Conditional Distributions for Continuous Random Variables

Recall that for discrete random variables $X$ and $Y$, the conditional pmf of $Y$ given $X = x$ is

$$p_{Y|X}(y \mid x) = \dfrac{p_{X,Y}(x,y)}{p_X(x)}.$$

For continuous random variables, the same idea applies, but with densities instead of probability masses.

The conditional probability density function of $Y$ given $X = x$ is

$$f_{Y|X}(y \mid x) = \dfrac{f_{X,Y}(x,y)}{f_X(x)},$$

defined whenever $f_X(x) > 0$. For each fixed value of $x$, this is a valid pdf in $y$: it is nonnegative and integrates to $1$ over $y$.

For example, suppose $X$ and $Y$ have joint pdf

$$f_{X,Y}(x,y) = 2, \quad 0 \le y \le x \le 1.$$

The marginal density of $X$ is

$$f_X(x) = \int_0^x 2\, dy = 2x, \quad 0 < x \le 1.$$

Therefore the conditional density of $Y$ given $X = x$ is

$$f_{Y|X}(y \mid x) = \dfrac{2}{2x} = \dfrac{1}{x}, \quad 0 \le y \le x.$$

Given $X = x$, the random variable $Y$ is uniformly distributed on $[0, x]$.