Engineering Probability Class 2 Mon 2019-01-14

W Randolph Franklin (WRF), RPI

2019-01-14 00:00

Table of contents::

1 Review from last class

Outcomes and events for die toss
1. Outcome: 1 or 2 or ...
2. Event: even, or prime, ...
3. Venn diagram. Probabilities.
Discrete outcome vs continuous:
1. Weight of random student
2. Number of cars in student union parking garage on random day

2 Chapter 1 ctd

Rossman-Chance coin toss applet demonstrates how the observed frequencies converge (slowly) to the theoretical probability.
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.
Example of text compression (page 13)
1. Simple way: Use 5 bits for each letter: A=00000, B=00001
2. In English, 'E' common, 'Q' rare
3. Use fewer bits for E than Q.
4. Morse code did this 170 years ago.
  1. E = .
  2. Q = _ _ . _
1. Aside: An expert Morse coder is faster than texting.
2. English can be compressed to about 1 bit per letter (with difficulty); 2 bits is easy.
3. Aside: there is so much structure in English text, that if you add the bit strings for 2 different texts bit-by-bit, they can usually mostly be reconstructed.
4. That's how cryptoanalysis works.
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.

3 Chapter 2

A random experiment (page 21) has 2 parts:
1. experimental procedure
2. set of measurements
Random experiment may have subexperiments and sequences of experiments.
Outcome or sample point $\zeta$: a non-decomposable observation.
Sample space S: set of all outcomes
$|S|$:
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.
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.
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.
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. $p = \frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \cdots = \frac{2}{3}$
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. $P[1.23 < x < 1.24]$ $= 2^{-1.23} - 2^{-1.24} = 0.003$
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
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
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.
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 $\cal F$.
2.2 Axioms of probability
1. An axiom system is a general set of rules. The probability axioms apply to all probabilities.
2. Axioms start with common sense rules, but get less obvious.
3. I: 0<=P[A]
4. II: P[S]=1
5. III: $A\cap B=\emptyset \rightarrow$ $P[A\cup B] = P[A]+P[B]$
6. III': For $A_1, A_2, ....$ if $\forall_{i\ne j} A_i \cap A_j = \emptyset$ then $P[\bigcup_{i=1}^\infty A_i]$ $= \sum_{i=1}^\infty P[A_i]$
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.
Example C=event that card is clubs. H and C are disjoint. Probabilities do sum.
1. $C\cap H = \emptyset$ (corrected).
2. P[C] = 13/52, P[H] = 1/4=13/52, P[Q $\cup$ H] = 26/52.
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 = $\sum_{i=10}^\infty 1/2^i$
Corollory 1
1. $P[A^c] = 1-P[A]$
2. E.g., P[heart] = 1/4, so P[not heart] = 3/4
Corollory 2: P[A] <=1
Corollory 3: P[$\emptyset$] = 0
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.

Engineering Probability Class 1 Thu 2019-01-10

W Randolph Franklin (WRF), RPI

2019-01-10 00:00

Table of contents::

1 Topics

Syllabus and Intro.
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 6/30/19 for 300 dollars? (Today, 1/10/19, it's 345.) It's complicated because you don't have to buy if TSLA is below $300 then.
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.
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.
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.
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?
Probability model
1. Roulette wheel: $p_i=\frac{1}{38}$ (ignoring http://www.amazon.com/Eudaemonic-Pie-Thomas-Bass/dp/0595142362 )
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.
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.
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.)
Axiomatic approach
1. Probability is between 0 and 1.
2. Probs sum to 1.
3. If the events are disjoint, then the probs add.
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.
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.
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.)
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?

4 Probability and gambling

Probability for blackjack: Beat the Dealer: A Winning Strategy for the Game of Twenty-One.
How MIT Students Won $8 Million in the Massachusetts Lottery.
Computer glitch leads Arizona Lottery to issue new Pick 3 tickets. It wasn't ever picking 8s or 9s in certain positions. I heard of another similar story for Arizona years ago but can't find the cite. They seem to have some serious competency problems.

Engineering Probability Homework 1 due Thurs 2019-01-17

W Randolph Franklin (WRF), RPI

2019-01-10 00:00

How to submit

Submit to LMS; see details in syllabus.

Questions

(6 pts) One of the hardest problems is forming an appropriate probability model. E.g., suppose you're working for Verizon deciding how much data capacity your network will need once it starts selling the iphone. Suppose that you know that each customer will use 5GB/month. Since a month has about 2.5M seconds, does that mean that your network will need to provide only 2KB/s per customer? What might be wrong with this model? How might you make it better? (This is an open-ended question; any reasonable answer that shows creativity gets full points.)
(3 pts) One hard problem with statistics is how they should be interpreted. For example, mental health care professionals observe that young men with schizophrenia usually smoke pot (marijuana). Assuming for the sake of argument that this correlation is real, does this mean that pot smoking causes schizophrenia?

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

The tobacco companies were maintaining that the strong correlation between smoking and lung cancer (1/4 of smokers died from cancer) 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
(6 pts) Do exercise 1.3 in the text on page 19.
(6 pts) Do exercise 1.6 on page 19.
(6 pts) Do exercise 1.11 on page 20.

Total: 27 pts.

