Suppose y is not observed but that we wish to

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ently rolled. Let Xk = 1 if one shows on the k th die 0 else Yk = 1 if two shows on the k th die 0 else. Let X = n=1 Xk , which is the number of one’s showing, and Y = n=1 Xk , which is the number k k of two’s showing. Note that if a histogram is made recording the number of occurrences of each of the six numbers, then X and Y are the heights of the first two entries in the histogram. (a) Find E [X1 ] and Var(X1 ). (b) Find E [X ] and Var(X ). (c) Find Cov(Xi , Yj ) if 1 ≤ i, j ≤ n (Hint: Does it make a difference if i = j ?) (d) Find Cov(X, Y ). (e) Find the correlation coefficient ρX,Y . Are X and Y positively correlated, uncorrelated, or negatively correlated? 4.8. MOMENTS OF JOINTLY DISTRIBUTED RANDOM VARIABLES 155 1 Solution (a) Each Xk is a Bernoulli random variable with parameter p = 6 , so E [Xk ] = 1 and 6 2 ] − E [X ]2 = p − p2 = p(1 − p) = 5 . Var(Xk ) = E [Xk k 36 (b) E [X ] = nE [X1 ] = n , and Var(X ) = nVar(X1 ) = 5n . 6 36 (c) If i = j then Cov(Xi , Xj ) = 0, because Xi and Yj are associated with different, independent, dice rolls. If i = j the situation...
View Full Document

This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

Ask a homework question - tutors are online