Conditional Probability and Expectation: Exam Questions Explained

Section A (Q1 -- Q11): Short-Answer Questions. Show your work as much as you can for partial points. Question 1 4/4pts Suppose that f(y) = y/2 for 0<y<2 and O otherwise. The conditional pdf is f(y | Y<1) = 2y for O<y<1 and O otherwise. Calculate E[Y|Y<1]. Give your answer up to the fourth decimal place (e.g., 1.2345). 0.6667 0.6666 (with margin: 0.0001)

Question 2 WiEEs | Suppose random variables X and Y have the following joint pdf: 1/6, 0<z<2,0<y< flz,y)=71/3, -1<z<0,-1<y<0; 0, otherwise. We also know that fx (z) = 1/3 for —1 < < 2;and 0, otherwise. Fill out the values/expressions for a, b, and c below: frx(lz =1) = {@: bl<y< [ 0, otherwise.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Answer 3: IR cvou et s bl | Correct Answer 2 | Correct Answer 2.0000 Question 3 2/4pts | A math exam is given to 500 students. Among 500 students, 300 are middle-school students, and 200 are high-school students. The mean exam score for the middle-school students is 50, and the mean test score for the high-school students is 80. What is the mean of all students? Hint: Double expectation

74 62 (with margin: 0) formula correct Question 4 4/4pts A random variable X is uniformly distributed from 10 to 20. Thus, it has pdf f(x) = 1/10 for 10 < x < 20 and O otherwise. Then we have f(z|X > 15) = [a]for 15 <x<20and 0 otherwise. Give the expression or value for [a]. Hint 1: We know that Pr(X>15) = 1/2.

Hint Z: Use the definition ot a conditional density. EE . 0.2 (with margin: 0) Question 5 3/6pts The number N of phone calls made by a person during one month has a geometric distribution with mean 20 variance 380. The lengths of the calls (denoted by X;) are independent exponential random variables with mean 10 minutes and variance 100 minutes. Let S be the total time on the phone in one month. Thatis, S = Zf\;l X; . Compute the mean and the standard deviation of S. mean = 200 and standard deviation =

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

194.94 Hint 1: Do NOT attempt to derive them step by step! You will run out of time. Hint 2: Do you recognize what S is? Note that N and X; are random variables. Use a theorem we learned in class for the quantity. Answer 1: Correct Answer 200.0000 Answer 2: | 194.94 Correct Answer 200 Correct Answer 200.0000

| Question 6 U/ 4pts There are three servers whose service times are independent exponential random variables with a mean 1 hour. Adam has been in service for 2 hours and Ben has been in service for 1 hour. They are still in service. Cindy's service just started. Then what is the probability that Cindy's service will be finished before Adam and Ben? Hint: Exponential distributions are special due to the memoryless property. 002 0.333 (with margin: 0.0003)

Question 7 4/4pts Consider a Markov chain with state space S = {0,1,...,7}. At each transition, the chain moves to a different state with equal probability. Thus, forall¢ = 0,1,2, .... ,7, P;; =0 for j = 1and P;; = 1/7 for j = 4. Calculate the stationary probability for state O. (Hint: You should NOT attempt to set up and solve balance equations! Instead, write down the transition matrix and check if it is a special case. This question does not involve any heavy calculation at all.) Yournswered _{ SV 0.125 (with margin: 0) [This is for Q7 -- Q10] Consider a Markov chain {X, :n=0,1,2,...} with a state space 5={1,2,3,4}, a transition matrix 1/10 1/2 0 2/5 0 1/2 1/2 0 0 0 0 1 1 0 0 0 distribution a®) = (1/4,1/4,1/4,1/4). Pi= and an initial

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Question 8 0/4pts Find Pr(X; =4). Hint: You need a(V). Also, note that the states are {1,2,3,4} not {0,1,2,3}. You Answered 0 0.35 (with margin: 0) Question 9 4/4pts Suppose that Pr(Xy=1)=08; Pr(Xi=2 =038, Pe(Xi=3)=016;, Pe(X;=4}=02L Then find Pr(X, = 2, X5 = 2). Hint: Use Pr(A,B) = Pr(A|B)Pr(B). You need to decide carefully which event will be A between X4=2 and Xs=2. Also, note that the states are {1,2,3,4} not {0,1,2,3}. I . 0.15 (with margin: 0)

Question 10 0/4pts 0.325 | i Find B[X7|Xo = 1. 2.7 (with margin: 0) To find the stationary distribution, we need to set up equations. The following is the equation for state 3. Fill values for (a) -- (e). @71'3:@7&-&-%2-0—@%3—0—@774. | Question 11 1.5/2.5 pts \ |

;ande = a= 0 d= 0 Answer 1: e reeves SR | Correct Answer 1. i Correct Answer 1.0000 T ‘ Answer 2: 0 ‘ Answer 3: vt S

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

| Correct Answer 1/2 Correct Answer 0.5 | Correct Answer 5 ] Correct Answer .5000 Correct Answer 0.5000 ‘ Answer 4: Comect! 0 ‘ Answer 5:

Section B (Q12 -- Q14): T/F or Multiple-Choice Questions Question 12 3/3pts Let X, represent the number of customers in a store at n minutes forn =0, 1, 2, ... The following is a transition diagram for this Markov chain with state space $={0,1,2,...}. 03 O OO OO 050-0-0-0 This Markov chain is positive recurrent .