Engineering Probability Syllabus

W Randolph Franklin (WRF), RPI

2019-01-09 00:00

Table of contents

This is the syllabus for ENGR-2500 Engineering Probability, Rensselaer Polytechnic Institute, Spring 2019.

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

Prerequisites/Corequisites: Corequisite: ECSE 2410.

When Offered: Fall and spring terms annually.

Credit Hours: 3.

CRN: 73727.

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:

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

4 Instructors

4.1 Professor

W. Randolph Franklin. BSc (Toronto), AM, PhD (Harvard)

Informal meetings:
Office:	Jonsson Engineering Center (JEC) 6026
Phone:	+1 (518) 276-6077 (forwards)
Email:	frankwr@YOUKNOWTHEDOMAIN Email is my preferred communication medium. Non-RPI accounts are fine, but please show your name, at least in the comment field. A subject prefix of #Prob is helpful. GPG encryption is fine.
Web:	https://wrf.ecse.rpi.edu/ A quick way to get there is to google RPIWRF.
Office hours:	After each lecture, usually as long as anyone wants to talk. Also by appointment.
	If you would like to lunch with me, either individually or in a group, just mention it. We can then talk about most anything legal and ethical.

4.2 Teaching assistants

Who:
1. Rui Li lir9@YOUKNOWTHEDOMAIN
2. Lingyu Zhang zhangl34@YOUKNOWTHEDOMAIN
Office hours:
1. ECSE Flip Flop lounge in JEC 6037.
2. times TBD
They will try to stay as long as there are students asking questions, but will leave after 15 minutes if no one has arrived.
If you need more time, or a different time, then write them.

5 Computer usage

5.1 Course wiki

This current page https://wrf.ecse.rpi.edu/Teaching/probability-s2019/ has lecture summaries, syllabus, homeworks, etc. You can also get to it from my home page.

5.2 Piazza

Piazza for discussion and questions.

5.3 LMS

RPI LMS will be used only for you to submit homeworks and for us to distribute grades.

5.4 Iclickers

Iclickers may be used in class for attendance and to give the prof feedback.

5.5 Matlab

Matlab may be used for computations.

6 Textbooks etc

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

Why I picked it (in spite of the price):
1. It is a good book.
2. This is the same book as we've used for several years.
3. This book is used at many other universities because it is good. Those courses' lecture notes are sometimes online, if you care to look.
There is also a lot of web material on probability.

7 Class times & places

Mon & Thurs, 4-5:20pm, in Darrin 337.
Important announcements will be posted on the class wiki.
I intend no class activities outside the scheduled times, except for a possible final exam review, a day or two before the exam.
You may miss classes. However you are still responsible for knowing what happened.
Except when some equipment fails, I post an copy of everything that I write in class.
You may use computers etc in class if you don't disturb others.
I welcome short questions that have short answers.

8 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.

8.1 Iclickers

Iclickers will be worth a little.

8.2 Exams

There will be a total of three exams of which the two best count towards the final grade.
Dates: Thur Feb 21, Thu Mar 21.
You may bring one 2-sided letter-size cheat sheet to the first exam, 2 sheets to the second, and 3 sheets to the third.
There are no make-up exams, as the one of the exams can be dropped.
If you're satisfied with your first two exam grades, then you may skip the final.

8.3 Homework

Homework will be assigned every 7-10 days.
Submit your completed homework assignments in LMS as PDF files by midnight on the due date.
Late homeworks receive a 50% reduction of the points if the homework is less than 24hrs late.
Homeworks will not be accepted more than 24hrs late except in cases of an excused absences.
Homework keys will be posted.
The homework sets can be done in groups of up to two students, as follows:
1. One student submits the homework, including the names of both group members.
2. The other student submits a brief note saying, "I'm with Joe Smith smithj34@rpi.edu".
The make-up of the groups is allowed to change from one homework set to the next.
Each member of a group working on a homework set will receive the same grade for this homework.
Some homework questions will be recycled as exam questions.
We will drop the lowest homework.

8.4 Participation in Discussion Forum

The class will make use of the Piazza class discussion system.

If you have questions about a particular item discussed in class or the homeworks then check if this question has already been posted and answered here. If not, then please post your question. If the question does not get answered then see the instructor. The reason for this procedure is that posting questions helps you organize your thoughts, but also that other students can learn a lot from thinking about your question and answering it. The instructor and TA will also monitor this forum and provide feedback. Piazza participation is determined by your activity level, including questions asked, questions answered, posts viewed and days online. This is measured at different times during the semester. The total number of points is then divided between your participation at these times. This implies that and excessive contribution at the end of the semester will not compensate for no participation during the rest of the semester.

Why does part of the grade depend on piazza participation? Success as an engineer is partly determined by your ability to express yourself in public and take part in group discussions.

8.5 Bonus knowitall points

You can earn an extra point by giving me a pointer to interesting material on the web, good enough to post on the class wiki.
Occasionally I make mistakes, either in class or on the web site. The first person to correct each nontrival error will receive an extra point on his/her grade.
One person may accumulate several of these knowitall points.

8.6 Weights and cutoffs

Relative weights of the different grade components
Component	Weight
All the homeworks together	20%
Iclickers	10%
Piazza participation	10%
Top 2 of the 3 exams (each)	30%

Even if the homeworks be out of different numbers of points, they will be normalized so that each homework has the same weight, except that the lowest homework will be dropped.

