Assignments_Fall2023_IE410CS481

CS481/IE410 Assignments Chapter 1 4. Let E, F, G be three events. Find expressions for the events that of E, F, G (a) only F occurs, (b) both E and F but not G occur, (c) at least one event occurs, (d) at least two events occur, (e) all three events occur, (f) none occurs, (g) at most one occurs, (h) at most two occur. 10. Show that a set of n events, E 1 , E 2 , …, E n , P( È i=1,2,…,n E i ) £ S i=1,2,…,n P(E i ) 15. Argue that E = EF U EF c , E U F = E U FE c . 18. Assume that each child who is born is equally likely to be a boy or a girl. If a family has two children, what is the probability that both are girls given that (a) the eldest is a girl, (b) at least one is a girl? 23. For events ࠵? ! , ࠵? " , … , ࠵? # show that ࠵?(࠵? ! ࠵? " ⋯ ࠵? # ) = ࠵?(࠵? ! )࠵?(࠵? " |࠵? ! )࠵?(࠵? $ |࠵? ! ࠵? " ) ⋯ ࠵?(࠵? # |࠵? ! ⋯ ࠵? #%! ) 31. Suppose two fair dice are rolled. What is the conditional probability that the first die is six given that the sum of the dice is seven? 35. A fair coin is continually flipped. What is the probability that the first four flips are (a) H, H, H, H? (b) T, H, H, H? (c) What is the probability that the pattern T, H, H, H occurs before the pattern H, H, H, H? Chapter 2 10. Suppose three fair dice are rolled. What is the probability at most one six appears? 12. On a multiple-choice exam with three possible answers for each of the five questions, what is the probability that a student would get four or more correct answers just by guessing? 30. Let X be a Poisson random variable with parameter λ . Show that P {X = ࠵? } increases monotonically and then decreases monotonically as ࠵? increases, reaching its maximum when ࠵? is the largest integer not exceeding λ .

CS481/IE410 Assignments 34. Let the probability density function of X be given by f ( x ) = - ࠵?(4࠵? − 2࠵? " ), 0 < ࠵? < 2 0, ࠵?࠵?ℎ࠵?࠵?࠵?࠵?࠵?࠵? (a) What is the value of c? (b) Compute P{1/2 < X < 3/2}. 59. Let X 1 , X 2 , X 3 and X 4 be independent continuous random variables with a common distribution function F and let p = P{X 1 < X 2 > X 3 < X 4 } (a) Argue that the value of p is the same for all continuous distribution functions F . (b) Find p by integrating the joint density function over the appropriate region. (c) Find p by using the fact that all 4! Possible orderings of X 1 , X 2 , X 3 , X 4 are equally likely. 76. Use Chebyshev’s inequality to prove the weak law of large numbers . Namely, if X 1 , X 2 , … are independent and identically distributed with mean ࠵? and variance ࠵? " , then for any ࠵? > 0, ࠵? ?@ ’ ! (’ " …(’ # # − ࠵?@ > ࠵?B ⇒ 0 as ࠵? ⇒ ∞ Chapter 3 5. An urn contains three white, six red, and five black balls. Six of these balls are randomly selected from the urn. Let X and Y denote respectively the number of white and black balls selected. Compute the conditional probability mass function of X given that Y = 3. Also compute E[X|Y = 1]. 15. The joint probability density function of X and Y is given by f ( x, y ) = * $ & + , 0 < x < y, 0 < y < ∞ Compute E[X 2 | Y = y]. 24. A coin, having probability p of landing heads, is continually flipped until at least one head and one tail have been flipped. (a) Find the expected number of flips needed. (b) Find the expected number of flips that land on heads. (c) Find the expected number of flips that land on tails. (d) Repeat part (a) in the case where flipping is continued until a total of at least two heads and one tail have been flipped. 33. If R i denotes the random amount that is earned in period i , then ࠵? , - = β i-1 R i , where 0 < β < 1 is a specified constant, is called the total discounted reward with discount factor β. Let T be a geometric random variable with parameter 1 – β that is independent of the R i . Show that the expected total discounted reward is equal to the expected total (undiscounted) reward earned by time T. That is, show that E GH ࠵? ,%! ࠵? , - .̇0! K = E G∑ ࠵? , 1 .̇0! K

CS481/IE410 Assignments 60. The following is the transition probability matrix of a Markov chain with states 1, 2, 3, 4 P = ! ." .$ .% .& .% .% .% ." .%’ .%’ .’ ( .% .& ." .$ " If X 0 = 1 (a) Find the probability that state 3 is entered before state 4; (b) Find the mean number of transitions until either state 3 or state 4 is entered. Chapter 5 Example 5.3 (used in setup for question 6): Consider a post office that is run by two clerks. Suppose that when Mr. Smith enters the system, he discovers that Mr. Jones is being served by one of the clerks and Mr. Brown by the other. Suppose also that Mr. Smith is told that his service will begin as soon as either Jones or Brown leaves. If the amount of time that a clerk spends with a customer is exponentially distributed with mean 1/λ, what is the probability that, of the three customers, Mr. Smith is the last to leave the post office? Solution: The answer is obtained by this reasoning: Consider the time at which Mr. Smith first finds a free clerk. At this point either Mr. Jones or Mr. Brown would have just left and the other one would still be in service. However, by the lack of memory of the exponential, it follows that the amount of time that this other man (either Jones or Brown) would still have to spend in the post office is exponentially distributed with mean 1/λ. That is, it is the same as if he were just starting his service at this point. Hence, by symmetry, the probability that he finishes before Smith must equal ½. 6. In example 5.3, if server i serves at an exponential rate ࠵? , , i = 1, 2, show that P {Smith is not last} = P 7 ! 7 ! (7 " Q " + P 7 " 7 ! (7 " Q " 12. If X i , i = 1, 2, 3, are independent exponential random variables with rates ࠵? , , i = 1, 2, 3, find (a) P {X 1 < X 2 < X 3 }, (b) P { X 2 < X 3 | max (X 1 , X 2 , X 3 ) = X 1 }, (c) E [max X i | X 1 < X 2 < X 3 ], (d) E [max X i ]. 30. The lifetimes of A’s dog and cat are independent exponential random variables with respective rates ࠵? 8 and ࠵? 9 . One of them has just died. Find the expected additional lifetime of the other pet.

