Conditional Density Function

Topics: Probability

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(definition)

Let $(X,Y)$ be a random vector of dimension $2$.

We define the conditional density function of $X$ given $Y$ as:

$$f_{X\mid Y}(x\mid y)=\mathbb{P}(X=x\mid Y=y)=\frac{f_{(x,y)}(x,y)}{f_Y(y)}$$

Similarly, the conditional density function of $Y$ given $X$ is defined as:

$$f_{Y\mid X}(y\mid x)=\mathbb{P}(X=x\mid Y=y)=\frac{f_{(x,y)}(x,y)}{f_X(x)}$$