Probability | Part IA, 2005

Let (S,T)(S, T) be uniformly distributed on [1,1]2[-1,1]^{2} and define R=S2+T2R=\sqrt{S^{2}+T^{2}}. Show that, conditionally on

R1,R \leqslant 1,

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

Introduce the new random variables

X=SR2log(R2),Y=TR2log(R2)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)(S, T) are uniformly distributed on the unit disc. The pair (X,Y)(X, Y) may be viewed as a (random) point in R2\mathbb{R}^{2} with polar coordinates (Q,Ψ)(Q, \Psi). Express QQ as a function of RR and deduce its density. Find the joint density of (Q,Ψ)(Q, \Psi). Hence deduce that XX and YY are independent normal random variables with zero mean and unit variance.

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