this happens for the first time on the 3rd toss?
   1. .025
   2. .05
   3. .081
   4. .1
   5. .333

4   Chapter 2 ctd

1. 2.6.4 p63 Geometric probability law

   1. Repeat Bernoulli experiment until 1st success.
   2. Define outcome to be # trials until that happens.
   3. Define q=(1-p).
   4. \(p(m) = (1-p)^{m-1}p = q^{m-1}p\) (p has 2 different uses here).
   5. \(\sum_{m=1}^\infty p(m) =1\)
   6. Probability that more than K trials are required = \(q^K\).

2. Example: probability that more than 10 tosses of a die are required to get a 6 = \(\left(\frac{5}{6}\right)^{10} = 0.16\)

3. Review: You have a coin where the probability of a head is p=2/3 What's the probability that the 1st head occurs on the 2nd toss?

   1. 1/2
   2. 1/3
   3. 2/9
   4. 5/9
   5. 4/9

4. Example 2.43 p64: error control by retransmission. A sent over a noisy channel is checksummed so the receiver can tell if it got mangled, and then ask for retransmission. TCP/IP does this.

   Aside: This works better when the roundtrip time is reasonable. Using this when talking to Mars is challenging.

5. 2.6.5 p64 Sequences, chains, of dependent experiments.

   1. This is an important topic, but mostly beyond this course.
   2. In many areas, there are a sequence of observations, and the probability of each observation depends on what you observed before.
   3. It relates to Markov chains.
   4. Motivation: speech and language recognition, translation, compression
   5. E.g., in English text, the probability of a u is higher if the previous char was q.
   6. The probability of a b may be higher if the previous char was u (than if it was x), but is lower if the previous two chars are qu.
   7. Need to look at probabilities of sequences, char by char.
   8. Same idea in speech recognition: phonemes follow phonemes...
   9. Same in language understanding: verb follows noun...

6. Example 2.44, p64. #. Example 2.45, p66.

5   Discrete Random Variables

1. Chapter 3, p 96. Discrete random variables
   1. From now on our random experiments will always produce numbers, called random variables, at least indirectly.
   2. Then we can compute, e.g., a fair value to pay for a gamble. What should you pay to play roulette so that betting on red breaks even on average?
   3. Discrete is different from discreet.
   4. Random experiment \(\rightarrow\) nonnumerical outcome \(\zeta\) \(\rightarrow\)
   5. Random Variable \(X(\zeta )\). Any real number.
   6. Random vars in general: X, Y, ...
   7. particular values: x, y, ...
   8. It's the outcome that's random, not the r.v., which is a deterministic function of the outcome.
2. Example 3.1 p97 Coin tosses
   1. Define X to be the number of heads from 3 tosses.
   2. \(\zeta\)
3. Example 3.2 Betting game addon to 3.1
   1. Define another random var Y to be payoff: 8 if X=3, 1 if X=2, 0 else.
   2. Y is derived from X
4. Example 3.3 add probs to 3.2, assuming fair coin. P[X=2], P[Y=8]
5. 3.1.1 Ignore since it's starred.
6. 3.2 Discrete r.v. and Probability mass function (pmf).
   1. The pmf shows the probability of every value of random variable X, and of every real number.
   2. If X cannot have the value x, then the pmf is 0 at x.
   3. \(p_X(x) = P[X=x] = P[\{\zeta:X(\zeta)=x\}]\)
7. p100: 3 properties of pmf. They're all common sense.
   1. Nonnegative.
   2. Sums to one.
   3. The probability of an event B is the sum of the probabilities of the outcomes in B.
8. Example 3.5 p101 probability of # heads in 3 coin tosses: \(p_X(0)=1/8\)
9. Example 3.6 betting game \(p_Y(1)=3/8\)
10. Fig 3.4. You can graph the pmf.
11. There are many types of random variables, depending on the shape of the pmf. These start out the same as the various probability laws in Chapter 2. However we'll see more types (e.g., Poisson) and more properties of each type (e.g., mean, standard deviation, generating function).
12. Example 3.7 random number generator
    1. produces integer X equally likely in range 0..M-1
    2. \(S_X=\{0, 1, ... M-1 \}\)
    3. pmf: \(p_X(k)=1/M\) for k in 0..M-1.
    4. X is a uniform random variable over that set.
13. Example 3.8 Bernoulli random variable
    1. indicator function \(I_A(\zeta)=1\) iff \(\zeta\in A\)
    2. pmf of \(I_A\): \(p_I(0)=1-p, p_I(1)=p\)
    3. \(I_A\) is a Bernoulli random variable.
14. Example 3.9 Message transmission until success
    1. \(p_X(k)=q^{k-1}p, k=1,2,3,...\)
    2. Geometric random variable
    3. What about P[X is even]?
15. Example 3.10 Number of transmission errors
    1. \(p_X(k) = {n \choose k} p^k q^{n-k}, k=0,1,...n\)
    2. binomial random variable
16. Fig 3.5 You can graph the relative frequencies from running an experiment repeatedly.
    1. It will approach the pmf graph (absent pathological cases like the Cauchy distribition that are beyond this course.)
17. 3.3 p104 Expected value and other moments
    1. This is a way to summarize a r.v., and capture important aspects.
    2. E.g., What's a fair price to pay for a lottery ticket?
    3. Mean or expected value or center of mass: \(m_X = E[X] = \sum_{x\in S_X} x p_X(x)\)
    4. Defined iff absolute convergence: \(\sum |x| p(x) < \infty\)
18. Example 3.11 Mean of Bernoulli r.v.
19. Example 3.12 Mean of Binomial r.v. What's the expected # of heads in 3 tosses?
20. Example 3.13 Mean of uniform discrete r.v.
21. Run an experiment n times and observe \(x(1), x(2), ...\)
    1. \(N_k(n)\) # times \(x_k\) was seen
    2. \(f_k(n) = N_k(n)/n\) frequencies
    3. Sample mean \(<X>_n = \sum x_kf_k(n)\)
    4. With lots of experiments, frequencies approach probabilities and sample mean converges to E[X]
    5. However it may take a long time, which is why stock market investors can go broke first.
22. Example 3.14 p 106 Betting game
23. Example 3.15 Mean of a geometric r.v.
24. Example 3.16 p107 St Petersburg paradox

6   Xkcd comic

Cell Phones

Questions

1. (5 points) Assume that it is known that one person in a group of 100 committed a crime. You're in the group, so there's a prior probability of 1/100 that you are it. There is a pretty good forensic test. It makes errors (either way) only 0.2% of the time. You are given the test; the result is positive. Using this positive test, what's the probability now that you are the criminal? (Use Bayes.)
2. (5 pts) Do exercise 2.58 on page 87 of the text.
3. (5 pts) Do exercise 2.65 on page 88 of the text.
4. (5 pts) Do 2.69, but use the interval [-3,3].
5. (5 pts) Do 2.72.
6. (5 pts) Do 2.76.
7. (5 pts) Do 2.82.
8. (5 pts) Do 2.97.
9. (5 pts) Do 2.102.
10. (5 pts) Do 2.106.

