HW8_10-Soln

HW8_10-Soln - IE 111 Fall 2010 Homework#8 Solutions...

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Unformatted text preview: IE 111 Fall 2010 Homework #8 Solutions Question 1 Consider the following joint Distribution: X-1 1---|---------------------------- 2 | 0.1 0.4 Y | 3 | 0.4 0.1 a) Find E(X) The marginal distribution of X is first calculated by summing over Y x-1 1------------------------------- P(x) 0.4 0.2 0.4 Then E(X) = (-1)(0.3) + (0)(0.4) + (1)(0.3) = 0 b) Find P(Y | X=0) y 2 3---------------------------------------------- P(Y | X=0) 0.1/0.2 0.1/0.2 Or y 2 3---------------------------------------------- P(Y | X=0) 0.5 0.5 c) Find the marginal distribution of Y y 2 3------------------------ P(y) 0.5 0.5 d) Are X and Y independent? Justify your answer The conditional of Y|X=1 is y 2 3-------------------------------- P(y|X=1) 1 Since the conditional distribution of y depends on x, they are dependent. We can also check if the joint equals the product of the marginals. For example P X,Y (X=1, Y=3) = 0 P X (X=1) = 0.4 P Y (Y=3) = 0.5 0 ≠ (0.4)(0.5) Question 2 Consider the following joint Distribution: X-1 1---|---------------------------- 1 | 0.1 | Y 2 | 0.4 0.4 | 3 | 0.1 a) Find E(X) The marginal of X is X-1 1------------------------------------- P X (x) 0.4 0.2 0.4 E(X) = (-1)(0.4) + (0)(0.2) + (1)(0.4) = 0 b) Find P(X | Y=3), the conditional distribution of X given that Y=3. Given Y=3, P(X=0) = 1 c) Find the marginal distribution of Y The marginal of Y is y 1 2 3------------------------------------- P Y (y) 0.1 0.8 0.1 d) Are X and Y independent? Justify your answer Since the marginal of X in part a) and the conditional of X in part b) are different, they are dependent. Question 3. Two friends, Clyde and Seymour, make reservations on a round-trip from San Diego to Orlando, Florida. But, one or the other or both might have to cancel. The probability that Clyde goes on the trip is 80%. If Clyde goes (=given that Clyde goes), the probability that Seymour goes is 87%; if Clyde doesn't go (=given that Clyde doesn't go), the probability that Seymour goes is 42%. Let C be a random variable that is 0 if Clyde cancels and 1 if Clyde travels; let S be a similar random variable for Seymour. (a) Write out a table of the joint PMF of C and S. Seymour 1 Clyde 0.116 0.084 0.2 1 0.104 0.696 0.8 0.22 0.78 1 (b) What is the probability that Seymour travels?...
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HW8_10-Soln - IE 111 Fall 2010 Homework#8 Solutions...

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