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.

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