Total: 50 pts.

1   Probability in the real world - enrichment

See examples in the random section below.

2   Homework 3

is online here , due in a week.

3   Chapter 2 ctd

1. Example 2.28, p51. Chip quality control.
   1. Each chip is either good or bad.
   2. P[good]=(1-p), P[bad]=p.
   3. If the chip is good: P[still alive at t] = \(e^{-at}\)
   4. If the chip is bad: P[still alive at t] = \(e^{-1000at}\)
   5. What's the probability that a random chip is still alive at t?
2. 2.4.1, p52. Bayes' rule. This lets you invert the conditional probabilities.
   1. \(B_j\) partition S. That means that
      1. If \(i\ne j\) then \(B_i\cap B_j=\emptyset\) and
      2. \(\bigcup_i B_i = S\)
   2. \(P[B_j|A] = \frac{B_j\cap A}{P[A]}\) \(= \frac{P[A|B_j] P[B_j]}{\sum_k P[A|B_k] P[B_k]}\)
   3. application:
      1. We have a priori probs \(P[B_j]\)
      2. Event A occurs. Knowing that A has happened gives us info that changes the probs.
      3. Compute a posteriori probs \(P[B_j|A]\)
3. In the above diagram, what's the probability that an undergrad is an engineer?
4. Example 2.29 comm channel: If receiver sees 1, which input was more probable? (You hope the answer is 1.)
5. Example 2.30 chip quality control: For example 2.28, how long do we have to burn in chips so that the survivors have a 99% probability of being good? p=0.1, a=1/20000.
6. Example: False positives in a medical test
   1. T = test for disease was positive; T' = .. negative
   2. D = you have disease; D' = .. don't ..
   3. P[T|D] = .99, P[T' | D'] = .95, P[D] = 0.001
   4. P[D' | T] (false positive) = 0.98 !!!

4   Bayes theorem ctd

1. Wikipedia on Bayes theorem.

   We'll do the examples.

2. We'll do these examples from Leon-Garcia in class.

3. Example 2.28, page 51. I'll use e=0.1.

   Variant: Assume that P[A0]=.9. Redo the example.

4. Example 2.30, page 53, chip quality control: For example 2.28, how long do we have to burn in chips so that the survivors have a 99% probability of being good? p=0.1, a=1/20000.

5. Event A is that a random person has a lycanthopy gene. Assume P(A) = .01.

   Genes-R-Us has a DNA test for this. B is the event of a positive test. There are false positives and false negatives each w.p. (with probability) 0.1. That is, P(B|A') = P(B' | A) = 0.1

   1. What's P(A')?
   2. What's P(A and B)?
   3. What's P(A' and B)?
   4. What's P(B)?
   5. You test positive. What's the probability you're really positive, P(A|B)?

5   Chapter 2 ctd: Independent events

1. 2.5 Independent events

   1. \(P[A\cap B] = P[A] P[B]\)
   2. P[A|B] = P[A], P[B|A] = P[B]

2. A,B independent means that knowing A doesn't help you with B.

3. Mutually exclusive events w.p.>0 must be dependent.

4. Example 2.33, page 56.

   

5. More that 2 events:

   1. N events are independent iff the occurrence of no combo of the events affects another event.
   2. Each pair is independent.
   3. Also need \(P[A\cap B\cap C] = P[A] P[B] P[C]\)
   4. This is not intuitive A, B, and C might be pairwise independent, but, as a group of 3, are dependent.
   5. See example 2.32, page 55. A: x>1/2. B: y>1/2. C: x>y

6. Common application: independence of experiments in a sequence.

7. Example 2.34: coin tosses are assumed to be independent of each other.

   P[HHT] = P[1st coin is H] P[2nd is H] P[3rd is T].

8. Example 2.35, page 58. System reliability

   1. Controller and 3 peripherals.
   2. System is up iff controller and at least 2 peripherals are up.
   3. Add a 2nd controller.

9. 2.6 p59 Sequential experiments: maybe independent

10. 2.6.1 Sequences of independent experiments

    1. Example 2.36

11. 2.6.2 Binomial probability

    1. Bernoulli trial flip a possibly unfair coin once. p is probability of head.
    2. (Bernoulli did stats, econ, physics, ... in 18th century.)

12. Example 2.37

    1. P[TTH] = \((1-p)^2 p\)
    2. P[1 head] = \(3 (1-p)^2 p\)

13. Probability of exactly k successes = \(p_n(k) = {n \choose k} p^k (1-p)^{n-k}\)

14. \(\sum_{k=0}^n p_n(k) = 1\)

15. Example 2.38

16. Can avoid computing n! by computing \(p_n(k)\) recursively, or by using approximation. Also, in C++, using double instead of float helps. (Almost always you should use double instead of float. It's the same speed.)

17. Example 2.39

18. Example 2.40 Error correction coding

6   Xkcd comic

Linear Regression

1   Probability in the real world - enrichment

Statistician Cracks Code For Lottery Tickets

Finding these stories is just too easy.

2   Chapter 2 ctd

1. Today: counting methods, Leon-Garcia section 2.3, page 41.

   1. We have an urn with n balls.
   2. Maybe the balls are all different, maybe not.
   3. W/o looking, we take k balls out and look at them.
   4. Maybe we put each ball back after looking at it, maybe not.
   5. Suppose we took out one white and one green ball. Maybe we care about their order, so that's a different case from green then white, maybe not.

