Conditional Distribution

Given two jointly distributed random variables, the conditional PMF/PDF of $X$ given $Y=y$ (where $f_Y(y)>0$) is defined as:

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

Similarly, the conditional distribution of $Y$ given $X=x$ is:

$f_{Y|X}(y|x)=\frac{f(x,y)}{f_X(x)}$

Finding Marginal from Conditional:

1. Find joint distribution: $f(x,y)=f_X(x)f(y|x)$
2. Integrate/Sum out $x$: $f_Y(y)={\int }_{\mathbb{R}}f(x,y)dx$