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# Multiple Random Variables (cont.)

## Marginal CDF, PMF/PDF

If $\left(X,Y\right)'$ is a bivariate random vector, then the cdf of $X$ (and of $Y$) is called the marginal cdf of $X$ $\left(Y\right)$.

For example, the marginal cdf of $X$ can be obtained via:

$F_{X}\left(x\right)=\lim_{y\rightarrow\infty}F_{X,Y}\left(x,y\right),\,\forall x\in\mathbb{R}$.

Notice that knowledge of $F_{X,Y}\left(x,y\right)$ implies knowledge of the marginal distributions. The converse is only true if $X$ and $Y$ are independent.

We can also obtain the marginal pmf/pdf in the following way.

• If $\left(X,Y\right)'$ is discrete, then

$f_{X}\left(x\right)=\sum_{y\in\mathbb{R}}f_{X,Y}\left(x,y\right),\,x\in\mathbb{R}$.

• If $\left(X,Y\right)'$ is continuous, then

$f_{X}\left(x\right)=\int_{-\infty}^{\infty}f_{X,Y}\left(x,y\right)dy,\,x\in\mathbb{R}$.

## Independence

Two random variables $X$ and $Y$ are independent if

$F_{X,Y}\left(x,y\right)=F_{X}\left(x\right)F_{Y}\left(y\right),\forall\left(x,y\right)'\in\mathbb{R}^{2}.$

Equivalently, two random variables $X$ and $Y$ are independent if

$f_{X,Y}\left(x,y\right)=f_{X}\left(x\right).f_{Y}\left(y\right),\forall\left(x,y\right)'\in\mathbb{R}^{2}.$