2. Applications:

   1. How many ways can we divide a class of 12 students into 2 groups of 6?
   2. How many ways can we pick 4 teams of 6 students from a class of 88 students (leaving 64 students behind)?
   3. We pick 5 cards from a deck. What's the probability that they're all the same suit?
   4. We're picking teams of 12 students, but now the order matters since they're playing baseball and that's the batting order.
   5. We have 100 widgets; 10 are bad. We pick 5 widgets. What's the probability that none are bad? Exactly 1? More than 3?
   6. In the approval voting scheme, you mark as many candidates as you please. The candidate with the most votes wins. How many different ways can you mark the ballot?
   7. In preferential voting, you mark as many candidates as you please, but rank them 1,2,3,... How many different ways can you mark the ballot?

3. Leon-Garcia 2.3: Counting methods, pp 41-46.

   1. finite sample space
   2. each outcome equally probable
   3. get some useful formulae
   4. warmup: consider a multiple choice exam where 1st answer has 3 choices, 2nd answer has 5 choices and 3rd answer has 6 choices.
      1. Q: How many ways can a student answer the exam?
      2. A: 3x5x6
   5. If there are k questions, and the i-th question has \(n_i\) answers then the number of possible combinations of answers is \(n_1n_2 .. n_k\)

4. 2.3.1 Sampling WITH replacement and WITH ordering

   1. Consider an urn with n different colored balls.
   2. Repeat k times:
      1. Draw a ball.
      2. Write down its color.
      3. Put it back.
   3. Number of distinct ordered k-tuples = \(n^k\)

5. Example 2.1.5. How many distinct ordered pairs for 2 balls from 5? 5*5.

6. Review. Suppose I want to eat one of the following 4 places, for tonight and again tomorrow, and don't care if I eat at the same place both times: Commons, Sage, Union, Knotty Pine. How many choices to I have where to eat?

   1. 16
   2. 12
   3. 8
   4. 4
   5. something else

7. 2.3.2 Sampling WITHOUT replacement and WITH ordering

   1. Consider an urn with n different colored balls.
   2. Repeat k times:
      1. Draw a ball.
      2. Write down its color.
      3. Don't put it back.
   3. Number of distinct ordered k-tuples = n(n-1)(n-2)...(n-k+1)

8. Review. Suppose I want to visit two of the following four cities: Buffalo, Miami, Boston, New York. I don't want to visit one city twice, and the order matters. How many choices to I have how to visit?

   1. 16
   2. 12
   3. 8
   4. 4
   5. something else

9. Example 2.1.6: Draw 2 balls from 5 w/o replacement.

   1. 5 choices for 1st ball, 4 for 2nd. 20 outcomes.
   2. Probability that 1st ball is larger?
   3. List the 20 outcomes. 10 have 1st ball larger. P=1/2.

10. Example 2.1.7: Draw 3 balls from 5 with replacement. What's the probability they're all different?

    1. P = \(\small \frac{\text{# cases where they're different}}{\text{# cases where I don't care}}\)
    2. P = \(\small \frac{\text{# case w/o replacement}}{\text{# cases w replacement}}\)
    3. P = \(\frac{5*4*3}{5*5*5}\)

11. 2.3.3 Permutations of n distinct objects

    1. Distinct means that you can tell the objects apart.

    2. This is sampling w/o replacement for k=n

    3. 1.2.3.4...n = n!

    4. It grows fast. 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720, 7!=5040

    5. Stirling approx:

       \begin{equation*} n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n\left(1+\frac{1}{12n}+...\right) \end{equation*}

    6. Therefore if you ignore the last term, the relative error is about 1/(12n).

12. Example 2.1.8. # permutations of 3 objects. 6!

13. Example 2.1.9. 12 airplane crashes last year. Assume independent, uniform, etc, etc. What's probability of exactly one in each month?

    1. For each crash, let the outcome be its month.
    2. Number of events for all 12 crashes = \(12^{12}\)
    3. Number of events for 12 crashes in 12 different months = 12!
    4. Probability = \(12!/(12^{12}) = 0.000054\)
    5. Random does not mean evenly spaced.

14. 2.3.4 Sampling w/o replacement and w/o ordering

    1. We care what objects we pick but not the order

    2. E.g., drawing a hand of cards.

    3. term: Combinations of k objects selected from n. Binomial coefficient.

       \begin{equation*} C^n_k = {n \choose k} = \frac{n!}{k! (n-k)!} \end{equation*}

    4. Permutations is when order matters.

15. Example 2.20. Select 2 from 5 w/o order. \(5\choose 2\)

16. Example 2.21 # permutations of k black and n-k white balls. This is choosing k from n.

17. Example 2.22. 10 of 50 items are bad. What's probability 5 of 10 selected randomly are bad?

    1. # ways to have 10 bad items in 50 is \(50\choose 10\)
    2. # ways to have exactly 5 bad is 3 ways to select 5 good from 40 times # ways to select 5 bad from 10 = \({40\choose5} {10\choose5}\)
    3. Probability is ratio.

18. Multinomial coefficient: Partition n items into sets of size \(k_1, k_2, ... k_j, \sum k_i=n\)

    \begin{equation*} \frac{n!}{k_1! k_2! ... k_j!} \end{equation*}

19. 2.3.5. skip

Reading: 2.4 Conditional probability, page 47-

3   Review questions

1. Retransmitting a very noisy bit 2 times: The probability of each bit going bad is 0.4. What is probability of no error at all in the 2 transmissions?
   1. 0.16
   2. 0.4
   3. 0.36
   4. 0.48
   5. 0.8
2. Flipping an unfair coin 2 times: The probability of each toss being heads is 0.4. What is probability of both tosses being tails?
   1. 0.16
   2. 0.4
   3. 0.36
   4. 0.48
   5. 0.8
3. Flipping a fair coin until we get heads: How many times will it take until the probability of seeing a head is >=.8?
   1. 1
   2. 2
   3. 3
   4. 4
   5. 5
4. This time, the coin is weighted so that p[H]=.6. How many times will it take until the probability of seeing a head is >=.8?
   1. 1
   2. 2
   3. 3
   4. 4
   5. 5

