HW10-Soln(2) - IE 111 Fall 2007 Homework #10 Solutions...

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IE 111 Fall 2007 Homework #10 Solutions Question 1 Consider the following joint Distribution: X -1 0 1 ---|---------------------------- 2 | 0 0.2 0.3 Y | 3 | 0.3 0.2 0 a) Find E(X) The marginal distribution of X is first calculated by summing over Y x -1 0 1 ------------------------------- P(x) 0.3 0.4 0.3 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) 0.2/0.4 0.2/0.4 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
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d) Are X and Y independent? Justify your answer The conditional of Y|X=1 is y 2 3 -------------------------------- P(y|X=1) 1 0 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) = 0.3 P Y (Y=3) = 0.5 0 (0.3)(0.5) Question 2 Consider the following joint Distribution: X -1 0 1 ---|---------------------------- 1 | 0 0.2 0 | Y 2 | 0.3 0 0.3 | 3 | 0 0.2 0 a) Find E(X) The marginal of X is X -1 0 1 ------------------------------------- P X (x) 0.3 0.4 0.3 E(X) = (-1)(0.3) + (0)(0.4) + (1)(0.3) = 0 b) Find P(X | Y=3), the conditional distribution of X given that Y=3. Given Y=3, P(X=0) = 1
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Find the marginal distribution of Y The marginal of Y is y 1 2 3 ------------------------------------- P Y (y) 0.2 0.6 0.2 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 70%. If Clyde goes (=given that Clyde goes), the probability that Seymour goes is 93%; if Clyde doesn't go (=given that Clyde doesn't go), the probability that Seymour goes is 35%. 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. Clyde 0 1 Seymour 0 0.195 0.049 0.244 1 0.105 0.651 0.756 0.3 0.7 1 (b) What is the probability that Seymour travels? 0.756 (c) What is the probability that Clyde goes, given that Seymour does? =0.651/0.756 = 0.861111 (d)Let T=C+S; it is the Total number of travellers in this group. What is the probability that T=0? How about T=1? T=2? P(T=0)=0.195
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This homework help was uploaded on 02/26/2008 for the course IE 111 taught by Professor Storer during the Spring '07 term at Lehigh University .

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HW10-Soln(2) - IE 111 Fall 2007 Homework #10 Solutions...

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