Jacod and Protter, problems 11.5, 12.4 and 12.11

11.15 Let be a continuous distribution function and the be uniform on . Define . Show that has distribution function .

12.4 Let denote the correlation coefficient for . Let , and . Show that

.

(This is useful because it shows that is independent of the scale of measurement for and .)

12.11 Let be independent normals, both with means and variances . Let

and

where . Show that has a Rayleigh distribution, that is uniform on , and and are independent.