4   Review

1. Followon to the meal choice review question. My friend and I wish to visit a hospital, chosen from: Memorial, AMC, Samaritan. We might visit different hospitals.
   1. If we don't care whether we visit the same hospital or not, in how many ways can we do this?
      1. 1
      2. 2
      3. 3
      4. 6
      5. 9
   2. We wish to visit different hospitals, to later write a Poly review. In how many ways can we visit different hospitals, where we care which hospital each of us visits?
      1. 1
      2. 2
      3. 3
      4. 6
      5. 9
   3. Modify the above, to say that we care only about the set of hospitals we two visit.
      1. 1
      2. 2
      3. 3
      4. 6
      5. 9
   4. We realize that Samaritan and Memorial are both owned by St Peters and we want to visit two different hospital chains to write our reviews. In how many ways can we pick hospitals so that we pick different chains?
      1. 1
      2. 2
      3. 3
      4. 4
      5. 5
   5. We each pick between Memorial and AMC with 50% probability, independently. What is the probability that each hospital is picked exactly once (in contrast to picking one twice and the other not at all).
      1. 0
      2. 1/4
      3. 1/2
      4. 3/4
      5. 1

5   Chapter 2 ctd

1. New stuff, pp. 47-66:

   1. Conditional probability - If you know that event A has occurred, does that change the probability that event B has occurred?
   2. Independence of events - If no, then A and B are independent.
   3. Sequential experiments - Find the probability of a sequence of experiments from the probabilities of the separate steps.
   4. Binomial probabilities - tossing a sequence of unfair coins.
   5. Multinomial probabilities - tossing a sequence of unfair dice.
   6. Geometric probabilities - toss a coin until you see the 1st head.
   7. Sequences of dependent experiments - What you see in step 1 influences what you do in step 2.

2. 2.4 Conditional probability, page 47.

   1. big topic
   2. E.g., if it snows today, is it more likely to snow tomorrow? next week? in 6 months?
   3. E.g., what is the probability of the stock market rising tomorrow given that (it went up today, the deficit went down, an oil pipeline was blown up, ...)?
   4. What's the probability that a CF bulb is alive after 1000 hours given that I bought it at Walmart?
   5. definition \(P[A|B] = \frac{P[A\cap B]}{P[B]}\)

3. E.g., if DARPA had been allowed to run its Futures Markets Applied to Prediction (FutureMAP) would the future probability of King Zog I being assassinated be dependent on the amount of money bet on that assassination occurring?

   1. Is that good or bad?
   2. Would knowing that the real Zog survived over 55 assassination attempts change the probability of a future assassination?

4. Consider a fictional university that has both undergrads and grads. It also has both Engineers and others:

   

5. Review: What's the probability that a student is an Engineer?

   1. 1/7
   2. 4/7
   3. 5/7
   4. 3/4
   5. 3/5

6. Review: What's the probability that a student is an Engineer, given that s/he is an undergrad?

   1. 1/7
   2. 4/7
   3. 5/7
   4. 3/4
   5. 3/5

7. \(P[A\cap B] = P[A|B]P[B] = P[B|A]P[A]\)

8. Example 2.26 Binary communication. Source transmits 0 with probability (1-p) and 1 with probability p. Receiver errs with probability e. What are probabilities of 4 events?

9. Total probability theorem

   1. \(B_i\) mutually exclusive events whose union is S
   2. P[A] = P[A \(\cap B_1\) + P[A \(\cap B_2\) + ...
   3. \(P[A] = P[A|B_1]P[B_1]\) \(+ P[A|B_2]P[B_2] + ...\)
   

   What's the probability that a student is an undergrad, given ... (Numbers are fictitious.)

10. Example 2.28. Chip quality control.

    1. Each chip is either good or bad.
    2. P[good]=(1-p), P[bad]=p.
    3. If the chip is good: P[still alive at t] = \(e^{-at}\)
    4. If the chip is bad: P[still alive at t] = \(e^{-1000at}\)
    5. What's the probability that a random chip is still alive at t?

11. 2.4.1, p52. Bayes' rule. This lets you invert the conditional probabilities.

    1. \(B_j\) partition S. That means that
       1. If \(i\ne j\) then \(B_i\cap B_j=\emptyset\) and
       2. \(\bigcup_i B_i = S\)
    2. \(P[B_j|A] = \frac{B_j\cap A}{P[A]}\) \(= \frac{P[A|B_j] P[B_j]}{\sum_k P[A|B_k] P[B_k]}\)
    3. application:
       1. We have a priori probs \(P[B_j]\)
       2. Event A occurs. Knowing that A has happened gives us info that changes the probs.
       3. Compute a posteriori probs \(P[B_j|A]\)

12. In the above diagram, what's the probability that an undergrad is an engineer?

13. Example 2.29 comm channel: If receiver sees 1, which input was more probable? (You hope the answer is 1.)

14. Example 2.30 chip quality control: For example 2.28, how long do we have to burn in chips so that the survivors have a 99% probability of being good? p=0.1, a=1/20000.

15. Example: False positives in a medical test

    1. T = test for disease was positive; T' = .. negative
    2. D = you have disease; D' = .. don't ..
    3. P[T|D] = .99, P[T' | D'] = .95, P[D] = 0.001
    4. P[D' | T] (false positive) = 0.98 !!!

6   Xkcd comic

Correlation

1   Homework 2

is online, due next Thurs.

2   Probability in the real world - enrichment

How MIT Students Won $8 Million in the Massachusetts Lottery.

3   Chapter 2 ctd

1. Corollory 6:

   \(\begin{array}{c} P\left[\cup_{i=1}^n A_i\right] = \\ \sum_{i=1}^n P[A_i] \\ - \sum_{i<j} P[A_i\cap A_j] \\ + \sum_{i<j<k} P[A_i\cap A_j\cap A_k] \cdots \\ + (-1)^{n+1} P[\cap_{i=1}^n A_i] \end{array}\)

   1. Example Q=queen card, H=heart, F= face card.
      1. P[Q]=4/52, P[H]=13/52, P[F]=12/52,
      2. P[Q \(\cap\) H]=1/52, P[Q \(\cap\) F] = ''you tell me''
      3. P[H \(\cap\) F]= ''you tell me''
      4. P[Q \(\cap\) H \(\cap\) F] = ''you tell me''
      5. So P[Q \(\cup\) H \(\cup\) F] = ?
   2. Example from Roulette:
      1. R=red, B=black, E=even, A=1-12
      2. P[R] = P[B] = P[E] = 16/38. P[A]=12/38
      3. \(P[R\cup E \cup A]\) = ?

2. Corollory 7: if \(A\subset B\) then P[A] <= P[B]

   Example: Probability of a repeated coin toss having its first head in the 2nd-4th toss (1/2+1/4+1/8) \(\ge\) Probability of it happening in the 3rd toss (1/4).