Grade cutoffs:
Percentage grade	Letter grade
>=95.0%	A
>=90.0%	A-
>=85.0%	B+
>=80.0%	B
>=75.0%	B-
>=70.0%	C+
>=65.0%	C
>=60.0%	C-
>=55.0%	D+
>=50.0%	D
>=0%	F

However, if that causes the class average to be lower than the prof and TA feel that the class deserves, based on how hard students appeared to work, then the criteria will be eased.

8.7 Grade distribution & verification

We'll post homework grading comments on LMS. We'll return graded midterm exams in class.
If you disagree with a grade, then
1. report it within one week,
2. in writing,
3. emailed to a TA, with a copy to the prof.
From time to time, we'll post your grades to LMS. Please report any missing grades within one week to the TA, with a copy to the prof.
It is not allowed to wait until the end of the semester, and then go back 4 months to try to find extra points.
We maintain standards (and the value of your diploma) by giving the grades that are earned, not the grades that are desired. Nevertheless, this course's average grade is competitive with other courses.
Appeal, in writing, first to the TA, then to the prof, to another prof acting as mediator if you wish, and then to the ECSE Head.

8.8 Mid-semester assessment

After the midterm, and before the drop date, we will compute an estimate of your performance to date.

8.9 Early warning system (EWS)

As required by the Provost, we may post notes about you to EWS, for example, if you're having trouble doing homeworks on time, or miss an exam. E.g., if you tell me that you had to miss a class because of family problems, then I may forward that information to the Dean of Students office.

9 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.

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.
The penalty for two teams handing in identical work is a zero for both.
Writing assistance from the Writing Center and similar sources in allowed, if you acknowledge it.
The penalty for plagiarism is a zero grade.
You must not communicate with other people or machines, exchange notes, or use electronic aids like computers and PDAs during exams.
The penalty is a zero grade on the exam.
Cheating will be reported to the Dean of Students Office.

10 Students with special accommodations

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

11 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.

Engineering Probability 2018 Exam 3 - Tues 2018-05-08

W Randolph Franklin (WRF), RPI

2018-05-07 00:00

Name, RCSID:

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Rules:

You have 80 minutes.
You may bring three 2-sided 8.5"x11" papers with notes.
You may bring a calculator.
You may not share material with each other during the exam.
No collaboration or communication (except with the staff) is allowed.
Check that your copy of this test has all nine pages.
Each part of a question is worth 5 points.
You may cross out three question parts, which will not be graded.
When answering a question, don't just state your answer, prove it.

Questions:

You toss two coins. Each comes up heads half of the time. However, for some funny reason, they both come up heads together, or both come up tails together. Intuitively, they not independent. This question is to prove that from the definition of independence.
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This time, you toss three coins, A, B, and C. These are the probabilities:

P[TTT] = P[THH] = P[HTH] = P[HHT] = 0

P[TTH] = P[THT] = P[HTT] = P[HHH] = 1/4

My notation is that TTH means that coin A is tails, coin B tails, and coin C heads. Etc.
1. Are the individual coins fair (i.e., heads half the time)?
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2. Are coins A and B independent?
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3. Are all 3 coins independent?
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This question is about a continuous probability distribution on 2 variables.

$$f_{XY}(x,y) = \begin{cases} c x y & \text{ if } (0\le x) \ \& \ (0\le y)\ \& \ (0\le x+y \le 1) \\ 0 & \text{ otherwise}\end{cases}$$

The nonzero region is the triangle with vertices (0,0), (1,0) and (0,1).

c is some constant, but I didn't tell you what it is.
1. What is c?
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2. What is $F_{XY}(x,y)$?
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3. What is $f_X(x)$?
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4. Are X and Y independent?
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5. What is $P[X\le Y]$ ?
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6. Define a new random variable $Z=X+Y$. What is $F_Z(z)$?
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7. What is $E[X]$?
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8. What is $COV[X,Y]$?
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9. What is $\rho_{X,Y}$?
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10. What is $f_Y(y|x)$?
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11. What is $E[Y|x]$?
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Compute $$\int_0^\infty e^{-x^2} dx$$ .
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What is the legal range for a correlation coefficient?
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What is the variance of the sum of 100 independent variables, each of which is N(0,1)?
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You have 10 independent random variables. Each is uniform on [0,1]. What is the expected value of the max?
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You toss 10 independent fair coins, one after the other.
1. What is the expected total number of heads, given that the first 5 coins came up heads?
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2. What is the probability of the total number of heads being 10, given that the first 5 coins came up heads?
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End of exam 3, total 90 points (considering that 3 questions aren't graded).

Engineering Probability 2018 Exam 3 solution - Tues 2018-05-08

W Randolph Franklin (WRF), RPI

2018-05-07 00:00

Name, RCSID: W. Randolph Franklin, frankwr

OK to give the formulas w/o working them out.

Rules:

You have 80 minutes.
You may bring three 2-sided 8.5"x11" papers with notes.
You may bring a calculator.
You may not share material with each other during the exam.
No collaboration or communication (except with the staff) is allowed.
Check that your copy of this test has all nine pages.
Each part of a question is worth 5 points.
You may cross out three question parts, which will not be graded.
When answering a question, don't just state your answer, prove it.

Questions:

You toss two coins. Each comes up heads half of the time. However, for some funny reason, they both come up heads together, or both come up tails together. Intuitively, they not independent. This question is to prove that from the definition of independence.

