Marginal Distributions

Marginal Mass Function

Definition

If $(X,Y)$ have joint distribution with mass function $f_{X,Y}$, then the marginal mass function for $X$ is defined by

$$f_X(x)=\mathbf{P}(X=x)=\sum _y\mathbf{P}(X=x,Y=y)=\sum _{y}f(x,y),$$

and the marginal mass function for $Y$ is defined by

$$f_Y(y)=\mathbf{P}(Y=y)=\sum _x\mathbf{P}(X=x,Y=y)=\sum _{x}f(x,y).$$

Marginal Density

Definition

For continuous random variables, the marginal densities are

$$f_X(x)=\int f(x,y)dy\text{and}{f}_Y(y)=\int f(x,y)dx.$$

The corresponding marginal distribution functions are denoted $F_X$ and $F_Y$.