I have two dependent Random Variables, having different distribution, and I want to calculate the sum of them. To be more specific, the dependence is given by the formula: Y = a * X, where x,y are RVs and a a positive number. What is the sum of $f_X$, $f_Y$?
I want to avoid a formula with convolution, because it is difficult to handle it.
$X+Y=aX+X=(a+1)X$, so $f_Y(x)=f_X(x/a)/a$ and $f_{X+Y}(x)=f_X(x/(a+1))/(a+1)$
$f_Y(x)dx=Pr(x<Y<x+dx)=Pr(x/a<X<x/a+dx/a)=f_X(x/a)dx/a$