# Homework 4, due Friday Oct 24

Jacod and Protter, problems 11.5, 12.4 and 12.11

11.15 Let $F$ be a continuous distribution function and the $U$ be uniform on $(0,1)$. Define $G(u) = \inf \{x \, : \, F(x) \geq u\}$. Show that $G(u)$ has distribution function $F$.

12.4 Let $\rho_{X,Y}$ denote the correlation coefficient for $(X,Y)$. Let $a > 0$, $c > 0$ and $b \in \mathbb{R}$. Show that

$\displaystyle \rho_{aX + b,cY+b} = \rho_{X,Y}$.

(This is useful because it shows that $\rho$ is independent of the scale of measurement for $X$ and $Y$.)

12.11 Let $(X,Y)$ be independent normals, both with means $\mu = 0$ and variances $\sigma^2$. Let

$\displaystyle Z = \sqrt{X^2 + Y^2}$ and $\displaystyle W = \arctan\Big(\frac{X}{Y}\Big)$

where $-\frac{\pi}{2} \leq W \leq \frac{\pi}{2}$. Show that $Z$ has a Rayleigh distribution, that $W$ is uniform on $(-\frac{\pi}{2},\frac{\pi}{2})$, and $Z$ and $W$ are independent.