1 Homework 11

1. The due date was accidentally set too late, and has been changed to Thurs. (Assignments are not allowed in reading period.)

2. As soon as possible after that, we'll calculate what letter grade you'd get if you didn't write the final, and upload it to LMS.

2 Misc statistics topics

Reviewing the videos.

3 Final exam

1. The material will go up to homework 11,

2. There will be no statistics or paradoxes, since we didn't have homeworks on that.

3. The final exam will be as specified by the registrar.

4. It will be in person, using gradescope.

5. Bring blank scratch paper.

6. You may have three (3) 2-sided crib sheets.

7. As specified in the syllabus, all 3 exams have the same weight, and the lowest will be dropped.

8. The lowest homework will also be dropped.

9. There was no final exam last year because RPI was shut down by the computer hack.

#. Here is from 2 years ago: Spring 2020 final exam. Answers .

1. However the material covered changes somewhat each year.

4 After the course

We have a professional relationship. I'm available to discuss any legal ethical topic even after you graduate.

Even after I retire, you have my non-RPI email.

Parting advice: look at the famous alumni on the Darrin windows. What can you do in later life, so your picture goes there also?

1 No lecture today

In place of today's lecture, watch these videos, and discuss them next Mon.

2 More counterintuitive stats

from current events.

Last year the number of deaths in the US increased by 15% from the year before.

Does that mean that life expectancy decreased by 15%?

(Answer: no. It decreased by a year or 2.)

3 Simpson's paradox

is the formal name for the school admission paradox that I showed you earlier.

https://en.wikipedia.org/wiki/Simpson%27s_paradox

It links to several other paradoxes.

1 Mathematica demo

The Mathematica demo I did in class 24 is online:

1. Printout

2. Mathematica session for people wanting to run the session themselves.

Again, this will not be on the exam. It is for students who feel that I'm going too slow and would like more material.

Nevertheless, Mathematica is an excellent tool that you should become familiar with. If I had my druthers, we'd use it in lots of courses.

2 Chapter 8: statistics

ctd.

Why statistics are important: just read the news about covid.

3 Class 27 videos

In place of Thurs's lecture, watch these videos, and discuss them next Mon.

4 More counterintuitive stats

from current events.

Last year the number of deaths in the US increased by 15% from the year before.

Does that mean that life expectancy decreased by 15%?

(Answer: no. It decreased by a year or 2.)

5 Simpson's paradox

is the formal name for the school admission paradox that I showed you earlier.

https://en.wikipedia.org/wiki/Simpson%27s_paradox

It links to several other paradoxes.

1 Chapter 8: statistics

ctd

2 Interesting Videos

From https://www.youtube.com/playlist?list=PLuh62Q4Sv7BXkeKW4J_2WQBlYhKs_k-pj

Time, number, title
9:57,        69, Significance Testing
9:34,        70, Hypothesis Testing
7:05,        71, A Hypothesis Testing Example

1 Chapter 8: statistics

ctd

1 Exam 2

This class was exam 2, on gradescope.

1 Exam 2

1. Hanjing will hold an extra virtual office hour on Wed 2-3pm.

2. Hanjing and Hao will proctor the exam.

3. Around 3:30, Hanjing will take people with accommodations to my lab to finish.

2 Mathematica

3 Radke videos

We might watch a few in class today from https://www.youtube.com/playlist?list=PLuh62Q4Sv7BXkeKW4J_2WQBlYhKs_k-pj

Time, number, title
8:08,        64, Confidence Intervals
9:34,        65, Maximum A Posteriori (MAP) Estimation
9:05,        66, Maximum Likelihood Estimation
5:55,        67, Minimum Mean-Square Estimation

4 Counterintuitive things in statistics

Statistics has some surprising examples, which would appear to be impossible. Here are some.

