Lecture 5. C) Multiple Random Variables
Multiple Random Variables
An n-dimensional vector [math]X=\left(X_{1}..X_{n}\right)'[/math] is a random vector if [math]X_{1}..X_{n}[/math] are random variables (defined on the same probability space).
We will mostly discuss bivariate distributions, but the results are mostly generalizable for the n-case.
Joint CDF
The joint cdf of random vector [math]\left(X,Y\right)[/math] is the function [math]F_{X,Y}:\mathbb{R}^{2}\rightarrow\left[0,1\right][/math], given by
[math]F_{X,Y\left(x,y\right)}=P\left(X\leq x,Y\leq y\right),\forall\left(x,y\right)'\in\mathbb{R}^{2}.[/math]
Joint PMF/PDF
- [math]\left(X,Y\right)'[/math] is discrete if [math]\exists f_{X,Y}:\mathbb{R}^{2}\rightarrow[0,1][/math] s.t. [math]F_{X,Y}\left(x,y\right)=\sum_{s\leq x}\sum_{t\leq y}f_{X,Y}\left(s,t\right),\,\forall\left(x,y\right)'\in\mathbb{R}^{2}.[/math]
- [math]\left(X,Y\right)'[/math] is continuous if [math]\exists f_{X,Y}:\mathbb{R}^{2}\rightarrow\mathbb{R}_{+}[/math] s.t. [math]F_{X,Y}\left(x,y\right)=\int_{-\infty}^{x}\int_{-\infty}^{y}f_{X,Y}\left(s,t\right)dtds,\,\forall\left(x,y\right)'\in\mathbb{R}^{2}.[/math]
The remaining properties we discussed in the univariate case extend: Functions with applicable domains and codomains that ‘sum up to one’ are pmfs/pdfs. In addition, expectations extend intuitively.
Let [math]g\left(x,y\right):\mathbb{R}^{2}\rightarrow\mathbb{R}[/math]. Its expected value is equal to
[math]E\left(g\left(x,y\right)\right)=\begin{cases} \sum_{s,t\in\mathbb{R^{2}}}g\left(s,t\right)f_{X,Y}\left(s,t\right), & \text{if}\,\left(X,Y\right)'\,\text{is discrete }\\ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g\left(s,t\right)f_{X,Y}\left(s,t\right)dtds & \text{if}\,\left(X,Y\right)'\,\text{is continuous} \end{cases}[/math]