P[HT] = 0. However P[A=H] = P[B=T] = 1/2, so P[HT] != P[A=H]P[B=T]. That's the def of not independent.
This time, you toss three coins, A, B, and C. These are the probabilities:

P[TTT] = P[THH] = P[HTH] = P[HHT] = 0

P[TTH] = P[THT] = P[HTT] = P[HHH] = 1/4

My notation is that TTH means that coin A is tails, coin B tails, and coin C heads. Etc.
1. Are the individual coins fair (i.e., heads half the time)?
  
  P[A=H] = 0+0+1/4+1/4 = 1/2 so fair. Ditto B and C.
2. Are coins A and B independent?
  
  P[A=H,B=H] = 1/4, good. P[A=H,B=T]=1/4, good. P[TH] = 1/4, good. P[TT] = 1/4, good. Yes independent.
3. Are all 3 coins independent?
  
  P[HHH] = 1/4 != P[A=H]P[B=H]P[C=H] = 1/8. Not independent.
This question is about a continuous probability distribution on 2 variables.

$$f_{XY}(x,y) = \begin{cases} c x y & \text{ if } (0\le x) \ \& \ (0\le y)\ \& \ (0\le x+y \le 1) \\ 0 & \text{ otherwise}\end{cases}$$

The nonzero region is the triangle with vertices (0,0), (1,0) and (0,1).

c is some constant, but I didn't tell you what it is.
1. What is c?
  
  $\int_0^1\int_0^{1-x} xy\ dy\ dx = 1/24$ so $c=24$
2. What is $F_{XY}(x,y)$?
  
  $$F_{XY}(x,y)=\begin{cases} 0 & \text{ if } x\le0 \cup y\le0 \\ 1 & \text{ if } x\ge 1 \cap y\ge1 \\ 6x^2y^2 & \text{ if } 0\le x \cap 0\le y \cap x+y\le1 \\ (\int_0^x\int_0^{1-x} + \int_0^{1-y}\int_{1-x}^y + \int_{1-y}^x\int_{1-x}^{1-x_0}) (12x_0y_0 dy_0dx_0) & \text{ otherwise}\end{cases}$$
  
  The last case above splits the nonzero integration region into two rectangles and a triangle.
  
  It's also acceptable to draw a figure and say something intelligent w/o being explicit about all the details.
3. What is $f_X(x)$?
  
  $f_X(x)= \int_0^{1-x}f_{XY}(x,y) dy = 12x(1-x)^2$
  
  Note that $\int_0^1 f_X(x)=1$, which is correct.
4. Are X and Y independent?
  
  $f_X(x)=12x(1-x)^2,f_Y(y)=12y(1-y)^2,f_X(x)f_Y(y)\ne f_{XY}(x,y)$
  
  No.
5. What is $P[X\le Y]$ ?
  
  $\int_0^1\int_0^{\min(x,1-x)} f_{XY}(x,y) dy\ dx$. However, since $f_{XY}(x,y) = f_{XY}(y,x)$, $P[X\le Y]=1/2$
6. Define a new random variable $Z=X+Y$. What is $F_Z(z)$?
  
  $f_Z(z) = \int_0^z f_{XY}(x,z-x) dx = 24\int_0^z x(z-x)dx$ for $0\le z\le 1$
  
  $F_Z(z) = \int f_Z(z)dz$
7. What is $E[X]$?
  
  $\int_0^1 xf_X(x)dx = 2/5$
8. What is $COV[X,Y]$?
  
  E[XY] = $\int_0^1\int_0^{1-x}xyf_{XY}dx dy=8\int_0^1x^2(1-x)^4 dx, E[X]=E[Y]=2/5$, COV[X,Y]=E[XY]-E[X]E[Y]. You don't have to work through the math.
9. What is $\rho_{X,Y}$?
  
  $\sigma_X=\sigma_Y= E[X^2]-E[X]^2$
  
  $\rho_{X,Y}=COV[X,Y]/(\sigma_X\sigma_Y)$
10. What is $f_Y(y|x)$?
  
  $f_Y(y|x)=f(x,y)/f(x) = 12xy/(4x^3) = 2\frac{y}{x^2}$
11. What is $E[Y|x]$?
  
  Integrate the above over $y$ to get $x^{-2}$
Compute $$\int_0^\infty e^{-x^2} dx$$ .

Consider $\sigma=1/\sqrt{2}$. Working a little, this will give $\int_0^\infty e^{-x^2} dx=\sqrt{\pi}/2=0.886$. It was also ok just to use a calculator.
What is the legal range for a correlation coefficient?

-1 to 1
What is the variance of the sum of 100 independent variables, each of which is N(0,1)?

100.
You have 10 independent random variables. Each is uniform on [0,1]. What is the expected value of the max?

Let $W=\max(X_i). F_W(w)=w^{10}. f_W(w)=10w^9.E[W]=10/11.$
You toss 10 independent fair coins, one after the other.
1. What is the expected total number of heads, given that the first 5 coins came up heads?
  
  The last 5 coins do not depend on the first 5. So the expectation is 7.5.
2. What is the probability of the total number of heads being 10, given that the first 5 coins came up heads?
  
  1/32

End of exam 3, total 90 points (considering that 3 questions aren't graded).

