Marginal Density Function

Topics: Probability - Density Function of a Random Vector

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(definition)

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

The marginal density function of $(X,Y)$ with respect to $X$ is defined as:

$$f_X(x)=\sum _{y\in R_Y}{f}_{(X,Y)}(x,y)$$

Similarly, the marginal density function of $(X,Y)$ with respect to $Y$ is defined as:

$$f_Y(y)=\sum _{x\in R_X}{f}_{(X,Y)}(x,y)$$

Properties §

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

2. ${\sum }_{x,y\in R_{(X,Y)}}{f}_{(X,Y)}(x,y)=1$