3
$\begingroup$

$X_1$ is a sample from a normal distribution with mean$=\mu$ and variance $= 1$. The joint distribution of $X_1$ and the sample mean is bivariate normal. I need to find the conditional distribution of $X_1$ given the sample mean. To do this I need to calculate the co-variance between $X_1$ and the sample mean. By doing some simulations I know that that co-variance between $X_1$ and the sample mean is $\frac{1}{n}$, but I'm not sure how to prove it. How do you prove that the co-variance between $X_1$ and the sample mean is $\frac{1}{n}$?

$\endgroup$
0
$\begingroup$

Suppose that $X_1,X_2,\dots,X_n$ are independent random variables, and suppose that the $X_i$ have variance $1$. By the bilinearity of covariance, we have $$\text{Cov}\left(X_1, \frac{1}{n}(X_1+\cdots+X_n)\right)=\frac{1}{n}\sum_1^n\text{Cov}(X_1,X_i).$$ Almost all the covariances on the right are $0$, by independence, and $\text{Cov}(X_1,X_1)=\text{Var}(X_1)=1$. The result follows.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.