1. Average income can increase faster in a whole country than in any part of the country.

   1. Consider a country with two parts: east and west.

   2. Each part has 100 people.

   3. Each person in the west makes \$100 per year; each person in the east \$200.

   4. The total income in the west is \$10K, in the east \$20K, and in the whole country \$30K.

   5. The average income in the west is \$100, in the east \$200, and in the whole country \$150.

   6. Assume that next year nothing changes except that one westerner moves east and gets an average eastern job, so he now makes \$200 instead of \$100.

   7. The west now has 99 people @ \$100; its average income didn't change.

   8. The east now has 101 people @ \$200; its average income didn't change.

   9. The whole country's income is \$30100 for an average of \$150.50; that went up.

2. College acceptance rate surprise.

   1. Imagine that we have two groups of people: Albanians and Bostonians.

   2. They're applying to two programs at the university: Engineering and Humanities.

   3. Here are the numbers. The fractions are accepted/applied.

      city/major	Engin	Human	Total
      Albanians	11/15	2/5	13/20
      Bostonians	4/5	7/15	11/20
      Total	15/20	9/20	24/40

      E.g, 15 Albanians applied to Engin; 11 were accepted.

   4. Note that in Engineering, a smaller fraction of Albanian applicants were accepted than Bostonian applicants.

   5. Ditto in Humanities.

   6. However in all, a larger fraction of Albanian applicants were accepted than Bostonian applicants.

3. I could go on.

5 Chapter 8, Statistics

1. We have a population. (E.g., voters in next election, who will vote Democrat or Republican).

2. We don't know the population mean. (E.g., fraction of voters who will vote Democrat).

3. We take several samples (observations). From them we want to estimate the population mean and standard deviation. (Ask 1000 potential voters; 520 say they will vote Democrat. Sample mean is .52)

4. We want error bounds on our estimates. (.52 plus or minus .04, 95 times out of 100)

5. Another application: testing whether 2 populations have the same mean. (Is this batch of Guiness as good as the last one?)

6. Observations cost money, so we want to do as few as possible.

7. This gets beyond this course, but the biggest problems may be non-math ones. E.g., how do you pick a random likely voter? In the past phone books were used. In a famous 1936 Presidential poll, that biased against poor people, who voted for Roosevelt.

8. In probability, we know the parameters (e.g., mean and standard deviation) of a distribution and use them to compute the probability of some event.

   E.g., if we toss a fair coin 4 times what's the probability of exactly 4 heads? Answer: 1/16.

9. In statistics we do not know all the parameters, though we usually know that type the distribution is, e.g., normal. (We often know the standard deviation.)

   1. We make observations about some members of the distribution, i.e., draw some samples.

   2. From them we estimate the unknown parameters.

   3. We often also compute a confidence interval on that estimate.

   4. E.g., we toss an unknown coin 100 times and see 60 heads. A good estimate for the probability of that coin coming up heads is 0.6.

10. Some estimators are better than others, though that gets beyond this course.

    1. Suppose I want to estimate the average height of an RPI student by measuring the heights of N random students.

    2. The mean of the highest and lowest heights of my N students would converge to the population mean as N increased.

    3. However the median of my sample would converge faster. Technically, the variance of the sample median is smaller than the variance of the sample hi-lo mean.

    4. The mean of my whole sample would converge the fastest. Technically, the variance of the sample mean is smaller than the variance of any other estimator of the population mean. That's why we use it.

    5. However perhaps the population's distribution is not normal. Then one of the other estimators might be better. It would be more robust.

11. (Enrichment) How to tell if the population is normal? We can do various plots of the observations and look. We can compute the probability that the observations would be this uneven if the population were normal.

12. An estimator may be biased. We have an distribution that is U[0,b] for unknown b. We take a sample. The max of the sample has a mean n/(n+1)b though it converges to b as n increases.

13. Example 8.2, page 413: One-tailed probability. This is the probability that the mean of our sample is at least so far above the population mean. $$\alpha = P[\overline{X_n}-\mu > c] = Q\left( \frac{c}{\sigma_x / \sqrt{n} } \right)$$ Q is defined on page 169: $$Q(x) = \int_x^ { \infty} \frac{1}{\sqrt{2\pi} } e^{-\frac{x^2}{2} } dx$$

14. Application: You sample n=100 students' verbal SAT scores, and see $ \overline{X} = 550$. You know that $\sigma=100$. If $\mu = 525$, what is the probability that $\overline{X_n} > 550$ ?

