Lecture 6. A) Multiple Random Variables (cont.)

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Multiple Random Variables (cont.)

Marginal CDF, PMF/PDF

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

For example, the marginal cdf of [math]X[/math] can be obtained via:

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

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

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

  • If [math]\left(X,Y\right)'[/math] is discrete, then

[math]f_{X}\left(x\right)=\sum_{y\in\mathbb{R}}f_{X,Y}\left(x,y\right),\,x\in\mathbb{R}[/math].

  • If [math]\left(X,Y\right)'[/math] is continuous, then

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

Independence

Two random variables [math]X[/math] and [math]Y[/math] are independent if

[math]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}.[/math]

Equivalently, two random variables [math]X[/math] and [math]Y[/math] are independent if

[math]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}.[/math]