This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: IE 111 Fall 2010 Homework #8 Solutions Question 1 Consider the following joint Distribution: X1 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 x1 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 YX=1 is y 2 3 P(yX=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: X1 1 1  0.1  Y 2  0.4 0.4  3  0.1 a) Find E(X) The marginal of X is X1 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 roundtrip 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?...
View
Full Document
 Spring '07
 Storer
 Probability theory, probability density function, density function, Marginal distribution, Clyde

Click to edit the document details