    Answer: Q(2.5) = 0.006

15. This means that if we take 1000 random sample of students, each with 100 students, and measure each sample's mean, then, on average, 6 of those 1000 samples will have a mean over 550.

16. This is often worded as the probability of the population's mean being under 525 is 0.006, which is different. The problem with saying that is that presumes some probability distribution for the population mean.

17. The formula also works for the other tail, computing the probability that our sample mean is at least so far below the population mean.

18. The 2-tail probability is the probability that our sample mean is at least this far away from the sample mean in either direction. It is twice the 1-tail probability.

19. All this also works when you know the probability and want to know c, the cutoff.

6 Hypothesis testing

1. Say we want to test whether the average height of an RPI student (called the population) is 2m.

2. We assume that the distribution is Gaussian (normal) and that the standard deviation of heights is, say, 0.2m.

3. However we don't know the mean.

4. We do an experiment and measure the heights of n=100 random students. Their mean height is, say, 1.9m.

5. The question on the table is, is the population mean 2m?

6. This is different from the earlier question that we analyzed, which was this: What is the most likely population mean? (Answer: 1.9m.)

7. Now we have a hypothesis (that the population mean is 2m) that we're testing.

8. The standard way that this is handled is as follows.

9. Define a null hypothesis, called H0, that the population mean is 2m.

10. Define an alternate hypothesis, called HA, that the population mean is not 2m.

11. Note that we observed our sample mean to be $0.5 \sigma$ below the population mean, if H0 is true.

12. Each time we rerun the experiment (measure 100 students) we'll observe a different number.

13. We compute the probability that, if H0 is true, our sample mean would be this far from 2m.

14. Depending on what our underlying model of students is, we might use a 1-tail or a 2-tail probability.

15. Perhaps we think that the population mean might be less than 2m but it's not going to be more. Then a 1-tail distribution makes sense.

16. That is, our assumptions affect the results.

17. The probability is Q(5), which is very small.

18. Therefore we reject H0 and accept HA.

19. We make a type-1 error if we reject H0 and it was really true. See http://en.wikipedia.org/wiki/Type_I_and_type_II_errors

20. We make a type-2 error if we accept H0 and it was really false.

21. These two errors trade off: by reducing the probability of one we increase the probability of the other, for a given sample size.

22. E.g. in a criminal trial we may prefer that a guilty person go free to having an innocent person convicted.

23. Rejecting H0 says nothing about what the population mean really is, just that it's not likely 2m.

24. (Enrichment) Random sampling is hard. The US government got it wrong here:

    http://politics.slashdot.org/story/11/05/13/2249256/Algorithm-Glitch-Voids-Outcome-of-US-Green-Card-Lottery

25. The above tests, called z-tests, assumed that we know the population variance.

26. If we don't know the population variance, we can estimate it by sampling.

27. We can combine estimating the population variance with testing the hypothesis into one test, called the t-test.

7 Dr Nic's videos

Enrichment; watch if you wish.

1. Understanding the Central Limit Theorem https://www.youtube.com/watch?v=_YOr_yYPytM

2. Variation and Sampling Error https://www.youtube.com/watch?v=y3A0lUkpAko

3. Understanding Statistical Inference https://www.youtube.com/watch?v=tFRXsngz4UQ

4. Understanding Hypothesis testing, p-value, t-test - Statistics Help https://www.youtube.com/watch?v=0zZYBALbZgg

1 Textbook

2 Exam 2

Next Thurs, in class.

Material up through Chap 6.

2 cheat sheets allowed.

1 Textbook

2 How to make money from probability

1. Cracking the Scratch Lottery Code

2. https://www.inquirer.com/philly/news/16_schemes_lottery_legend_Joan_Ginther_likely_didnt_use.html

1 If you see LaTeX source code instead of math formulae in my blog

have you perhaps disabled Javascript?

