4. [BONUS] Let X and Y be two independent random variables with PDFs fx and fy respectively. We are often interested in the PDF of their sum, Z = X + Y. (a) (2 pts) Write the expression of the CDF of Z as the doube integral of the joint PDF of X and Y on the region X + Y s z. Don't solve for this except for simpli- fying the inner integral by recognizing it as a CDF. (b) (2 pts) Show that the PDF of Z looks like the convolution of the PDFs of X and Y, which is defined as follows: fz (z) = fx(x) fr(z - x)dx - oo or equivalently fz ( z ) = [ fx (y ) fx ( z - y) dy. [Hint: To get one of these, use the Leibniz rule to bring the derivative into the integral in (a) and differentiate the inner CDF. Which one you get depends on the order of integration that you chose in (a).]