3. 2.2.1 Discrete sample space

   1. If sample space is finite, probabilities of all the outcomes tell you everything.
   2. sometimes they're all equal.
   3. Then P[event]} \(= \frac{\text{#. outcomes in event}}{\text{total # outcomes}}\)
   4. For countably infinite sample space, probabilities of all the outcomes also tell you everything.
   5. E.g. fair coin. P[even] = 1/2
   6. E.g. example 2.9. Try numbers from random.org.
   7. What probabilities to assign to outcomes is a good question.
   8. Example 2.10. Toss coin 3 times.
      1. Choice 1: outcomes are TTT ... HHH, each with probability 1/8
      2. Choice 2: outcomes are # heads: 0...3, each with probability 1/4.
      3. Incompatible. What are probabilities of # heads for choice 1?
      4. Which is correct?
      5. Both might be mathematically ok.
      6. It depends on what physical system you are modeling.
      7. You might try doing the experiment and observing.
      8. You might add a new assumption: The coin is fair and the tosses independent.

4. Example 2.11: countably infinite sample space.

   1. Toss fair coin, outcome is # tosses until 1st head.
   2. What are reasonable probabilities?
   3. Do they sum to 1?

5. 2.2.2 Continuous sample spaces

   1. Usually we can't assign probabilities to points on real line. (It just doesn't work out mathematically.)
   2. Work with set of intervals, and Boolean operations on them.
   3. Set may be finite or countable.
   4. This set of events is a ''Borel set''.
   5. Notation:
      1. [a,b] closed. includes both. a<=x<=b
      2. (a,b) open. includes neither. a<x<b
      3. [a,b) includes a but not b, a<=x<b
      4. (a,b] includes b but not a, a<x<=b
   6. Assign probabilities to intervals (open or closed).
   7. E.g., uniform distribution on [0,1] \(P[a\le x\le b] = \frac{1}{b-a}\)
   8. Nonuniform distributions are common.
   9. Even with a continuous sample space, a few specific points might have probabilities. The following is mathematically a valid probability distribution. However I can't immediately think of a physical system that it models.
      1. \(S = \{ x | 0\le x\le 1 \}\)
      2. \(p(x=1) = 1/2\)
      3. For \(0\le x_0 \le 1, p(x<x_0) = x_0/2\)

6. For fun: Heads you win, tails... you win. You can beat the toss of a coin and here's how....

7. Example 2.13, page 39, nonuniform distribution: chip lifetime.

   1. Propose that P[(t, \(\infty\) )] = \(e^{-at}\) for t>0.
   2. Does this satisfy the axioms?
   3. I: yes >0
   4. II: yes, P[S] = \(e^0\) = 1
   5. III here is more like a definition for the probability of a finite interval
   6. P[(r,s)] = P[(r, \(\infty\) )] - P[(s, \(\infty\) )] = \(e^{-ar} - e^{-as}\)

8. Probability of a precise value occurring is 0, but it still can occur, since SOME value has to occur.

9. Example 2.14: picking 2 numbers randomly in a unit square.

   1. Assume that the probability of a point falling in a particular region is proportional to the area of that region.
   2. E.g. P[x>1/2 and y<1/10] = 1/20
   3. P[x>y] = 1/2

10. Recap:

    1. Problem statement defines a random experiment
    2. with an experimental procedure and set of measurements and observations
    3. that determine the possible outcomes and sample space
    4. Make an initial probability assignment
    5. based on experience or whatever
    6. that satisfies the axioms.

4   Xkcd comic

P-Values

Questions

1. (6 pts) Do exercise 2.2, page 81 of Leon-Garcia.

2. (6 pts) Do exercise 2.4, page 81.

3. (6 pts) Do exercise 2.6, page 82.

4. (6 pts) Do exercise 2.21, page 84.

5. (6 pts) Do exercise 2.25, page 84.

6. (6 pts) Do exercise 2.35(a), page 85. Assume the "half as frequently" means that for a subinterval of length d, the probability is half as much when the subinterval is in [0,2] as when in [-1,0).

7. (6 pts) Do exercise 2.39, page 86. Ignore any mechanical limitations of combo locks. Good RPI students should know what those limitations are.

   (Aside: A long time ago, RPI rekeyed the whole campus with a more secure lock. Shortly thereafter a memo was distributed that I would summarize as, "OK, you can, but don't you dare!")

8. (6 pts) Do exercise 2.59, page 87. However, make it 21 students and 3 on each day of the week. Assume that there is no relation between birthday and day of the week.

9. (6 pts) Find a current policy issue where you think that probabilities are being misused, and say why, in 100 words. Full points will be awarded for a logical argument. I don't care what the issue is, or which side you take. Try not to pick something too too inflammatory; follow the Page 1 rule that an NSF lawyer taught me when I was there. (Would you be willing to see your answer on page 1 of tomorrow's paper?)

Total: 54 pts.

1   Note the handwritten_notes button in the top bar

2   Probability in the real world - enrichment

1. How Did Economists Get It So Wrong? is an article by Paul Krugman (2008 Nobel Memorial Prize in Economic Science). It says, "the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth." You might see a certain relevance to this course. You have to get the model right before trying to solve it.

   Though I don't know much about it, I'll cheerfully try to answer any questions about econometrics.

   Another relevance to this course, in an enrichment sense, is that some people believe that the law of large numbers does not apply to certain variables, like stock prices. They think that larger and larger sample frequencies do not converge to a probability, because the variance of the underlying distribution is infinite. This also is beyond this course.

3   Chapter 1 ctd

1. Rossman-Chance coin toss applet demonstrates how the observed frequencies converge (slowly) to the theoretical probability.

2. Example of unreliable channel (page 12)

   1. Want to transmit a bit: 0, 1
   2. It arrives wrong with probability e, say 0.001
   3. Idea: transmit each bit 3 times and vote.
      1. 000 -> 0
      2. 001 -> 0
      3. 011 -> 1
   1. 3 bits arrive correct with probability \((1-e)^3\) = 0.997002999
   2. 1 error with probability \(3(1-e)^2e\) = 0.002994
   3. 2 errors with probability \(3(1-e)e^2\) = 0.000002997
   4. 3 errors with probability \(e^3\) = 0.000000001
   5. corrected bit is correct if 0 or 1 errors, with probability \((1-e)^3+3(1-e)^2e\) = 0.999996999
   6. We reduced probability of error by factor of 1000.
   7. Cost: triple the transmission plus a little logic HW.

