[[Conditional probability]]
# Conditional distribution

When one applies [[Conditional probability]] to multiple [[Real random variable|random variables]] one obtains a so-called **conditional distribution**. #m/def/prob 
For a [[Discrete random variable]] we define
$$
\begin{align*}
p_{X \mid Y}(x \mid y) =  \mathbb{P}(\{ X = x \} \mid \{ Y = y \}) = \frac{\mathbb{P}(\{ X = x \} \cap \{ Y=y \})}{\mathbb{P}(\{ Y=y \})}
\end{align*}
$$
and for a [[Continuous random variable]]
$$
\begin{align*}
f_{Y \mid X}( y \mid x) = \frac{f_{X,Y}(x,y)}{f_{X}(x)}
\end{align*}
$$


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