Cfrgtkky

This is the problem i'm currently tackling from my textbook:

Let $P$ have a uniform distribution on $[0,1]$, and, conditional on $P=p$, let $X$ have a Bernoulli distribution with parameter $p$. Find the conditional distribution of $P$ given $X$.

However, while I would be grateful for the solution to this question as a means to check when I (hopefully) finish the question. I'm more interested in the general approach to questions like these. Something of the sort: Let $X$ have a [distribution], conditional on $X=x$, let $Y$ have a [different distribution]. Find the conditional distribution of $X$ given $Y$.

Maybe i'm being too vague, but is there a general/systematic approach to these sort of questions? Like, I know that we can have it so that:

$f_{X|Y}(x|y)=frac{f_{X,Y}(x,y)}{f_Y(y)}$

but then what would you do afterwards and how would you get a solution from that? Would you just sub in the appropriate density into $f_Y(y)$, e.g. if $Y$ had an exponential distribution, would $f_Y(y)=λe^{-λy}$? And what would you do with the joint distribution?

I hope what i'm saying makes sense...please let me know if there's anything I should clarify.

1. We have
   $$ Xsim f_X$$
   and
   $$ Y|X=x sim f_{Y|X}(cdot|x).$$
   We want
   $$ X|Y=ysim f_{X|Y}(cdot|y)=?$$




2. We know
   $$ f_{X|Y=y}(cdot|y)=frac{f_{X,Y}(cdot,y)}{f_Y(y)}.$$
   It is enough to determine the joint distribution.




3. Of (1) we have
   $$ f_{X,Y}(x,cdot)= f_{Y|X=x}(cdot)f_X(x).$$