Question 13 6/6pts This chain is irreducible and has none transient state(s).

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

ARRR State 1 is positive recurrent and has period 4 . P does not exist and/but a stationary distribution exists . Answer 1: irreducible Answer 2: none Answer 3: positive recurrent Answer 4: Answer 5: does not exist

Question 14 0/3pts For the following finite-state Markov chains, each transition is marked with arrows and the transition probability for each arrow is nonzero. For each chain, identify all classes, determine the period of each class, and specify whether each class is recurrent or transient. @) @) OnOxONOS02040) Transient states are {1,2,3,4,5}.

‘ Answer 1: | I | Correct Answer {0,1,2,34,5} Section C (Q15 -- Q19): Show your work.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Question 15 5/15pts Consider a bank with two tellers. Teller 1 has an exponential service time with mean 3 minutes, and Teller 2 has an exponential service time with mean 6 minutes. Alice, Betty, and Carol enter the bank at the same time. Alice goes to teller 1 and Betty goes to teller 2 while Carol waits for the first available teller. Show your work. Double-check if your service rates are correct. (a) What is the expected wait time (in minutes) that Carol spends in the waiting line? 4 (b) What is the probability that Alice leaves before Betty? 0.6667 (c) What is the expected service time (in minutes) that Carol spends with a teller? 2

Correct Answer 2 Correct Answer 2.0000 Correct Answer 2/3 Correct Answer 6667 Correct Answer 0.6666 Correct Answer 6666

‘ Answer 3: Cosrosves B | Correct Answer 4 Correct Answer 4.0000 [This is for Q16 -- Q18] Consider two urns Aand B in a casino game. Initially, urn A contains two white balls, and B contains three black balls. The balls are then “shuffled' repeatedly at each round according to the following rule: pick at random one ball from each run, and swap them. Let X, represent the number of white balls in A. Then the three possible states are shown below:

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Question 16 OIS Find one-step transition probability. Pr(Xnn =1 X, =2) Pr(Xni1 =2| Xn = 1) Pr(X,;1 =0]| X, =1) = Pr(X,11=1|X,=0) = Show work on your scratch paper. Answer 1: | (You left this blank)

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Correct Answer Correct Answer L 1.0000 Answer 2: | (You left this blank) | Correct Answer Correct Answer Correct Answer 1/6 0.1667 1667 T ‘ Answer 3: | (You left this blank) | Correct Answer Correct Answer 1/3 T 0.3333 T | Correct Answer T .3333 Answer 4: | (You left this blank) Correct Answer Correct Answer L Correct Answer 2/3 0.6667 6667

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Question 17 0/4pts Suppose that the stationary distribution is (g, m1,m) = (0.07,0.41,0.52) (Note that this is not necessarily a stationary distribution for this problem but we will pretend that it is a correct stationary distribution). Given that Xy = 2, how many rounds does it take on average to have two white balls in A again (for the first time)? Show work on the scratch paper. (You should not round up or down an average/expectation. So if your answer is not terminating, then report up to the 0.0001th digit.) Hint: This is supposed to be an easy question with a very simple calculation! 2439 14.2857 (with margin: 0)

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Question 18 5/5pts We still pretend that the stationary distribution is (ma,m1,m) = (0.07,0.41, 0.52). A banker decides to gamble on the above process. At each step, the banker wins $500 if there are two white balls in urn A, but has to pay $10 if not. If the banker plays 200 rounds, what is the expected amount of dollars in his pocket? The sign of your answer matters. If the banker earns money, then it should be positive. If not, your answer should be negative. 5,140 5,140 (with margin: 0)

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Question 19 8/9pts Awoman is trapped in a mine having two doors. If she takes the left door, then she will wander around and return to the same position after 3 hours. If she takes the right door, the probability that she will be free after 4 hours is 1/3; the probability that she will return to the same position after 2 hours is 2/3. If she takes one door and returns to the original position, she picks the other door next time. Suppose that she picks the left door on the first attempt. What is the expected duration (in hours) before she reaches freedom? (Hints: Define a = the expected duration before she reaches freedom given that she takes the left door. Also, define b = the expected duration before she reaches freedom given that she takes the right door. Then derive expressions for two expectations a and b and solve the two linear equations.) [ Yousnswered ] S 17 (with margin: 0) almost

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

6650 exam 2

Related Documents