Solved – 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

There are two options:

  1. either the two variables are correlated (or might be), and in this case the sample values must have been collected in pairs, which means that the size of both samples must be the same. The fact that the size of the samples is indexed with two different variables n and m indicates otherwise. Please read this post discussing the calculation of the covariance of two samples with different sizes for further clarification.
  2. or the variables are not correlated, and in this case:

$$E(X-Y)= E(X) – E(Y) = mu_1 – mu_2 $$ Regarding the Variance: $$Var(X−Y) = Var(X) + Var(Y) – 2* COV(X,Y) = 2sigma^2 – 2COV(X, Y)= 2sigma^2$$ 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

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