FYI, the math is rendered client-side by modifying the web page DOM after it's downloaded. The package is MathJaX , and was originally written by a prof at Union in Schenectady.

2 Textbook material

1 Exam 2

Will be in class on Thurs, April 7. Same rules as exam 1, except you can have 2 2-sided crib sheets.

2 Trivia question

Today is the spring equinox. However, today in Albany, sunrise is at 6:56am, sunset 7:08pm. That makes the day 12 hours and 12 minutes long, not 12 hours. Why?

3 Section 5.7, page 261. Conditional pdf, ctd

1. There is nothing majorly new here; it's an obvious extension of 1 variable.

   1. Discrete: Work out an example with a pair of 3-sided loaded dice.

   2. Continuous: a triangular dart board. There is one little trick because for P[X=x]=0 since X is continuous, so how can we compute P[Y=y|X=x] = P[Y=y &amp; X=x]/P[x]? The answer is that we take the limiting probability P[x<X<x+dx] etc as dx shrinks, which nets out to using f(x) etc.

2. Example 5.31 on page 264. This is a noisy comm channel, now with Gaussian (normal) noise. This is a more realistic version of the earlier example with uniform noise. The application problems are:

   1. what input signal to infer from each output,

   2. how accurate is this, and

   3. what cutoff minimizes this?

   In the real world there are several ways you could reduce that error:

   1. Increase the transmitted signal,

   2. Reduce the noise,

   3. Retransmit several times and vote.

   4. Handshake: Include a checksum and ask for retransmission if it fails.

   5. Instead of just deciding X=+1 or X=-1 depending on Y, have a 3rd decision, i.e., uncertain if $|Y|<0.5$, and ask for retransmission in that case.

3. Section 5.8 page 271: Functions of two random variables.

   1. We already saw how to compute the pdf of the sum and max of 2 r.v.

4. What's the point of transforming variables in engineering? E.g. in video, (R,G,B) might be transformed to (Y,I,Q) with a 3x3 matrix multiply. Y is brightness (mostly the green component). I and Q are approximately the red and blue. Since we see brightness more accurately than color hue, we want to transmit Y with greater precision. So, we want to do probabilities on all this.

5. Functions of 2 random variables

   1. This is an important topic.

   2. Example 5.44, page 275. Tranform two independent Gaussian r.v from (X,Y) to (R, $\theta$} ).

   3. Linear transformation of two Gaussian r.v.

   4. Sum and difference of 2 Gaussian r.v. are independent.

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.

7. 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.

8. 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.

1 Chapter 5, Two Random Variables, 3

1. Example 5.9, page 242.

2. Example 5.16, page 252.

3. Example 5.17, page 253.

2 Sect 5.5 Independence, page 254

1. Example 5.19 on page 255.

2. Example 5.20, page 255.

3. Independence: Example 5.22 on page 256. Are 2 normal r.v. independent for different values of $\rho$ ?

3 Sect 5.6 Joint moments, p 257

1. Example 5.24, sum.

2. Example 5.25, product.

3. Example 5.26 page 259. Covariance of independent variables.

4. Correlation coefficient.

5. Example 5.27 page 260 uncorrelated but dependent. Correlation measures linear dependence. If the dependence is more complicated, the variables may be dependent but not correlated.

4 Sect 5.7 Conditional page 261

1. Example 5.29 loaded dice

2. 5.6.2 Joint moments etc

   1. Work out for 2 3-sided dice.

   2. Work out for tossing dart onto triangular board.

3. Example 5.30.

4. Covariance, correlation coefficient.

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 Chapter 5, Two Random Variables, 2

1. Today's reading: Chapter 5, page 233-242.

2. 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.

3. Example 5.3 on page 235. There's no calculation here, but this topic is used for several future problems.

4. Example 5.5 on page 238.

5. Example 5.6 on page 240. Easy, look at it yourself.

6. Example 5.7 on page 241. Easy, look at it yourself.

7. Example 5.8 on page 242. Easy, look at it yourself.

8. Example 5.9 on page 242.

9. 5.3 Joint CDF page 242.

