Marginal Distribution

The marginal distribution isolates the probability distribution of one variable while ignoring the value of the other variable.

Discrete Case (Marginal PMF)

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

Continuous Case (Marginal PDF)

• $f_X(x)={\int }_{-\infty }^{\infty }f(x,y)dy$
• $f_Y(y)={\int }_{-\infty }^{\infty }f(x,y)dx$

Marginal CDF

• $F_X(x)=P(X\le x)={\lim }_{y\to \infty }F(x,y)$
• $F_Y(y)=P(Y\le y)={\lim }_{x\to \infty }F(x,y)$