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