10. Example 5.11 on page 245. What is f(x,y)?

11. Example 5.12 p 246

12. 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]?

13. Example 5.14 on page 247.

14. Example 5.16 on page 252.

1 Matlab

1. Matlab, Mathematica, and Maple all will help you do problems too big to do by hand. Today I'll demo Matlab.

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).

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

3 Tutorial on probability density - 2 variables

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.

4 Comic

Dilbert

1 Midterm 1

Answer it directly in gradescope on your laptop.

1 TA office hours

1. Will meet on their webex meeting places.

2. Hao Lu, luh6@, 2pm Tues and 3pm Sat.

3. Hanjing Wang, wangh36@, 3pm Fri and 9pm Sun.

4. If no one has joined in 15 minutes, they will leave.

5. Write them for other meeting times.

2 Homework 6

is on gradescope.

3 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.

4 Comic

Conditional Risk

1 Continuing Chapter 4

Continue Chapter 4; see the handwritten notes.

2 Comic

Null Hypothesis

1 Midterm exam 1: Feb 28

1. Reminder that it will be on Mon Feb 28

2. In class,

   1. unless you are quarantined etc.

   2. If so, remind me before, and you'll do it remotely.

3. It will use gradescope.

4. So bring your computers.

5. If you have an accommodation, we'll move you to my lab for the rest of the time.

6. The exam is not intended to be a speed contest, but opinions may differ.

7. FWIW, when I graph finish time vs grade, there has been no correlation.

8. You are allowed one 2-sided, 8.5"x11" crib sheet. It may be produced mechanically. You may form consortia and mass produce and sell crib sheets. If so, give me a copy.

9. You are expected not to use your computers for anything except putting answers into gradescope.

2 In-class vs remote lectures

1. I don't personally care which you do.

2. Except exams, which are in person.

3. Suggestions for improving the technical quality of the videos are welcome.

4. However I've already thought of most of the obvious things.

5. I don't want to reduce the quality for the students that do attend in person.

3 Continuing Chapter 4

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?).

4 Examples

4.11, p153.

5 4.3.2 Variance

p160

6 Memoryless Exponential Distn

p 166.

7 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.

8 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.

9 Functions of a r.v.

1. Example 4.29 page 175.

2. Linear function: Example 4.31 on page 176.

10 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.

11 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 50 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.

12 4.9 Generating r.v

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

13 4.10 Entropy

Ignore since it's starred.

14 Xkcd comic

Frequentists vs. Bayesians

1 Programming tools I use

1. I like writing in goodnotes on the ipad, then exporting the day's notes as a PDF file.

2. Then I export the PDF file into MS onedrive.

3. On my linux laptop, I use rclone to mount my onedrive account as a virtual filesystem.

4. Onedrive works better in linux than in windows! In linux, I can mount several personal onedrive accounts. Not so in windows.

5. Then I copy the pdf into the Files directory of the class's nikola installation.

6. Then rebuild the class nikola blog.

7. Then use git to sync the compiled blog over to wrf.ecse.rpi.edu

8. Today I'm trying to share the ipad's screen to a window on my linux laptop (thinkpad x12).

9. Then, from the x12, I can run webex to broadcast the class, and also project to the classroom.

10. What didn't work before was connecting the ipad to the x12, since its integrated graphics causes problems.

11. Also this config is powerful when it works. Sometimes it doesn't.

2 Continuing Chapter 4

1 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.

2 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.

13. Text 4.2 p 148 pdf

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

15. 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.

16. Properties

    1. Memoryless.

    2. \(f(x) = \lambda e^{-\lambda x}\) if \(x\ge0\), 0 otherwise.

    3. Example: time for a radioactive atom to decay.

17. Skip 4.2.1 for now.

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

19. 4.2.2 p 152. Conditional probabilities work the same with continuous distributions as with discrete distributions.

20. 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

21. Text 4.3 p 156 Expected value

3 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).

4 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).

