The covariance of two random variables $X$ and $Y$ is defined by: $$\mathrm{Cov}(X,Y)= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\operatorname{E}(Y)$$
Another related definition is correlation coefficient $$\rho(X,Y) = \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}$$
It can be proved that the correlation coefficient $\rho(X,Y)$ always lies between −1 and +1. $X$ and $Y$ are two independent standard normal random variables. We now define another random variable $Z$ by $Z=\rho X+\sqrt{1-\rho^2}\cdot Y$ where $\rho \in [−1,1]$.
How can one prove hat $\rho(X,Z) = \rho$?