Suppose that X is a random variable (say, a normal random variable with mean a and variance b). Then is the sum $X + X$ equal to $2X$?
I am asking this because I know that $2X$ has mean $2a$ and variance $4b$. If we just apply $var(X + X) = var(X) + var(X) = 2b$, we get a different answer because $var$ cannot be applied this way to dependent random variables?
Best Answer
It is because you missed the covariance in between. $$ Var[X + X] = Var[X] + 2Cov[X, X] + Var[X] = b + 2b + b = 4b $$