5 Comic

Broomhilda

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 worked there as a Program Director for a few years. (Would you be willing to see your answer on page 1 of tomorrow's paper?)

Total: 54 pts.

1 Continuing Chapter 3

1. Example 3.28 p119 Variance of a Binomial Random Variable

2. Example 3.29 Redundant Systems

1 Exam one

1. Will be in class on Feb 28.

2. The test platform will be gradescope.

3. The questions will mostly be multiple choice and short answer.

4. Students with accommodations will start at 2 and run late in my lab.

5. If you're quarantined, tell me, and do it at home.

6. I cannot possibly block all the ways you might cheat. So I make a reasonable effort, and beyond that have to trust you. That's how it will work after you graduate.

7. If you submit a design for, say a new airport terminal, your customer may not be able to double check your calculations. They have to assume you did it right. However:

   1. The 2004 Collapse at Airport Charles de Gaulle

   2. https://en.wikipedia.org/wiki/Ponte_Morandi

2 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?

3 Chapter 3 Discrete Random Variables

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. 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: \(VAR[cX] = c^2 VAR[X]\)

12. Derive variance for Bernoulli.

13. 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.

14. Geometric distribution: review mean and variance.

15. 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.

16. 3.4 page 111 Conditional pmf

17. 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. \(p_X(m+j|X>m) = \frac{P[X =m+j]}{P[X>m]} = \frac{1/L}{(L-m)/L} = 1/(L-m)\)

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

19. 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?

20. Example 3.26, p114.

21. 3.5 p115 More important discrete r.v

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

23. 3.5.1 p 117 The Bernoulli Random Variable

    We'll do mean and variance.

24. Example 3.28 p119 Variance of a Binomial Random Variable

25. Example 3.29 Redundant Systems

26. 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).

27. 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.

28. Example 3.32 p123 Errors in Optical Transmission

29. 3.5.5 p124 The Uniform Random Variable

4 Xkcd comic

Seashell

1 TA office hours

1. On their webex meeting places.

2. Hao Lu, luh6@, 2pm Tues

3. Hanjing Wang, wangh36@, 3pm Fri

4. If no one has joined in 15 minutes, they will leave.

5. Write them for other meeting times.

2 Midterm exam 1

Will be in class on Feb 24 or 28. Which do you prefer?

Update: you preferred Feb 28.

3 Leon Garcia, chapter 2, ctd

1. This is a summary of some of what I presented last week.

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 https://en.wikipedia.org/wiki/Policy_Analysis_Market, CNN: Amid furor, Pentagon kills terrorism futures market would the future probability of fictional 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:

   

   Compute P[E|U], P[E|U'], P[U|E], etc.

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

6. 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?

7. 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.)

8. 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?

9. 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 \(P[B_j|A]\)

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

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

12. 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.

13. 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 !!!

14. 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*}

15. Example 2.41 p63 dartboard.

16. Example 2.42 p63 random phone numbers.

17. 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. The Mersenne Twister is a common hi-quality RNG.

    4. Arizona lottery got it wrong in 1998.

    5. 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.

    6. 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.

    7. Computer random number routines usually return the same sequence of number each time you run your program, so you can reproduce your results.

    8. You can override this by seeding the generator with a genuine random number from linux /dev/random.

18. 2.8 and 2.9 p70 Fine points: Skip.

19. 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] ?

20. 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.

4 Bayes theorem ctd

1. Wikipedia on Bayes theorem.

   We'll do the examples.

2. We'll (re)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 Bayes theorem ctd

1. Wikipedia on Bayes theorem.

   We'll do the defective item example, using both numbers and Bayes rule.

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

3. Example 2.28, page 51. Assume P[A0]=1/3. P[e]=.1 What is P[A0|B0]?

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. 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: \(VAR[cX] = c^2 VAR[X]\)

6. Derive variance for Bernoulli.

7. 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.

8. Geometric distribution: review mean and variance.

9. 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.

10. 3.4 page 111 Conditional pmf

11. 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. \(p_X(m+j|X>m) = \frac{P[X =m+j]}{P[X>m]} = \frac{1/L}{(L-m)/L} = 1/(L-m)\)

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

13. 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?

