I have a random variable $X$, whose c.d.f. is known. Now I want to estimate the c.d.f of $Y = X \cdot Z$, where the only information I have about $Z$ is that it is a discrete r.v. which takes on three values.
What is the best estimate(in MMSE sense??) for the c.d.f. of $Y$.
My intuition says that it must be the c.d.f. of $X$, but I cant show it rigorously?