Suppose we have $X$ and $Y$ are jointly distributed with density of $2$ if $0 < x < y < 1$ (and $0$ otherwise). I'd like to find the PDF of the sum, $X + Y$.
I read the Jacobian method here: pdf of sum of two dependent random variables
and set $Z = X + Y, W = X$ to apply that method.
However, I am having trouble writing the bounds for the joint density function. I know that $0 < x < y < 1$ and thus $2x < z < 1 + x$, and since $w = x$ we know that $2w < z < 1 + w$ and $0 < w < z$.
Finally when I integrate the density function $2$ from $0$ to $z$ to find PDF of $Z$ the solution is simply…$2z$ for (what are the bounds?) How can I finish up the problem?
Best Answer
The joint density is
$$f_{WZ}(w,z)=2$$
because tha jacobian is 1. Thus it is still uniform. The only difficulty is to derive the joint support of $(W;Z)$
considering that you have
$$0<w<z-w<1$$
that is
$$0<2w<z<1+w$$
the following region is the joint support $(W;Z)$
in order to get $f_Z(z)$ you have to integrate wrt $w$ obtaining
When $0<z<1$
$$f_Z(z)=\int_0^{z/2}2dw=z$$
and when $1<z<2$
$$f_Z(z)=\int_{z-1}^{z/2}2dw=2-z$$