Problem Set 6 - Probability with Engineering Applications | ECE 313 | Assignments Statistics

University Problem Set #6 ECE 313

of Illinois Page 1 of 3 Spring 1999

Assigned: Wednesday, February 24, 1999

Due: Wednesday, March 3, 1999

Reading: Ross, Chapter 3

Noncredit Exercises: Ross pp. 104-117: 53, 58, 59, 62, 63, 70-74, 78, 81

Reminder: Hour Exam I is scheduled for Wednesday March 3, 7:00 pm to 8:00 pm,

in Room 269 Everitt Laboratory.

One 8.5" by 11" sheet of notes is allowed.

Calculators, laptop computers, Palm Pilots etc are not allowed.

The material covered on Problem Sets 2-5 is included on the exam. This Problem Set

also has material on decision theory, and thus, working these problems will help you

prepare for the exam. Coverage of material from Ross is as follows:

Chapter 1 (except Section 1.6)

Chapter 2 (except Section 2.6)

Chapter 3 (except for Section 3.4 on independent events. However, note that we

considered Example 4f on pp. 86-87 in the context of random variables)

Chapter 4.1, 4.3–4.5, 4.7 upto and including Proposition 7.1 on p. 150,

4.8 through p. 156, and 4.9.1-4.9.3. Material on CDF and variance is not

included. However, you must know (or have on your sheet of notes) the

pmfs and means of the binomial, Poisson, and geometric random variables.

Additional material (not always in Ross but covered in class and on homework) can

be found in my Lecture Notes in Chapters 1, 2 (except for decision making involving

costs on pp. 32-33), pp. 54-59 of Chapter 3. Material on random variables can be

found on pp. 63-67, 71-72, and 92-95 of Chapter 4.

Problems:

1. [“I’m leaving on a prop plane”] Consider again Problem #2 of Problem Set #4. Suppose

that 15 of the 105 passengers who hold reservations are arriving in Chicago on a

connecting flight. If the connecting flight is on time, all 15 show up for the flight to

Champaign (nobody stops off at a bar and misses the flight!); else, obviously none of the

15 shows up. Let Y denote the number of nonconnecting passengers who actually show

up for the flight. Let H0 denote the hypothesis that the connecting flight is late, and H1 the

hypothesis that the connecting flight is on time. It is reasonable to assume that the pmf of

Y is the same regardless of which hypothesis is true, and hence we model Y as a binomial

random variable with parameters (90, 0.9). On the other hand, X, the total number of

passengers showing up for the flight, equals Y if H0 is true, while if H1 is true, then

X = 15 + Y, and thus the pmf of X does depend on which hypothesis is true.

(a) Suppose that the gate agent observes that X = 86. What is P{X = 86} when H0 is the true

hypothesis? What is P{X = 86} when H1 is the true hypothesis? What is the value of the

likelihood ratio when X = 86, and what is the agent’s maximum-likelihood decision as to

whether the connecting flight is late?

(b) Repeat part (a) for the case when the gate agent observes that X = 96.

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of Illinois Page 1 of 3 Spring 1999

Assigned: Wednesday, February 24, 1999 Due: Wednesday, March 3, 1999 Reading: Ross, Chapter 3 Noncredit Exercises: Ross pp. 104-117: 53, 58, 59, 62, 63, 70-74, 78, 81

Reminder: Hour Exam I is scheduled for Wednesday March 3, 7:00 pm to 8:00 pm,

in Room 269 Everitt Laboratory. One 8.5" by 11" sheet of notes is allowed. Calculators, laptop computers, Palm Pilots etc are not allowed. The material covered on Problem Sets 2-5 is included on the exam. This Problem Set also has material on decision theory, and thus, working these problems will help you prepare for the exam. Coverage of material from Ross is as follows: Chapter 1 (except Section 1.6) Chapter 2 (except Section 2.6) Chapter 3 (except for Section 3.4 on independent events. However, note that we considered Example 4f on pp. 86-87 in the context of random variables) Chapter 4.1, 4.3–4.5, 4.7 upto and including Proposition 7.1 on p. 150, 4.8 through p. 156, and 4.9.1-4.9.3. Material on CDF and variance is not included. However, you must know (or have on your sheet of notes) the pmfs and means of the binomial, Poisson, and geometric random variables. Additional material (not always in Ross but covered in class and on homework) can be found in my Lecture Notes in Chapters 1, 2 (except for decision making involving costs on pp. 32-33), pp. 54-59 of Chapter 3. Material on random variables can be found on pp. 63-67, 71-72, and 92-95 of Chapter 4.