CS481/IE410 Assignments 31. A doctor has schedules two appointments, one at 1 P.M. and the other at 1:30 P.M. The amounts of time that appointments last are independent exponential random variables with mean 30 minutes. Assuming that both patients are on time, find the expected amount of time that the 1:30 appointment spends at the doctor’s office. 34. Two individuals, A and B, both require kidney transplants. If she does not receive a new kidney, then A will die after an exponential time with rate µ A , and B after an exponential time with rate µ B . New kidneys arrive in accordance with Poisson process having rate ࠵? . It has been decided that the first kidney will go to A (or to B if B is alive and A is not at that time) and the next one to B (if still living). (a) What is the probability that A obtains a new kidney first? (b) What is the probability that B obtains a new kidney first? (c) What is the probability that neither A nor B obtains a new kidney? (d) What is the probability that both A and B obtain new kidneys? 56. An event independently occurs on each day with probability p . Let N( n ) denote the total number of events that occur on the first n days, and let T r denote the day on which the rth event occurs. (a) What is the distribution of N( n )? (b) What is the distribution of T 1 ? (c) What is the distribution of T r ? (d) Given that N( n ) = r, show that the set of r days on which events occurred has the same distribution as a random selection (without replacement) of r of the values 1, 2, . . . , n . Chapter 6 5. There are N individuals in a population, some of whom have a certain infection that spreads as follows. Contacts between two members of this population occur in accordance with a Poisson process having rate ࠵? . When a contact occurs, it is equally likely to involve any of the P ࠵? 2 Q pairs of individuals in the population. If a contact involves an infected and a noninfected individual, then with probability p the noninfected individual becomes infected. Once infected, an individual remains infected throughout. Let X ( t ) denote the number of infected members of the population at time t . (a) Is {X ( t ), t ≥ 0} a continuous-time Markov chain? (b) Specify its type. (c) Starting with a single infected individual, what is the expected time until all members are infected? 6. Consider a birth and death process with birth rates ࠵? , = (i + 1) ࠵? , i ≥ 0, and death rates µ i = iµ, i ≥ 0. (a) Determine the expected time to go from state 0 to state 4. (b) Determine the expected time to go from state 2 to state 5.

CS481/IE410 Assignments 15. Consider a miner trapped in a room that contains three doors. Door 1 leads him to freedom after two days of travel; door 2 returns him to his room after a four-day journey; and door 3 returns him to his room after a six-day journey. Suppose at all times he is equally likely to choose any of the three doors, and let T denote the time it takes the miner to become free. (a) Define a sequence of independent and identically distributed random variables X 1 , X 2 , . . . and a stopping time of N such that T = ∑ ࠵? , : ,0! Note: You may have to imagine that the miner continues to randomly choose doors even after he reaches safety. (b) Use Wald’s equation to find E[T]. (c) Compute E G∑ ࠵? , : ,0! Z࠵? = ࠵?K and note that it is not equal to E [∑ ࠵? , # ,0! ] . (d) Use part (c) for a second derivation of E[T]. 23. In a serve and rally competition involving players A and B, each rally that begins with a serve by player A is won by player A with probability p a and is won by player B with probability q a = 1 - p a, whereas each rally that begins with a serve by player B is won by player A with probability p b and is won by player B with probability q b = 1 - p b . The winner of the rally earns a point and becomes the server of the next rally. (a) In the long run, what proportion of points are won by A? (b) What proportion of points are won by A if the protocol is that the players alternate service? That is, if the service protocol is that A serves for the first point, then B for the second, then A for the third point, and so on. (c) Give the condition for which A wins a higher percentage of points under the winner serves protocol than under the alternating service protocol. 26. Consider a train station to which customers arrive in accordance with a Poisson process having rate ࠵? . A train is summoned whenever there are N customers waiting in the station, but it takes K units of time for the train to arrive at the station. When it arrives, it picks up all waiting customers. Assuming that the train station incurs a cost at a rate of nc per unit of time whenever there are n customers present, find the long-run average cost. 37. There are three machines, all of which are needed for a system to work. Machine i functions for an exponential time with rate ࠵? i before it fails, i = 1, 2, 3. When a machine fails, the system is shut down and repair begins on the failed machine. The time to fix machine 1 is exponential with rate 5; the time to fix machine 2 is uniform (0, 4); and the time to fix machine 3 is a gamma random variable with parameters n = 3 and ࠵? = 2. Once a failed machine is repaired, it is as good as new and all machines are restarted. (a) What proportion of time is the system working? (b) What proportion of time is machine 1 being repaired? (c) What proportion of time is machine 2 in a state of suspended animation (that is, neither working nor being repaired)?