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.
Best Answer
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.