Conditional Distribution

Definition

The conditional distribution of a random variable $Y$ given $X = x$ describes the distribution of $Y$ when $X$ is known to equal $x$.

  • Discrete case: The conditional PMF is
$$ p_{Y|X}(y \mid x) = \frac{p_{X,Y}(x, y)}{p_X(x)}, \quad p_X(x) > 0. $$
  • Continuous case: The conditional PDF is
$$ f_{Y|X}(y \mid x) = \frac{f_{X,Y}(x, y)}{f_X(x)}, \quad f_X(x) > 0. $$

Relation to Bayes’ Rule

$$ f_{Y|X}(y \mid x) = \frac{f_{X|Y}(x \mid y) f_Y(y)}{f_X(x)}. $$