IE523_Hw2 - IE 523 Probabilistic Analysis Homework 2 Due...

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Unformatted text preview: IE 523 Probabilistic Analysis Homework # 2 Due: Thursday, October 08, 17:00. To be submitted to EA 323. Question 1: Alvin shops for probability books for K hours, where K is a random variable that is equally likely to be 1, 2, 3 or 4. The number of books N that he buys is random and depends on how long he shops according to the conditional pmf pN |K ( n | k ) = a. Find the joint pmf of K and N. b. Find the marginal pmf of N. c. Find the conditional pmf of K given that N = 2. d. Find the conditional mean and variance of K, given that 2 ≤ N ≤ 3 e. The cost of each book is a random variable with mean $30. What is the expected value of his total expenditure? Question 2: Joe Lucky plays the lottery on any given week with probability p, independently of whether he played on any other week. Each time he plays, he has a probability q of winning, again independently of everything else. During a fixed time period of n weeks, let X be the number of weeks that he played the lottery and Y be the number of weeks that he won. a. What is the probability that he played the lottery on any particular week, given that he did not win on that week? b. Find the conditional pmf pY | X ( y | x ) . c. Find the joint pmf pY , X ( y, x ) . d. Find the marginal pmf pY ( y ) . e. Find the conditional pmf p X |Y ( x | y ) . Do this algebraically using the preceding answers. f. Rederive the answer to part-e by thinking as follows: for each one of the (n – Y) weeks that he did not win, the answer to part-a should tell you something. 1 , n = 1,..., k . k 1 Question 3: If X is a geometric random variable, show analytically that P { X = n + k | X > n} = P { X = k } . Give an argument using the interpretation of a geometric random variable as to why the preceding equation is true. Question 4: An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let X be the selection number of the first chosen ball that is blue. For instance, if the first selection is red and the second is blue, then X is equal to 2. a. Find P { X > i} , i ≥ 1 . b. Show that with probability 1, a blue ball is eventually chosen. (That is, show that P { X < ∞} = 1 .) c. Find E [ X ] . Question 5: Suppose that the number of events that occur in a given time period is a Poisson random variable with parameter λ . If each event is classified as type i event with probability pi , i = 1,..., n, ∑p i = 1 , independently of other events, show that the numbers of type i events that occur, i = 1,..., n, are independent Poisson random variables with respective parameters λ pi , i = 1,..., n . Hint: see Chapter2-Problem37 of your course textbook. Question 6: A coin, which lands on heads with probability p, is continually flipped. Compute the expected number of flips that are made until a string of r heads in a row is obtained. Hint: You may condition on the time of the first occurrence of tails. Question 7: An urn contains a white and b black balls. After a ball is drawn, it is returned to the urn if it is white; but if it is black, it is replaced by a white ball. Let M n denote the expected number of white balls in the urn after the foregoing operation has been repeated n times. a. Derive the recursive equation M n +1 = ⎜1 − ⎛ ⎝ 1⎞ ⎟ M n +1 a+b⎠ n 1⎞ ⎛ b. Use part-a to prove that M n = a + b − b ⎜1 − ⎟. ⎝ a+b⎠ c. What is the probability that the (n + 1)st ball drawn is white? 2 ...
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