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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c) 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 P(T=1)=0.049+0.105 = 0.154 P(T=2) = 0.651
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Question 4. Yew-Haul, a company that transports trees, has 13 crews available to work on an upcoming business day. Each crew can do one job per day. Their marketing team will make some effort to sign up customers. Let X be a measurement of the success of the marketing effort, on a scale from 0 to 10. Let Y be the number of customers that sign up, from 0 to 13 (since we can't accept job orders beyond 13). The joint distribution of X and Y is given in the spreadsheet file.
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