# Stochastics

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Final Exam ISE 575 Summer 2012.

Problem 1. Joe is a fool with probability of 0.6, a thief with probability 0.7, and neither with probability 0.25.

a) Determine the probability that he is a fool or a thief but not both.

b) Determine the probability that he is a thief, given that he is not a fool.

Problem 2. The number of steaks Shrek has for dinner is a discrete uniform random variable S with range {1, 2, 3, 4, 5}. The conditional PMF for the number of gallons of beer he drinks for dinner, given that he eats s stakes with this dinner, is discrete uniform with range {1, 2, ¡K, s}. Find the conditional PMF for the number of steaks Shrek eats for dinner, given that he drinks three gallons of beer with this dinner.

Problem 3. Consider N independent random variables distributed identically and uniformly on the interval [0, 1]. Find the PDF of the random variable M, the maximum of these random variables (i.e., M = max {Xi } where Xi are i.i.d. U[0, 1], i = 1,2,¡K,N).

Problem 4. Let X and Y be independent continuous random variables, both distributed uniformly on the interval (-1, 1). Compute the PDF of random variable V = (X + Y)2.

Problem 5. Three novice jugglers ¡V Al, Bob and Cho – play a practice game: they begin juggling at the same time and whoever drops a ball last wins. For each juggler, the time to drop a ball is exponentially distributed with mean 60 seconds. Find the probability that Bob wins such a game given that he does not drop a ball in the first 10 seconds of the game.

Problem 6. A Poisson process with an average arrival rate of £] arrivals per hour begins at time 0. Immediately after each arrival, the average arrival rate for the process is doubled. Define X as the time at which the first arrival occurs, Y as time at which the second arrival occurs, and Z as the time at which the third arrival occurs.

a) Determine the expectation, variance and third moment of variable K = Z – X

b) Determine either the PDF or the transform for K + Y

c) Find P(A), where A is the event that the time between the first and third arrivals is greater than the time from 0 until the second arrival.

Problem 7. Taxis arrive to a taxi stop sequentially, with i.i.d. inter-arrival times exponentially distributed with mean 1/£f. Two persons are in a queue at the taxi stop; the two persons have different destinations (i.e., they cannot share a taxi) and Person 1 has priority over Person 2. The time each of them is willing to wait for a taxi before giving up and walking away is an exponentially distributed random variable, with mean 1/£ for Person 1 and mean 1/£] for Person 2 (each independent of the taxi¡¦s arrival distribution and the other person¡¦s wait time distribution). What is the conditional probability that Person 1 leaves the stop in a taxi, given that Person 2 leaves the stop in a taxi?

Problem 8. A certain part is loaded on a manufacturing machine for processing. The time to complete the processing is distributed exponentially with mean 1/£g. A plant worker supervises the process. He performs checks on the part according to the Poisson process with parameter ƒÜ. After any such check, he either removes the part from the machine (if the check reveals the part has been completed), or performs some tweak in the machine¡¦s setup (if the part has not been completed yet) and leaves. Assume that the checks and the tweaks are done instantaneously.

a) After the part has been loaded on the machine to begin processing, how soon is it expected to be removed?

b) How will your answer to Question (a) change if each tweak doubles the parameter of the exponential function for the time until process completion (originally, £g)? (Provide an expression.)

Problem 9. Two independent production lines each make tennis balls in a Poisson manner. One line makes white tennis balls at an average rate of £fw balls per hour. The other production line makes yellow tennis balls at an average rate of £fy balls per hour. Tennis balls are put into cans as soon as they are produced. Different types of cans are used for white and for yellow balls; colors are never mixed. Each can holds two tennis balls.

a) What is the distribution (PMF) of the number of full cans of white balls produced over one hour?

b) What is the probability that the first can of tennis balls to be filled will be a can of white balls?

c) Find the probability that among the first 10 produced balls, at least 8 will be yellow.

d) If we arrive at a random instant, what is the expectation of T, the time between the moment when the last ball was produced and the moment when the next ball will be produced (independently of color)?