14. Example 3.26, p114.

15. 3.5 p115 More important discrete r.v

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

17. 3.5.1 p 117 The Bernoulli Random Variable

    We'll do mean and variance.

18. Example 3.28 p119 Variance of a Binomial Random Variable

19. Example 3.29 Redundant Systems

20. 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).

21. 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.

22. Example 3.32 p123 Errors in Optical Transmission

23. 3.5.5 p124 The Uniform Random Variable

7 Xkcd comic

Cell Phones

1 New stuff

1. See my notes handwritten during class.

2. We basically finished Chapter 2 and started 3.

3. One new idea was independence. It is a possible property of any set of 2 or more events.

4. In a set of 3 events, it can happen that each pair of events is independent, but that all 3 are dependent.

1 Teaching assistants

We have 2 10-hour grad TAs and an undergrad grader:

1. grad: Hao Lu, luh6@

2. grad: Hanjing Wang, wangh36@

3. ugrad: Zehao Li, liz32@

Hao and Hanjing are eager to talk to you and help. Zehao is not allowed to (by RPI's rules); he just grades. We'll assign office hours soon.

2 Alternate emails

If you would like also to receive emails at another address, tell me.

3 Probability in the real world - enrichment

Statistician Cracks Code For Lottery Tickets

Finding these stories is just too easy.

4 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?

5 Leon Garcia, chapter 2, ctd

1. 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 \(n_1n_2 .. n_k\)

2. 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 = \(n^k\)

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

4. 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

5. 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)

6. 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

7. 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.

8. 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 = \(\frac{5*4*3}{5*5*5}\)

9. 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).

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

11. 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.

12. 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.

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

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

15. 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.

16. 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*}

17. 2.3.5. skip

Reading: 2.4 Conditional probability, page 47-

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.

6 Rich Radke's Probability Bites

This is an excellent set of videos for students who want a second viewpoint.

https://www.youtube.com/playlist?list=PLuh62Q4Sv7BXkeKW4J_2WQBlYhKs_k-pj

7 Xkcd comic

Linear Regression

8 Xkcd comic

Correlation

1 What I wrote on the whiteboard in class 2

is visible in the lecture video.

2 The files accessible under the FILES button

now have a new naming scheme to make them sort better.

3 Nuclear reactor fault tree analysis

Basics of Nuclear Power Plant Probabilistic Risk Assessment has more info.

This is enrichment material for students who are interested. It will not be on an exam.

4 Probability in the real world - enrichment

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

5 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.

6 Xkcd comic

P-Values

1 Note the files button in the top bar

In that dir:

class??.mp4

has videos of the remote classes.

transcript??.txt

generated by Webex for those videos

handwritten??.txt

my handwritten notes during the class

misc other files

used in class

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 = _ _ . _

   5. An expert Morse coder is faster than texting.

   6. English can be compressed to about 1 bit per letter (with difficulty); 2 bits is easy.

   7. 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.

   8. 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. Another such failure once crashed the long distance telephone network.

4 Chapter 2: Basic Concepts of Probability Theory

1. Main concepts presented here:

   Set theory

   spec the sample space and events

   Axioms of probability

   rules for computing events probabilities

   Conditional probability

   how partial info affects probabilities. independence of events.

   Sequential random experiments

   a sequence of simpler random subexperiments

2. A random experiment (page 21) has 2 parts:

   1. experimental procedure

   2. set of measurements

3. Random experiment may have subexperiments and sequences of experiments.

4. Outcome or sample point \(\zeta\): a non-decomposable observation.

5. Sample space S: set of all outcomes

6. \(|S|\):

   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.

7. 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.

8. 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.

9. 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. \(p = \frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \cdots = \frac{2}{3}\)

10. 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. \(P[1.23 < x < 1.24]\) \(= 2^{-1.23} - 2^{-1.24} = 0.003\)

11. 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

12. 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

13. 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.

14. 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 \(\cal F\).

