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
Best Answer
There are two options:
$$E(X-Y)= E(X) - E(Y) = \mu_1 - \mu_2 $$ 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