Engineering Probability 2018 Exam 2 - Thu 2018-03-29

W Randolph Franklin (WRF), RPI

2018-03-29 00:00

Name, RCSID:

.



.

Rules:

You have 80 minutes.
You may bring two 2-sided 8.5"x11" paper with notes.
You may bring a calculator.
You may not share material with each other during the exam.
No collaboration or communication (except with the staff) is allowed.
Check that your copy of this test has all eleven pages.
Each part of a question is worth 5 points.
You may cross out two questions, which will not be graded.
When answering a question, don't just state your answer, prove it.

Questions:

Consider this probability distribution:

$$f_X(x)= \begin{cases} a(2-x) & \text{if } 0\le x\le1\\ 0&\text{otherwise}\end{cases}$$
1. What is $a$?
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2. What is $F_X(x)$?
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3. What is E[X]?
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4. What is the reliability, R[x]?
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5. What is the MTTF?
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6. What is the failure rate, r(x)?
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7. What is $f_X(x|x>.5)$?
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Define a new r.v. Y=2X, where X is the r.v. in the previous question.
1. What is $f_Y(y)$?
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2. What is $F_Y(y)$?
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3. What is E[Y]?
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Your web server gets on the average 1 hit per second. The possible clients are independent of each other.
1. What is the name of appropriate distribution for the number of hits per second?
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2. What is the probability that it gets exactly one hit in the next two seconds?
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3. What is the name of appropriate probability distribution for the time between successive hits?
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4. What is the probability that the time between two successive hits is less than two seconds?
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Let X be an exponential random variable with mean 1.
1. Using the Markov inequality, what's P[X>3]?
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2. Using the Chebyshev inequality, what's P[X>3]?
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3. What's the exact P[X>3]?
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Let X be a normal random variable with mean 100 and standard deviation 10. Give the following numbers, using the supplied table.
1. P[X>100].
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2. P[80<X<100].
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You're tossing 10000 fair coins. What's the probability of getting between 5000 and 5100 heads? Use the table.
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Evaluate $$\int_0^\infty e^{-2 x^2} dx$$
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Let $f_X(x) = 1$ and $f_Y(y)=2y$, both in the range $0\le x, y\le1$.

Let Z=max(X,Y).