15. 2.2 Axioms of probability

    1. An axiom system is a general set of rules. The probability axioms apply to all probabilities.

    2. The idea is to capture rules that occur repeatedly in different applications.

    3. Then, anything we can prove with those rules will apply in all those different applications.

    4. Axioms start with common sense rules, but get less obvious.

    5. I: 0<=P[A]

    6. II: P[S]=1

    7. III: \(A\cap B=\emptyset \rightarrow\) \(P[A\cup B] = P[A]+P[B]\)

    8. III': For \(A_1, A_2, ....\) if \(\forall_{i\ne j} A_i \cap A_j = \emptyset\) then \(P[\bigcup_{i=1}^\infty A_i]\) \(= \sum_{i=1}^\infty P[A_i]\)

16. 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.

17. 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.

18. 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 = \(\sum_{i=10}^\infty 1/2^i\)

19. Corollory 1

    1. \(P[A^c] = 1-P[A]\)

    2. E.g., P[heart] = 1/4, so P[not heart] = 3/4

20. Corollory 2: P[A] <=1

21. Corollory 3: P[\(\emptyset\)] = 0

22. 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.

23. Prove de Morgan's law (page 28)

24. 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

6 Xkcd comic

Significant

7 To read

Leon-Garcia, chapter 2.

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 6/30/23 for 1500 dollars? (Today, 1/10/22, it is 1053) The answer is not simply 1500-1053=447. It is complicated because you don't have to buy if TSLA is below 1500 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 To read

Leon-Garcia, chapter 1.

3 Extra optional material

Prof Rich Radke has an excellent set of 75 short youtube videos called Probability Bites, to help if you're having trouble with the material.

https://www.youtube.com/playlist?list=PLuh62Q4Sv7BXkeKW4J_2WQBlYhKs_k-pj

These are also on Mediasite at:

https://mediasite.mms.rpi.edu/Mediasite5/Channel/probabilitybites

4 Homework 1

Homework 1 available, due Jan 20.

5 New ideas on modeling

Digital twin

6 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 10GB/month. Since a month has about 2.5M seconds, does that mean that your network will need to provide only 4KB/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? Alteratively, maybe schizophrenia causes a desire to smoke, or maybe something else causes both.

   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 forced to not 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 1.2, from the text on page 19.

4. (12 pts) Do exercise 1.6 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.

When Offered: Fall and spring terms annually.

Credit Hours: 3.

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, which keeps increasing each year):

   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, 12:00 pm - 1:20 pm

   1. virtual, and

   2. LOW 3116, 3 cr.

2. Probability will be a mix of in-class and virtual, trying to follow RPI's policy.

   1. If a class has to change to virtual at the last minute, I'll email students.

   2. For the virtual lectures, i.e., at least the first 2 weeks:

      I intend to lecture live, with some supplementary material using screen sharing from my blog, the text and the web . Students will be encouraged to ask questions with chat during the class.

      I'll attempt to record the lectures but recording does not always work with screen sharing.

      I'll also try to save and post the chat window.

   3. When the lectures are in class, attendance is required.

      There may be in-class quizzes, which will serve as a form of attendance taking.

      There will be no recordings since it is impossible to record the mix of talking, and computer stuff like videos, browsing, and notes.

      You are welcome to record my in-class lectures. You may even do this in an organized manner, with one person recording and distributing to others. If so, I might also like a copy.

3. I will maintain a class blog that briefly summarizes each class. Important points will be written down. FYI, see last year's blog https://wrf.ecse.rpi.edu/Teaching/probability-s2021/ .

4. How exams including the final exam will occur will be decided later.

5. I intend no class activities outside the scheduled times, except for a possible final exam review, a day or two before the exam.

6. I intend to use goodnotes on my ipad for any writing, either virtual or in-class, and to post the files.

7. While talking, I welcome short questions that have short answers.

8. 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

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 Other RPI rules

You've seen them in your other classes. They're incorporated here by reference.

12 Students with special accommodations

Please send me your authorizing memo.

13 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.

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.