Marginal Distributions for Continuous Random Variables

For discrete random variables, the marginal PMF of $X$ is obtained by summing the joint PMF over all values of $Y$. For continuous random variables, the analogous operation is integration.

If $X$ and $Y$ are continuous random variables with joint PDF $f_{X,Y}(x,y)$, then the marginal probability density functions of $X$ and $Y$ are

$$f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y)\,dy, \qquad f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x,y)\,dx.$$

In practice, the joint PDF is nonzero only on some region, so we integrate only over the values of the other variable for which $f_{X,Y}$ is nonzero.

For example, suppose

$$f_{X,Y}(x,y) = 4xy, \quad 0\le x\le 1,\ 0\le y\le 1.$$

For each fixed $x \in [0,1]$, $y$ ranges over $[0,1]$, so

$$f_X(x) = \int_0^1 4xy\,dy = 4x\cdot\dfrac{y^2}{2}\bigg|_0^1 = 2x, \quad 0\le x\le 1.$$