Consider three independently and identically distributed N(µ, $σ^2$) random variables. Let us call them X1, X2 and X3. You are to randomly select two out of these three random variables with replacement and compute the covariance of the two selected random variables.
(a) Find the probability mass function of the covariance of the selected random variables.
(b) Find the expected value of the covariance of the selected random variables.
Here's what I got so far:
With 2 different variables the covariance is 0 with probability 2/3.
With the same variables, the covariance is the variance which equals $σ^2$ with probability 1/3. But how do I get the probability mass function from this for part (a) and totally lost for part (b). Any help would be much appreciated!
You have already solved the questions!!
Probability Mass function of Covariance:
$$Pr[Cov = 0] = \frac{2}{3} \\ Pr[Cov= \sigma^2 ] =\frac{1}{3}$$
Expectation:
$$E[cov] = 0.\frac{2}{3} + \sigma^2.\frac{1}{3} \\= \frac{\sigma^2}{3}$$