What is E[Z]?
```
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```

Normal distribution:

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

Normal distribution:

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

End of exam 1, total 100 points (considering that 2 questions aren't graded).

Engineering Probability 2018 Exam 2 Solution - Thu 2018-03-29

W Randolph Franklin (WRF), RPI

2018-03-29 00:00

Name, RCSID:

WRF solution

Rules:

You have 80 minutes.
You may bring two 2-sided 8.5"x11" paper with notes.
You may bring a calculator.
You may not share material with each other during the exam.
No collaboration or communication (except with the staff) is allowed.
Check that your copy of this test has all eleven pages.
Each part of a question is worth 5 points.
You may cross out two questions, which will not be graded.
When answering a question, don't just state your answer, prove it.

Questions:

Consider this probability distribution:

$$f_X(x)= \begin{cases} a(2-x) & \text{if } 0\le x\le1\\ 0&\text{otherwise}\end{cases}$$
1. What is $a$?
  
  We require that $\int_0^1 f(x) = 1$.
  
  So, $a=2/3$. You don't have to write it, but this gives $f_X(x) = \frac{4}{3}-\frac{2}{3}x$ if $0<x<1$.
2. What is $F_X(x)$?
  
  $$F(x) = \int_0^x f(w) dw = \begin{cases} 0 & \text{if } x\le0 \\ \frac{4}{3}x - \frac{x^2}{3} & 0<x<1 \\ 1 & 1\le x \end{cases} $$
  
  You can use any placeholder variable (other than $x$) in place of $w$.
  
  It doesn't matter where you say $<$ versus $\le$.
3. What is E[X]?
  
  $$E[X] = \int_0^1 x f(x) dx = \int (4/3 x -2/3 x^2)dx = \left. \left(\frac{2 x^2}{3} - \frac{2 x^3}{9}\right)\right|_0^1 = \frac{4}{9}$$
4. What is the reliability, R[x]?
  
  $$R(x) = 1 - F(x) = \begin{cases} 1 - \frac{4}{3}x + \frac{x^2}{3} & 0<x<1 \\ 0 & 1\le x \end{cases} $$
  
  Lifetimes are nonnegative, so I deleted the case for $x<0$, but it doesn't matter.
5. What is the MTTF?
  
  MTTF = $$\int_0^1 R(x) dx = \int \left(1 - \frac{4}{3}x + \frac{x^2}{3}\right) dx = \frac{4}{9}$$
  
  MTTF=E[X]. The integral goes up to 1 because R(x) is 0 when x>1.
6. What is the failure rate, r(x)?
  
  For $0<x<1$,
  
  $$r(x) = \frac{-R'(x)}{R(x)} = \frac{\frac{4}{3}-\frac{2x}{3}}{1 - \frac{4}{3}x + \frac{x^2}{3}} = \frac{4-2x}{3 - 4 x + x^2}$$
  
  You don't need to simplify it.
  
  Note that $r(x)$ grows to infinity as $x$ approaches 1.
7. What is $f_X(x|x>.5)$?
  
  $P[x>.5] = 1-F(.5) = 5/12$.
  
  $$f_X(x|x>.5) = f(x)/P[x>.5] = \begin{cases} \frac{8}{5}(2-x) & \text{if } 0.5\le x\le1\\ 0&\text{otherwise}\end{cases}$$
  
  As a check, you can see that $\int f(x|x>.5) dx = 1$.
Define a new r.v. Y=2X, where X is the r.v. in the previous question.
1. What is $f_Y(y)$?
  
  The nonzero domain for $f_X(x)$ is $0<x<1$, and Y=2X.
  
  So the nonzero domain for $f_Y(y)$ will be $0<y<2$.
  
  $dy/dx = 2$, so
  
  $$f_Y(y) = f_X(y/2)/2 = \begin{cases} \frac{2}{3} - \frac{y}{6} & 0<y<2\\ 0 & \text{otherwise}\end{cases}$$
2. What is $F_Y(y)$?
  
  $$F_Y(y) = F_X(y/2)$$
  
  $$F_Y(y)= \begin{cases} \frac{2}{3} y - \frac{y^2}{12} & 0<y<2\\ 0 & y<0 \\ 1 & y>2 \end{cases}$$
3. What is E[Y]?
  
  $$ E[Y] = \int_0^2 y\left(\frac{2}{3} - \frac{y}{6}\right) dy = \left.\left(y^2/3-y^3/18\right)\right|_0^2 = \frac{8}{9}$$
  
  Alternatively, E[Y] = 2 E[X].
Your web server gets on the average 1 hit per second. The possible clients are independent of each other.
1. What is the name of appropriate distribution for the number of hits per second?
  
  Poisson
2. What is the probability that it gets exactly one hit in the next two seconds?
  
  That r.v. is Poisson with $\alpha=2$ so $$P[X=1] = \frac{2^1 e^{-2}}{1!} = 2 e^{-2} = .27$$
  
  full points for $2 e^{-2}$.
3. What is the name of appropriate probability distribution for the time between successive hits?
  
  Exponential
4. What is the probability that the time between two successive hits is less than two seconds?
  
  Mean: $1/\lambda= 1$. $F(x) = 1-e^{-x}$. $F(2) = 1-e^{-2}=.14=.86$.
  
  full points for $1-e^{-2}$.
Let X be an exponential random variable with mean 1.
1. Using the Markov inequality, what's P[X>3]?
  
  See page 181. $\mu=1$. $P\le 1/3$.
2. Using the Chebyshev inequality, what's P[X>3]?
  
  $\mu=\sigma=1$, P[X<0]=0, so $$P[X>3]= P[|X-1|>2] \le 1/4$$
3. What's the exact P[X>3]?
  
  $F(x) = 1-e^{-x}$ so $P[X>3] = 1-F[3] = e^{-3} = .05$
  
  full points for $e^{-3}$.
Let X be a normal random variable with mean 100 and standard deviation 10. Give the following numbers, using the supplied table.
1. P[X>100].
  
  0.5
2. P[80<X<100].
  
  Converted to $\mu=0,\ \sigma=1$, this is P[-2<Y<0] = F(0)-F(-2) = .5 - .02 = .48.
You're tossing 10000 fair coins. What's the probability of getting between 5000 and 5100 heads? Use the table.

This is a Bernoulli r.v. with $\mu=5000,\ \sigma=\sqrt{npq}=50$.

Use a normal approximation; the answer is F(2)-F(0) = .98-.5 = .48.
Evaluate $$\int_0^\infty e^{-2 x^2} dx$$

For $\mu=0$, what value of $\sigma$ would make $f(x) = c e^{-2 x^2}$?

$$f(x) = \frac{1}{\sqrt{2\pi} \cdot \sigma} \exp\left(\frac{-x^2}{2\sigma^2}\right)$$

so let $\sigma=1/2$ and

$$f(x) = \sqrt{\frac{2}{\pi} } e^{\left(-2 x^2\right)}$$

So $$\int_\infty^\infty e^{-2 x^2} = \sqrt{\frac{\pi}{2} }$$

and $$\int_0^\infty e^{-2 x^2} dx$$ = $$\sqrt{\frac{\pi}{8} }$$

which could be written various ways.
Let $f_X(x) = 1$ and $f_Y(y)=2y$, both in the range $0\le x, y\le1$.

Let Z=max(X,Y).

What is E[Z]?

$F_X(x) = \int f_X(x) dx = x$,

$F_Y(y) = y^2$.

$F_Z(z) = F_X(z) F_Y(z) = z^3$,

$f_Z(z) = 3 z^2$,

$$E[Z] = \int_0^1 3 z^3 dz = 3/4 z^4|_0^1 = \frac{3}{4}$$.

Normal distribution:

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

Normal distribution:

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