3. Example of text compression (page 13)

   1. Simple way: Use 5 bits for each letter: A=00000, B=00001
   2. In English, 'E' common, 'Q' rare
   3. Use fewer bits for E than Q.
   4. Morse code did this 170 years ago.
      1. E = .
      2. Q = _ _ . _
   1. Aside: An expert Morse coder is faster than texting.
   2. English can be compressed to about 1 bit per letter (with difficulty); 2 bits is easy.
   3. Aside: there is so much structure in English text, that if you add the bit strings for 2 different texts bit-by-bit, they can usually mostly be reconstructed.
   4. That's how cryptoanalysis works.

4. Example of reliable system design (page 13)

   1. Nuclear power plant fails if
      1. water leaks
      2. and operator asleep (a surprising number of disasters happen in the graveyard shift).
      3. and backup pump fails
      4. or was turned off for maintenance
   1. What's the probability of failure? This depends on the probabilities of the various failure modes. Those might be impossible to determine accurately.
   2. Design a better system? Coal mining kills.
   3. The backup procedures themselves can cause problems (and are almost impossible to test). A failure with the recovery procedure was part of the reason for a Skype outage.

4   Chapter 2

1. A random experiment (page 21) has 2 parts:
   1. experimental procedure
   2. set of measurements
2. Random experiment may have subexperiments and sequences of experiments.
3. Outcome or sample point \(\zeta\): a non-decomposable observation.
4. Sample space S: set of all outcomes
5. \(|S|\):
   1. finite, e.g. {H,T}, or
   2. discrete = countable, e.g., 1,2,3,4,... Sometimes discrete includes finite. or
   3. uncountable, e.g., \(\Re\), aka continuous.
6. Types of infinity:
   1. Some sets have finite size, e.g., 2 or 6.
   2. Other sets have infinite size.
   3. Those are either countable or uncountable.
   4. A countably infinite set can be arranged in order so that its elements can be numbered 1,2,3,...
   5. The set of natural numbers is obviously countable.
   6. The set of positive rational numbers between 0 and 1 is also countable. You can order it thus: \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \ \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \ \cdots\)
   7. The set of real numbers is not countable (aka uncountable). Proving this is beyond this course. (It uses something called diagonalization.
   8. Uncountably infinite is a bigger infinity than countably infinite, but that's beyond this course.
   9. Georg Cantor, who formulated this, was hospitalized in a mental health facility several times.
7. Why is this relevant to probability?
   1. We can assign probabilities to discrete outcomes, but not to individual continuous outcomes.
   2. We can assign probabilities to some events, or sets of continuous outcomes.
8. E.g. Consider this experiment to watch an atom of sodium-26.
   1. Its half-life is 1 second (Applet: Nuclear Isotope Half-lifes)
   2. Define the outcomes to be the number of complete seconds before it decays: \(S=\{0, 1, 2, 3, \cdots \}\)
   3. \(|S|\) is countably infinite, i.e., discrete.
   4. \(p(0)=\frac{1}{2}, p(1)=\frac{1}{4}, \cdots\) \(p(k)=2^{-(k+1)}\)
   5. \(\sum_{k=0}^\infty p(k) = 1\)
   6. We can define events like these:
      1. The atom decays within the 1st second. p=.5.
      2. The atom decays within the first 3 seconds. p=.875.
      3. The atom's lifetime is an even number of seconds. \(p = \frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \cdots = \frac{2}{3}\)
9. Now consider another experiment: Watch another atom of Na-26
   1. But this time the outcome is defined to be the real number, x, that is the time until it decays.
   2. \(S = \{ x | x\ge0 \}\)
   3. \(|S|\) is uncountably infinite.
   4. We cannot talk about the probability that x=1.23 exactly. (It just doesn't work out.)
   5. However, we can define the event that \(1.23 < x < 1.24\), and talk about its probability.
   6. \(P[x>x_0] = 2^{-x_0}\)
   7. \(P[1.23 < x < 1.24]\) \(= 2^{-1.23} - 2^{-1.24} = 0.003\)
10. Event
    1. collection of outcomes, subset of S
    2. what we're interested in.
    3. e.g., outcome is voltage, event is V>5.
    4. certain event: S
    5. null event: \(\emptyset\)
    6. elementary event: one discrete outcome
11. Set theory
    1. Sets: S, A, B, ...
    2. Universal set: U
    3. elements or points: a, b, c
    4. \(a\in S, a\notin S\), \(A\subset B\)
    5. Venn diagram
    6. empty set: {} or \(\emptyset\)
    7. operations on sets: equality, union, intersection, complement, relative complement
    8. properties (axioms): commutative, associative, distributive
    9. theorems: de Morgan
12. Prove deMorgan 2 different ways.
    1. Use the fact that A equals B iff A is a subset of B and B is a subset of A.
    2. Look at the Venn diagram; there are only 4 cases.
13. 2.1.4 Event classes
    1. Remember: an event is a set of outcomes of an experiment, e.g., voltage.
    2. In a continuous sample space, we're interested only in some possible events.
    3. We're interested in events that we can measure.
    4. E.g., we're not interested in the event that the voltage is exactly an irrational number.
    5. Events that we're interested in are intervals, like [.5,.6] and [.7,.8].
    6. Also unions and complements of intervals.
    7. This matches the real world. You can't measure a voltage as 3.14159265...; you measure it in the range [3.14,3.15].
    8. Define \(\cal F\) to be the class of events of interest: those sets of intervals.
    9. We assign probabilities only to events in \(\cal F\).
14. 2.2 Axioms of probability
    1. An axiom system is a general set of rules. The probability axioms apply to all probabilities.
    2. Axioms start with common sense rules, but get less obvious.
    3. I: 0<=P[A]
    4. II: P[S]=1
    5. III: \(A\cap B=\emptyset \rightarrow\) \(P[A\cup B] = P[A]+P[B]\)
    6. III': For \(A_1, A_2, ....\) if \(\forall_{i\ne j} A_i \cap A_j = \emptyset\) then \(P[\bigcup_{i=1}^\infty A_i]\) \(= \sum_{i=1}^\infty P[A_i]\)
15. Example: cards. Q=event that card is queen, H=event that card is heart. These events are not disjoint. Probabilities do not sum.
    1. \(Q\cap H \ne\emptyset\)
    2. P[Q] = 1/13=4/52, P[H] = 1/4=13/52, P[Q \(\cup\) H] = 16/52!=17/52.
16. Example C=event that card is clubs. H and C are disjoint. Probabilities do sum.
    1. \(C\cap H = \emptyset\).
    2. P[C] = 13/52, P[H] = 1/4=13/52, P[Q \(\cup\) H] = 26/52.
17. Example. Flip a fair coin \(A_i\) is the event that the first time you see heads is the i-th time, for \(i\ge1\).
    1. We can assign probabilities to these countably infinite number of events.
    2. \(P[A_i] = 1/2^i\)
    3. They are disjoint, so probabilities sum.
    4. Probability that the first head occurs in the 10th or later toss = \(\sum_{i=10}^\infty 1/2^i\)
18. Corollory 1
    1. \(P[A^c] = 1-P[A]\)
    2. E.g., P[heart] = 1/4, so P[not heart] = 3/4
19. Corollory 2: P[A] <=1
20. Corollory 3: P[\(\emptyset\)] = 0
21. Corollory 4:
    1. For \(A_1, A_2, .... A_n\) if \(\forall_{i\ne j} A_i \cap A_j = \emptyset\) then \(P\left[\bigcup_{i=1}^n A_i\right] = \sum_{i=1}^n P[A_i]\)
    2. Proof by induction from axiom III.
22. Prove de Morgan's law (page 28)
23. Corollory 5 (page 33): \(P[A\cup B] = P[A] + P[B] - P[A\cap B]\)
    1. Example: Queens and hearts. P[Q]=4/52, P[H]=13/52, P[Q \(\cup\) H]=16/52, P[Q \(\cap\) H]=1/52.
    2. \(P[A\cup B] \le P[A] + P[B]\)

