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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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