Problems: 1. [“I’m leaving on a prop plane”] Consider again Problem #2 of Problem Set #4. Suppose that 15 of the 105 passengers who hold reservations are arriving in Chicago on a connecting flight. If the connecting flight is on time, all 15 show up for the flight to Champaign (nobody stops off at a bar and misses the flight!); else, obviously none of the 15 shows up. Let Y denote the number of nonconnecting passengers who actually show up for the flight. Let H 0 denote the hypothesis that the connecting flight is late, and H 1 the hypothesis that the connecting flight is on time. It is reasonable to assume that the pmf of Y is the same regardless of which hypothesis is true, and hence we model Y as a binomial random variable with parameters (90, 0.9). On the other hand, X , the total number of passengers showing up for the flight, equals Y if H 0 is true, while if H 1 is true, then X = 15 + Y , and thus the pmf of X does depend on which hypothesis is true. (a) Suppose that the gate agent observes that X = 86. What is P{ X = 86} when H 0 is the true hypothesis? What is P{ X = 86} when H 1 is the true hypothesis? What is the value of the likelihood ratio when X = 86, and what is the agent’s maximum-likelihood decision as to whether the connecting flight is late? (b) Repeat part (a) for the case when the gate agent observes that X = 96. (c) The gate agent knows that ∏ 0 = P{H 0 is the true hypothesis} = 1/3. For each of the two

of Illinois Page 2 of 3 Spring 1999

observations considered in parts (a) and (b), what is the agent’s MAP (or Bayesian or minimum-probability-of-error} decision as to whether the connecting flight is late? (d) What is the probability that all passengers who show up get a seat? Given that all passengers who showed up got a seat, find the (conditional) probability that the connecting flight was late.

2. [“It a’in’t about bipartisan politics; it’s about …”] The Senate of a certain country has 100 members consisting of 43 Conservative Republicans, 21 Conservative Democrats, 12 Liberal Republicans, and 24 Liberal Democrats. Before each vote, the groups caucus separately. Each group decides independently of the other groups whether to support or oppose the motion. All members of the group then vote in accordance with the caucus decision. For those who think that this is the way politics works, I have this beautiful skyscraper on Wacker Drive in Chicago that I am willing to sell to you at a bargain price… (a) Let A, B, C, and D respectively denote the events that the four groups vote for a spending plan that will lead to a balanced budget in seven years. Suppose that the probabilities of these independent events are P(A) = 0.9, P(B) = 0.6 P(C) = 0.5 and P(D) = 0.2. What is the probability that the bill passes? (b) Let X denote the number of votes in favor of the bill. Then, X is a discrete random variable. Explain why X takes on 16 different values in the range [0,100] and find the pmf of X. Compute P{ X > 50} from the pmf. Is it the same as the answer obtained in part (a)? (Using a spreadsheet/MATLAB/Mathematica will help considerably in doing this part) (c) The President vetoes the bill. Let E, F, G, and H respectively denote the independent events that the four groups support the motion to override the veto. If these events have probabilities P(E) = 0.99, P(F) = 0.4, P(G) = 0.6, and P(H) = 0.1, what is the probability that the motion to override the veto passes? (d) Find the pmf of X , the number of votes in favor of overriding the veto. (e) The Clerk of the Senate, being new on the job, has forgotten whether the vote currently being taken is to pass a bill or to override a veto. The Clerk counts the votes and thus knows the value of X. Specify the maximum-likelihood decision rule (as to what kind of vote was just taken) in terms of the observed value of X. Thus, your answer should be “If X ∈ A then decide that it was a bill, while if X ∈ B then decide that it was an override”

where A and B are disjoint sets of numbers with |A ∪ B| = 16.

Political innocents are reminded that a simple majority (51 or more votes) is required to pass a bill, and a two-thirds majority (67 or more votes) to override a veto.

3. [“Party of Five”] A QMR (quintuple modular redundancy) system is a fancier and more expensive version of the TMR system studied in class. It uses 5 identical circuits. (a) If each circuit has probability p of failing, what is the probability that the majority gate output is incorrect? Ignore the possibility that the majority gate has failed. (Hint: condition on IV and V both failed, one of IV and V failed, and neither IV nor V failed; combine results using the theorem of total probability) (b) A graph model for the TMR system was discussed in class where it was shown that we must replicate links, e.g. each circuit is represented by more than one link, and if the circuit fails, all these links are removed from the graph. Consider the graph model of the QMR system. If there are no failures, how many paths are there from In to Out? How many links represent each of the circuits?

4. [“Reach out and touch someone”] MiddleEast Bell, a division of NYAAHNYUCKS Corp., has built a telephone network as shown below. Terrorists attack each of the seven links. The attacks may be considered to be independent events, and the attack on a link succeeds in severing the link with probability p. If a link is severed, switches automatically re-route calls so as to avoid the failed link (if possible).

Problem Set 6 - Probability with Engineering Applications | ECE 313, Assignments of Statistics

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where A and B are disjoint sets of numbers with |A ∪ B| = 16.