Fall 2011 Strategic Practice 8: Section 2 (Transformations) - Question 2
Let $U\sim Unif(0, 2\pi )$ and let $T\sim Expo(1)$ be independent of U. Define $X = \sqrt{2\pi }cosU$ and $Y = \sqrt{2T }sinU$. Find the joint PDF of (X, Y). Are they independent? What are their marginal distributions?
Solution: Using Jacobians to go from the joint pdf of U, T to that of X, Y, we see the latter factors into a function of x times a function of y, so X and Y are independent, and they each have the N(0, 1) distribution. Thus, X and Y are i.i.d. standard Normal r.v.s; this result is called the Box-Muller method for generating Normal r.v.s.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."