Here’s another interesting problem I came across on math.stackexchange: Suppose is a continuous random variable with density and where and are independent and
Show that is a continuous random variable, and that the density of is even, i.e. for each .
Solution: If then
As the distribution function of is differentiable, is continuous, so we may compute the density by differentiation:
By symmetry it is evident that .
Note that this does not depend on the distribution of !