End of exam 1, total 100 points (considering that 2 questions aren't graded).

Engineering Probability 2018 Exam 1 - Mon 2018-02-26

W Randolph Franklin (WRF), RPI

2018-02-26 00:00

Name, RCSID:

.




.

Rules:

You have 80 minutes.
You may bring one 2-sided 8.5"x11" paper with notes.
You may bring a calculator.
You may not share material with each other during the exam.
No collaboration or communication (except with the staff) is allowed.
Check that your copy of this test has all seven pages.
Do any 14 of the 17 questions or subquestions. Cross out the 3 that you don't do.
When answering a question, don't just state your answer, prove it.

Questions:

(5 pts) Ten people run a race for gold, silver, bronze. How many ways can the medals be won, w/o any ties?
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Two teams, the Albanians and the Bostonians, are playing a 7 game series. The first team to win 4 games wins the series, and no more games are played. In any game, the Albanians have a 60% chance of winning. The games are independent, and there are no ties.
1. (5 pts) What's the probability that the series will run to 7 games?
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2. (5 pts) What's the probability that the Albanians win the series?
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(5 pts) Imagine two coins. Coin A has two heads. Coin B has the usual one head and one tail, and it is fair. You pick a coin at random (p=.5 to pick either coin) and toss it. It comes up heads.

What is the probability that you picked coin A?
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(5 pts) Consider S={1,2,3,...22}. Is the set of even numbers independent of the set of multiples of 4?
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You are sitting an online multiple choice exam in thraumaturgy, about which you know nothing. Each question has 5 possible answers. You answer each question randomly. The exam ends when you get your first correct answer.
1. (5 pts) What's the relevant probability distribution?
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2. (5 pts) What's the expected number of questions you will need to answer?
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3. (5 pts) What's the standard deviation?
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You are designing a check bit system for transmitting 8-bit bytes over a noisy channel. Since it's really noisy, you append two check bits to each byte, transmitting ten bits in total for each byte. Each bit, independently, can be wrong with probability $10^{-6}$.

You're using a Reed-Solomon error correction scheme, which we teach in another class. If there is only one bad bit in the ten transmitted bits, it will correct the byte. If there are two bad bits, it will report the error, but can't correct it. Three or more bad bits are unlikely enough that we assume they never occur.
1. (5 pts) What's the probability that the receiver can deduce the correct byte?
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2. (5 pts) What's the probability that the receiver will receive a byte that it knows is bad, but can't correct?
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Pretend that we divide the 86 field into a grid of 100 by 100 squares. 5000 students toss 10 paper airplanes each off the JEC roof. Each paper airplane has an independent and uniform probability of hitting each square. Each airplane falls into exactly one square.
1. (5 pts) What's the mean number of airplanes to hit a particular square?
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2. (5 pts) What's the exact probability that a particular square gets zero airplanes? It's ok to give an expression; you don't need to evaluate it.
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3. (5 pts) What's a faster very good approximate formula? An expression is ok.
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4. (5 pts) What's the very good approximate standard deviation for the number of airplanes to hit a particular square? An expression is ok.
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Chip quality control:
1. Each chip is either good or bad.
2. P[good]= 0.9.
3. If the chip is good: P[still alive at t] = $2^{-t}$
4. If the chip is bad: P[still alive at t] = $3^{-t}$
Questions:
1. (5 pts) What's the probability that a random chip is still alive at t=2? Give an expression and evaluate it to give a number.
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2. (5 pts) If a random chip is still alive at t=2, what's the probability that it's a good chip?
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3. (5 pts) If a random chip is still alive at t=2, what's the probability that it will still be alive at t=3?
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End of exam 1, total 70 points.

Engineering Probability 2018 Exam 1 Solution - Mon 2018-02-26

W Randolph Franklin (WRF), RPI

2018-02-25 00:00

Name, RCSID: WRF solutions

Note: Full points will be given for an expression with the numbers, w/o computing the answer.

Rules:

You have 80 minutes.
You may bring one 2-sided 8.5"x11" paper with notes.
You may bring a calculator.
You may not share material with each other during the exam.
No collaboration or communication (except with the staff) is allowed.
Check that your copy of this test has all seven pages.
Do any 14 of the 17 questions or subquestions. Cross out the 3 that you don't do.
When answering a question, don't just state your answer, prove it.

Questions:

(5 pts) Ten people run a race for gold, silver, bronze. How many ways can the medals be won, w/o any ties?

10*9*8 = 720.
Two teams, the Albanians and the Bostonians, are playing a 7 game series. The first team to win 4 games wins the series, and no more games are played. In any game, the Albanians have a 60% chance of winning. The games are independent, and there are no ties.
