Find the joint pdf of X and Y for two uniform distributions

Question

Let X have a uniform distribution on the interval $(0,1)$. Given that X = x, let Y have a uniform distribution on the interval $(0,x+1)$.

Find the joint pdf of X and Y. Sketch the region where $f(x,y) > 0$.

Find f_Y$(y)$, the marginal pdf of Y. Be sure to include the domain.

I'm not really sure where to start. Is f_Y$(y)$ just $\frac{1}{(x+1)-0}$ since that's the pdf of a uniform distribution? And I have no idea how to find the joint pdf of X and Y.

Answer 1

The conditional density of $Y$ given $X=x$ is $f_{Y|X}(y|x)=\frac{1}{x+1}1_{0<y<x+1}$, hence the joint density of $X$ and $Y$ is $$ f(x,y)=f_{Y|X}(y|x)f_X(x)=\frac{1}{x+1}1_{0<x<1,0<y<x+1}$$

The marginal density can then be obtained by "integrating out" the $x$-variable: $$ f_Y(y)=\int f(x,y)\;dx$$