I am given 2 independent random variables $\mathit X_1$, $\mathit X_2$ that both follow uniform distribution $\mathit U(0,1)$. I am also given $\mathit Y = X_1+X_2$. How would I go about finding the expected value and the variance of $\mathit Y?$
So far I have $\mathit E(Y) = E(X_1+X_2) = E(X_1)+E(X_2) = \frac 12+\frac 12 = 1$, but not quite sure if this is correct. For variance, I'm not really sure where to start.
Also, in general, how would I go about finding the expected value and variance for any number of iid random variables (for any distribution)?
Seems like a pretty simple question, but I just can't find the answer anywhere. Any help is appreciated. Thanks in advance.
Your calculation for the expectation is correct.
More generally, the expected value satisfies linearity, i.e. for constants $c_1,...,c_n$ and random variables $X_1,...,X_n$,
$$E[c_1X_1+...+c_nX_n]=c_1E[X_1]+...+c_nE[X_n].$$
The variance is not linear. In the special case $X_1,...,X_n$ are uncorrelated (a sufficient condition for this is that they are independent), then
$$\text{Var}(c_1X_1+...+c_nX_n)=c_1^2\text{Var}(X_1)+...+c_n^2\text{Var}(X_n).$$
You can check out further properties of mean and variance here and here.