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MATH 255, Fall 2011 D. Akta s Due on Thursday, Nov 3rd, 2011 @ 17:00 Submit to the box in front of EE-506 HOMEWORK 3 Problem 1: Let HC and HB be the...

I found third and forth questions; but I am not able to find the first and second. Could you give me the first and second questions'answers?
MATH 255, Fall 2011 Due on Thursday, Nov 3rd, 2011 @ 17:00 D. Akta¸ s Submit to the box in front of EE-506 HOMEWORK 3 Problem 1: Let H C and H B be the number of heads obtained when a coin is flipped twice by Can and Bora, respectively. Define X = max( H C ,H B ). Assume that the probability of getting a head for the coin used is p . a) Describe the sample space Ω for this random experiment and compute the probabilities of each outcome. b) Describe the mapping from Ω to range of X . c) Compute the probability mass function (PMF) of X . d) Define Y as the number of heads in two tosses of the coin used. Compare the PMF of Y to PMF of X . Explain the difference. e) Compare E [ Y ] to E [ X ]. Compare var( X ) to var( Y ). f) Find the conditional PMF of X , given that Can got one head in the first toss. g) Let Z = min( H C ,H B ). Determine the joint PMF of X and Z . h) Find P ( X = Z ). i) Compute the conditional PMF of Z given X . j) Compute the conditional expectation E [ Z | X = x ] and then find E [ Z ]. Problem 2: Let X be a discrete random variable and let Y = | X | . a) Assume that the PMF of X is given as p X ( x ) = ± Kx 2 , for x = - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , 0 , otherwise. where K is a constant. Determine the value of K . b) Using a), compute the PMF of Y . c) For an arbitrary PMF of X , determine the PMF of X as a function of PMF of X . Problem 3: Suppose that X and Y are independent discrete random variables with the same geometric PMF p X ( k ) = p Y ( k ) = (1 - p ) ( k - 1) p, for k = 1 , 2 ,..., 1
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for 0 < p < 1 and p X,Y ( k 1 ,k 2 ) = p X ( k 1 ) p Y ( k 2 ) since X and Y are independent. Compute the conditional PMF P ( X = k | X + Y = n ) for n 2. Problem 4: 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 p N | K ( n | k ) = 1 k , for n = 1 , 2 ,...,k. a) Find the joint PMF of K and N . b) Find the marginal PMF of N . c) Find the conditional PMF p K | N ( k | N = 2). d) Find the conditional mean and variance of K , given that he bought at least 2 but no more than 3 books. e) The cost of each book is a random variable with mean 30TL. What is the expected value of his total expenditure? Hint: Condition on the events { N = 1 } ,..., { N = 4 } , and use the total expectation theorem. 2
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