The pdf of $X+Y$

Question

$X,Y$ are independent. $X\sim U(0,1)$ and $$f_Y(y)=\cases{2y,\;0<y<1\\ 0,\;Else.}$$ What is the pdf of $X+Y$? (i.e. $f_{X+Y}$)

I know that $$f_X(x)=\cases{1,\;0<x<1\\ 0,\;Else.}$$ But how I continue from here?
Thank you!!

Answer 1

The pdf of $Z=X+Y$ is the convolution of $f_X$ and $f_Y$: $$f_Z(z) = \int_{-\infty}^{+\infty}f_X(x)\,f_Y(z-x)dx = \int_{0}^{1}f_Y(z-x)\,dx = \int_{z-1}^{z}f_Y(x)\,dx,$$ hence: $$f_Z(x)=\left\{\begin{array}{ll}x^2 &\mbox{when } x\in[0,1]\\2x-x^2 &\mbox{when } x\in[1,2]\\0&\mbox{otherwise}.\end{array}\right.$$