# finding variance and expected value - multivariate case

I would like to ask you a question - I bumped into a problem that I do not know how to solve- Let $X_1;\dots;X_n$ and $Y_1;\dots; Y_m$, be two random samples from distributions with means $\mu_1$ and $\mu_2$, respectively, and with the same variance $\sigma^2$.

I would like to know how can I calculate $E(X-Y)$ and $\operatorname{ Var} (X-Y)$ - could you give me a hint?

I think that the problem is having insufficient information, would be grateful for any tips. Thanks

• What is the relationship between "$X$" and the $X_i$ and between "$Y$" and the $Y_j$? – whuber Nov 9 '15 at 0:35

$$E(X-Y)= E(X) - E(Y) = \mu_1 - \mu_2$$ Regarding the Variance: $$Var(X−Y) = Var(X) + Var(Y) - 2* COV(X,Y) = 2\sigma^2 - 2COV(X, Y)= 2\sigma^2$$ Note that COV(X, Y) = 0 because the two variables are assumed independent
P.S: I am assuming $\mu_1, \mu_2, \sigma^2$ are known or that you know how to estimate them. Let me know if otherwise and I will clarify further