1. (5 pts) What's the probability that the series will run to 7 games?
  
  The first 6 games must have exactly 3 Albanian wins.
  
  $p = {6 \choose 3} .6^3 .4^3 = .276$
2. (5 pts) What's the probability that the Albanians win the series?
  
  They might win in
  1. 4 games with prob $.6^4=0.130$ or in
  2. 5 games with prob ${4 \choose 3} .6^4 .4=0.201$
  3. 6 games with prob ${5 \choose 3} .6^4 .4^2=0.207$
  4. 7 games with prob ${6 \choose 3} .6^4 .4^3=0.166$
  The sum is 0.704 .
(5 pts) Imagine two coins. Coin A has two heads. Coin B has the usual one head and one tail, and it is fair. You pick a coin at random (p=.5 to pick either coin) and toss it. It comes up heads.

What is the probability that you picked coin A?

P[A] = P[A'] = 1/2. P[H|A] = 1. P[H|A'] = 1/2.

P[A & H] = P[A] P[H|A] = 1/2.

P[A' & H] = P[A'] P[H|A'] = 1/4.

P[H] = P[A&H]+P[A'H] = 3/4.

P[A|H] = P[A&H]/P[H] = (1/2)/(3/4) = 2/3.
(5 pts) Consider S={1,2,3,...22}. Is the set of even numbers independent of the set of multiples of 4?

There are 11 even numbers in S, 5 multiples of 4, and 5 both.

P[even] = 11/22 =1/2. P[mult] = 5/22. P[even]*P[mult] = 5/44.

P[even and mult] = 5/22 not = 5/44.

They are not independent.
You are sitting an online multiple choice exam in thraumaturgy, about which you know nothing. Each question has 5 possible answers. You answer each question randomly. The exam ends when you get your first correct answer.
1. (5 pts) What's the relevant probability distribution?
  
  Geometric.
2. (5 pts) What's the expected number of questions you will need to answer?
  
  p=1/5
  
  E = 1/p = 5.
3. (5 pts) What's the standard deviation?
  
  STD = sqrt(1-p)/p = 4.47.
You are designing a check bit system for transmitting 8-bit bytes over a noisy channel. Since it's really noisy, you append two check bits to each byte, transmitting ten bits in total for each byte. Each bit, independently, can be wrong with probability $10^{-6}$.

You're using a Reed-Solomon error correction scheme, which we teach in another class. If there is only one bad bit in the ten transmitted bits, it will correct the byte. If there are two bad bits, it will report the error, but can't correct it. Three or more bad bits are unlikely enough that we assume they never occur.
1. (5 pts) What's the probability that the receiver can deduce the correct byte?
  
  That would be the probability that 9 or 10 bits are ok.
  
  Let p = prob a given bit is bad. $p=10^{-6}$
  
  Let q=1-p = 0.999999.
  
  P[all 10 bits ok] = $q^{10}$
  
  P[exactly 9 ok] = ${10 \choose 1} p q^9$
  
  P[9 or 10 ok] = $q^{10}+{10 \choose 1} p q^9 = q^9 (q+10p)= .999999999955000$
2. (5 pts) What's the probability that the receiver will receive a byte that it knows is bad, but can't correct?
  
  prob of exactly 2 errors of the 10 bits.
  
  ${10 \choose 2} p^2 q^8 = 45 \cdot 10^{-12} \cdot 0.999999^8 = 4.5\cdot10^{-11}$
  
  Note: These probabilities are small, but if you are transmitting millions of bytes, then they're significant. If you didn't add extra bits, the probability of an 8-bit byte having a bad bit is $1-q^8=8\cdot10^{-6}$. The error correction reduced the probability of a bad byte by a factor of 50,000, at a cost of 25% more transmission and some computation. More, if the 8-bit byte is bad, you don't know it. However, if the 10-bit byte has 2 bad bits, you do know it. Those are the advantages of error correcting codes.
Pretend that we divide the 86 field into a grid of 100 by 100 squares. 5000 students toss 10 paper airplanes each off the JEC roof. Each paper airplane has an independent and uniform probability of hitting each square. Each airplane falls into exactly one square.
1. (5 pts) What's the mean number of airplanes to hit a particular square?
  
  50000 airplanes, 10000 squares. mean = 5.
2. (5 pts) What's the exact probability that a particular square gets zero airplanes? It's ok to give an expression; you don't need to evaluate it.
  
  Let p = prob this square gets a particular airplane = 1/10000 .
  
  Let q=1-p.
  
  Prob this particular square gets no airplane = $q^{50000}=.0067362626$ .
3. (5 pts) What's a faster very good approximate formula? An expression is ok.
  
  Poisson is appropriate here. Call the mean a. a=5.
  
  $P[0] = a^0 e^{-a}/ 0! = e^{-5}\approx .007$
4. (5 pts) What's the very good approximate standard deviation for the number of airplanes to hit a particular square? An expression is ok.
  
  Variance equals mean, so std = $\sqrt{5}$
Chip quality control:
1. Each chip is either good or bad.
2. P[good]= 0.9.
3. If the chip is good: P[still alive at t] = $2^{-t}$
4. If the chip is bad: P[still alive at t] = $3^{-t}$
Questions:
1. (5 pts) What's the probability that a random chip is still alive at t=2? Give an expression and evaluate it to give a number.
  
  Let G = good. Let A = event that chip is alive at 2.
  
  P[A|G] = 1/4. P[A|G'] = 1/9. P[A&G] = .25 * .9 = .225.
  
  P[A&G'] = 1/9 * .1 = .01111.
  
  P[A] = P[A&G] + P[A&G'] = .236 .
2. (5 pts) If a random chip is still alive at t=2, what's the probability that it's a good chip?
  
  Use Bayes.
  
  P[G|A] = P[A&G]/P[A] = .225/.236 = .95
3. (5 pts) If a random chip is still alive at t=2, what's the probability that it will still be alive at t=3?
  
  Let B = event that chip is alive at 3.
  
  B implies A, so the event (B&A) is the event B.
  
  P[B|A] = P[B & A] / P[A] = P[B] / P[A]
  
  P[B|G] = 1/8. P[B|G'] = 1/27.
  
  P[B&G] = P[B|G] P[G] = 1/8 * .9 = .1125
  
  P[B&G'] = P[B|G'] P[G'] = 1/27 * .1 = 0.0037.
  
  P[B] = P[B&G] + P[B&G'] = 0.1125 + 0.0037 = 0.1162 .
  
  P[B|A] = P[B] / P[A] = 0.1162 / .236 = 0.492.

End of exam 1, total 70 points.