You are given two independent random variables: $W \sim \mathrm{Exp}(1)$, $Q \sim U([0; 2\pi ])$.
Also, $a$ is a constant, chosen from $[-\pi/2; \pi/2]$.
You build following random variables, based on $R, W, Q, a$:
$R = \sqrt{2W}$,
$U = R \cos Q$,
$V = R \sin (Q + a)$.
The task is to prove that vector $(U, V)$ has bivariate normal distribution with correlation $\rho = \sin a$.
It is almost obvious that both $U$ and $V$ have zero expectation and odd variance:
$EU = EV = 0$, ${\sigma}_{U} = {\sigma}_{V} = 1$,
so it's not very difficult to prove, that correlation of $U$ and $V$ is $\sin a$. Though, I find it hard to prove, that the vector has normal distribution, so I'd really appreciate your help.
P.S.: Don't you find it amazing, that we can construct normal distribution based on exponential and uniform ones? :)