Product of exponentially distributed and uniformly distributed random variables

Question

Let $X$ be an exponentially distributed random variable, and let $V$ be a uniformly distributed random variable on $\{-1,+1\}$ that is independent from $X$. Furthermore, let $Y = X \cdot V$.

I want to calculate $P(\small|Y\small| > b)$, $P(Y > b)$, $E(Y)$, and $var(Y)$, and I also want to find the density of $Y$.

Excuse me if that task is simple for you, for me it certainly is not. Help appreciated!

Answer 1

Hints:

• $|Y|=|XV|=|X||V|$ what can be said about $|V|$?
• $P(Y\in A)=P(Y\in A\mid V=-1)P(V=-1)+P(Y\in A\mid V=1)P(V=1)$
• $P(Y\in A\mid V=v)=P(Xv\in A\mid V=v)=P(Xv\in A)$. The last equality because $X$ and $V$ are independent.
• $\mathbb EXV=\mathbb EX\mathbb EV$ again on base of independence.
• $\text{Var}(Y)=\mathbb EY^2-(\mathbb EY)^2$. Work that out.
• The PDF of $Y$ can be found as the derivative of the CDF of $Y$.