# Help with Independent Random Variables question

• February 21st 2011, 10:47 AM
Champ83
Help with Independent Random Variables question
Let
X1, · · · ,Xn be independent random variables with E(Xi) = μ and V (Xi) = sigma^2. Also,let ¯X be the sample mean. Then Cov(Xi ¯X , ¯X) = c. What is the numerical value of c ? (Your answer should not depend on n, μ, or sigma)

I have no clue how to do this question...any help will be greatly appreciated!

• February 22nd 2011, 11:58 AM
Moo
Hello,

$Cov(X_i-\bar X,\bar X)=Cov(X_i,\bar X)-Var[\bar X]$

The variance is easy since they're all independent (hence the variance of their sum is the sum of their variance)

$Cov(X_i,\bar X)=\frac 1n \left[Cov(X_i,X_i)+\sum_{j\neq i} Cov(X_i,X_j)\right]=\frac{\sigma^2}{n}$