I was looking at the proof of Two Random Variables, Each Correlated to a Third.
I can't understand some parts of the proof regarding inner product. For example, in the proof, it mentions that let's assume that $X$, a random variable, has a mean of $0$ and a variance of $1$, so does $Y$. Why is that $\left<X, X\right>=1$ and that $\left<X, Y\right>$ is the correlation coefficient of $X$ and $Y$? For $\left<X, X\right>$, I think that $\mathrm{Var}(X) = 1 = E(X^2) - 0 = \left<X, X\right>/n$ but not $\left<X, X\right>$.