Question:
X follows uniform [0, 1] and Y|X follows uniform[0,X]. What is the distribution of X|Y?
My Try: $$f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)} = \frac{1}{x}, \, with\, f_X(x)=1$$ $$f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}$$
I am not sure how to get $f_Y(y)$, any hints or help are appreciated!
Suppose $X\sim \mathcal{U}(0,1)$ and $Y|X\sim \mathcal{U}(0,X)$.
First notice $f_{X,Y}(x,y)=f_{Y|X}(y|x)f_{X}(x)=\frac{1}{x}\cdot 1_{\{0 < y < x,0< x < 1\}}$ so for any $x\in (y,1)$ we have $$f_{X|Y}(x|y)=\frac{f_{X,Y}(x,y)}{f_{Y}(y)}=\frac{f_{X,Y}(x,y)}{\int_{y}^1f_{X,Y}(x,y)\mathrm{d}x}=-\frac{1}{x \ln (y)}$$