Addition corresponds to convolution and subtraction?

Question

We know that if two random variables have proper densities, than the density of the sum of them is given by the convolution.

But what can we say about the difference of two random variables? $X-Y$ for two random variables $X,Y$? Basically this reduces to the question of finding the density of $-Y$ and then calculating $X+(-Y)$ by convolution. Does anybody here have an idea?

Answer 1

Note that $P(-Y<y) = P(Y>-y) = 1 - F_Y(-y)$, so that $$ f_{-Y}(y) = f_{Y}(-y). $$ Now use the convolution formula on $f_{-Y}(y)$ and $f_X(x)$.