Conditional Distributions

\begin{aligned} f_{X | Y} (x | y) = P (A | B) & = \frac{P (A \cap B)}{P (B)} \\ = \frac{P (X = x, Y = y)}{P (Y = y)} \\ f_{X | Y} (x | y) & = \frac{f_{X, Y} (x, y)}{f_{Y} (y)} \end{aligned}

And:

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

f (x, y) = \frac{\partial F (x)}{\partial x} \frac{\partial F (y)}{\partial y}

Independency

If $X$ and $Y$ are independent:

\begin{array}{r} f_{X, Y} = \end{array}

The conditional cumulative distributions of $X$ given $y$

\begin{aligned} F_{X | Y} (x | y) & = P (X \leq x | Y = y) \\ = {\begin{cases} \sum_{t \leq x} f_{X | Y} (t | y) discrete case \\ \int_{- \infty}^{x} f_{X | Y} (t | y) d t continuous case \end{cases} \end{aligned}