5   Questions

Continuous probability:

1. S is the real interval [0,1].
2. P([a,b]) = b-a if 0<=a<=b<=1.
3. Event A = [.2,.6].
4. Event B = [.4,1].

Questions:

1. What is P[A]?
   1. .2
   2. .4
   3. .6
   4. .8
2. What is P[B]?
   1. .2
   2. .4
   3. .6
   4. .8
3. What is P[A \(\cup\) B]?
   1. .2
   2. .4
   3. .6
   4. .8
4. What is P[A \(\cap\) B]?
   1. .2
   2. .4
   3. .6
   4. .8
5. What is P[A \(\cup\) B \(^c\) ]?
   1. .2
   2. .4
   3. .6
   4. .8
6. Retransmitting a noisy bit 3 times: Set e=0.1. What is probability of no error in 3 bits:
   1. 0.1
   2. 0.3
   3. 0.001
   4. 0.729
   5. 0.9
7. Flipping a fair coin until we get heads: How many times will it take until the probability of seeing a head is >=.8?
   1. 1
   2. 2
   3. 3
   4. 4
   5. 5
8. This time, the coin is weighted so that p[H]=.6. How many times will it take until the probability of seeing a head is >=.8?
   1. 1
   2. 2
   3. 3
   4. 4
   5. 5

1   Topics

1. Syllabus and Intro.

2. Why probability is useful

   1. AT&T installed bandwidth to provide level of iphone service (not all users want to use it simultaneously).
   2. also web servers, roads, cashiers, ...
   3. What is a fair price for a car or health or life insurance?
   4. Will a pension plan go broke?
   5. What would you pay today for the right to buy a share of Tesla (TSLA) on 6/30/20 for 400 dollars? (Today, 1/10/19, it's 478.) The answer is not simply $78. It's complicated because you don't have to buy if TSLA is below $400 then.

3. To model something

   1. Real thing too expensive, dangerous, time-consuming (aircraft design).
   2. Capture the relevant, ignore the rest.
   3. Coin flip: relevant: it's fair? not relevant: copper, tin, zinc, ...
   4. Validate model if possible.

4. Computer simulation model

   1. For systems too complicated for a simple math equation (i.e., most systems outside school)
   2. Often a graph of components linked together, e.g., with
      1. Matlab Simulink
      2. PSPICE
   1. many examples, e.g. antilock brake, US economy
   2. Can do experiments on it.

5. To make public policy: "Compas (Correctional Offender Management Profiling for Alternative Sanctions), is used throughout the U.S. to weigh up whether defendants awaiting trial or sentencing are at too much risk of reoffending to be released on bail." Slashdot.

6. Deterministic model

   1. Resistor: V=IR
   2. Limitations: perhaps not if I=1000000 amps. Why?
   3. Limitations: perhaps not if I=0.00000000001 amps. Why?

7. Probability model

   1. Roulette wheel: \(p_i=\frac{1}{38}\) (ignoring http://www.amazon.com/Eudaemonic-Pie-Thomas-Bass/dp/0595142362 )

8. Terms

   1. Random experiment: different outcomes each time it's run.
   2. Outcome: one possible result of a random experiment.
   3. Sample space: set of possible outcomes.
      1. Discrete, or
      2. Continuous.
   1. Tree diagram of successive discrete experiments.
   2. Event: subset of sample space.
   3. Venn diagram: graphically shows relations.

9. Statistical regularity

   1. \(lim_{n\rightarrow\infty}f_k(n) =p_k\)
   2. law of large numbers
   3. weird distributions (e.g., Cauchy) violate this, but that's probably beyond this course.

10. Properties of relative frequency

    1. the frequencies of all the possibilities sum to 1.
    2. if an event is composed of several outcomes that are disjoint, the event's probability is the sum of the outcomes' probabilities.
    3. E.g., If the event is your passing this course and the relevant outcomes are grades A, B, C, D, with probabilities .3, .3, .2, .1, then \(p_{pass}=0.9\) . (These numbers are fictitious.)

11. Axiomatic approach

    1. Probability is between 0 and 1.
    2. Probs sum to 1.
    3. If the events are disjoint, then the probs add.

12. Building a model

    1. Want to model telephone conversations where speaker talks 1/3 of time.
    2. Could use an urn with 2 black, 1 white ball.
    3. Computer random number generator easier.

13. Detailed example in more detail - phone system

    1. Design telephone system for 48 simultaneous users.

    2. Transmit packet of voice every 10msecs.

    3. Only 1/3 users are active.

    4. 48 channels wasteful.

