2.II.11F

Let $(S, T)$ be uniformly distributed on $[-1,1]^{2}$ and define $R=\sqrt{S^{2}+T^{2}}$. Show that, conditionally on

$R \leqslant 1,$

the vector $(S, T)$ is uniformly distributed on the unit disc. Let $(R, \Theta)$ denote the point $(S, T)$ in polar coordinates and find its probability density function $f(r, \theta)$ for $r \in[0,1], \theta \in[0,2 \pi)$. Deduce that $R$ and $\Theta$ are independent.

Introduce the new random variables

$X=\frac{S}{R} \sqrt{-2 \log \left(R^{2}\right)}, \quad Y=\frac{T}{R} \sqrt{-2 \log \left(R^{2}\right)}$

noting that under the above conditioning, $(S, T)$ are uniformly distributed on the unit disc. The pair $(X, Y)$ may be viewed as a (random) point in $\mathbb{R}^{2}$ with polar coordinates $(Q, \Psi)$. Express $Q$ as a function of $R$ and deduce its density. Find the joint density of $(Q, \Psi)$. Hence deduce that $X$ and $Y$ are independent normal random variables with zero mean and unit variance.

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