    5. Alloc only M<48 channels.

    6. In the next 10msec block, A people talked.

    7. If A>M, discard A-M packets.

    8. How good is this?

    9. n trials

    10. \(N_k(n)\) trials have k packets

    11. frequency \(f_k(n)=N_k(n)/n\)

    12. \(f_k(n)\rightarrow p_k\) probability

    13. We'll see the exact formula (Poisson) later.

    14. average number of packets in one interval:

        \(\frac{\sum_{k=1}^{48} kN_k(n)}{n} \rightarrow \sum_{k=1}^{48} kp_k = E[A]\)

    15. That is the expected value of A.

14. Probability application: unreliable communication channel.

    1. Transmitter transmits 0 or 1.
    2. Receiver receives 0 or 1.
    3. However, a transmitted 0 is received as a 0 only 90% of the time, and
    4. a transmitted 1 is received as a 1 only 80% of the time, so
    5. if you receive a 0 what's the probability that a 0 was transmitted?
    6. ditto 1.
    7. (You don't have enough info to answer this; you need to know also the probability that a 0 was transmitted. Perhaps the transmitter always sends a 0.)

15. Another application: stocking spare parts:

    1. There are 10 identical lights in the classroom ceiling.
    2. The lifetime of each bulb follows a certain distribution. Perhaps it dies uniformly anytime between 1000 and 3000 hours.
    3. As soon as a light dies, the janitor replaces it with a new one.
    4. How many lights should the janitor stock so that there's a 90% chance that s/he won't run out within 5000 hours?

2   Reading

Leon-Garcia, chapter 1.

3   Homework 1

Homework 1 available.

4   Fun with probability

Questions

1. (7 pts) One of the hardest problems is forming an appropriate probability model. E.g., suppose you're working for Verizon deciding how much data capacity your network will need once it starts selling the iphone. Suppose that you know that each customer will use 5GB/month. Since a month has about 2.5M seconds, does that mean that your network will need to provide only 2KB/s per customer? What might be wrong with this model? How might you make it better? (This is an open-ended question; any reasonable answer that shows creativity gets full points.)

2. (7 pts) One hard problem with statistics is how they should be interpreted. For example, mental health care professionals observe that young men with schizophrenia are usually pot (marijuana) smokers. Assuming for the sake of argument that this correlation is real, does this mean that pot smoking causes schizophrenia?

   Historical note: In 1974, the question of whether cigarette smoking causes lung cancer was answered by forcing some dogs in a lab to smoke and observing that they got cancer more than otherwise identical dogs not forced to smoke.

   The tobacco companies were maintaining that the strong correlation between smoking and lung cancer (1/4 of smokers died from cancer, and almost everyone who died from lung cancer was a smoker) did not demonstrate a causal relation. Maybe there was a common cause for both a desire to smoke and a likelihood to later get cancer. These experiments refuted that claim.

   Mary Beith, the journalist who broke the 'smoking beagles' story

3. (12 pts) Do exercise 1.1 (a-b for each of 3 experiments) in the text on page 18.

4. (12 pts) Do exercise 1.5 (a-c) on page 19.

5. (12 pts) Do exercise 1.10 (a-c) on page 20.

Total: 50 pts.

1   Catalog description

ENGR-2500 Engineering Probability

Axioms of probability, joint and conditional probability, random variables, probability density, mass, and distribution functions, functions of one and two random variables, characteristic functions, sequences of independent random variables, central limit theorem, and laws of large numbers. Applications to electrical and computer engineering problems.

Prerequisites/Corequisites: Corequisite: ECSE 2410.

When Offered: Fall and spring terms annually.

Credit Hours: 3.

CRN: 93683

2   Course Goals / Objectives

To understand basic probability theory and statistical analysis and be able to apply them to modeling typical computer and electrical engineering problems such as noisy signals, decisions in the presence of uncertainty, pattern recognition, network traffic, and digital communications.

3   Student Learning Outcomes

Students will be able to:

1. Be able to apply basic probability theory.
2. Be able to apply concepts of probability to model typical computer and electrical engineering problems.
3. Be able to evaluate the performance of engineering systems with uncertainty.

4   Instructors

5   Identifying yourself in email

If you use a non-RPI email account, please make your name part of the address, so, when I scan my inbox, it's obvious who the sender is. Tagging the subject with #Prob is also helpful. So is including your RCSID in the message. Your RCSID is letters and possibly numbers, not nine digits. Mine is FRANKWR.

6   Computer usage

7   Textbooks etc

1. Leon-Garcia, Probability, Statistics, and Random Processes for Electrical Engineering, 3rd Ed., Pearson/Prentice-Hall, 2008. ISBN 978-0-13-147122-1.

   Why I picked it (in spite of the price):

   1. It is a good book.
   2. This is the same book as we've used for several years.
   3. This book is used at many other universities because it is good. Those courses' lecture notes are sometimes online, if you care to look.

2. There is also a lot of web material on probability. Wikipedia is usually good.

8   Class times & places

1. Mon & Thurs, 4-5:20pm, in Darrin 337.
2. Important announcements will be posted on the class blog.
3. I intend no class activities outside the scheduled times, except for a possible final exam review, a day or two before the exam.
4. You may miss classes. However you are still responsible for knowing what happened.
5. Except when some equipment fails, I post an copy of everything that I write in class.
6. You may use computers etc in class if you don't disturb others.
7. However please do not talk in class.
8. I welcome short questions that have short answers.
9. I will usually stay after class so long as anyone wants to meet me.

9   Assessment measures, i.e., grades

You are welcome to put copies of exams and homeworks in test banks, etc, if they are free to access. However since I put everything online, it's redundant.

10   Academic integrity

1. See the Student Handbook for the general policy. The summary is that students and faculty have to trust each other. After you graduate, your most important possession will be your reputation.

Specifics for this course are as follows.

1. You may collaborate on homeworks, but each team people must write up the solution separately (one writeup per team) using their own words. We willingly give hints to anyone who asks.
2. The penalty for two teams handing in identical work is a zero for both.
3. Writing assistance from the Writing Center and similar sources in allowed, if you acknowledge it.
4. The penalty for plagiarism is a zero grade.
5. You must not communicate with other people or machines, exchange notes, or use electronic aids like computers and PDAs during exams.
6. The penalty is a zero grade on the exam.
7. Cheating will be reported to the Dean of Students Office.

11   Students with special accommodations

Please send me your authorizing memo at least a week before the exam.

12   Student feedback

Since it's my desire to give you the best possible course in a topic I enjoy teaching, I welcome feedback during (and after) the semester. You may tell me or write me or the TAs, or contact a third party, such as Prof John Wen, the ECSE Dept head.