Random vectors

Summary

Definition

  • \(X_1,…,X_n\) are \(n\) random variables. We define

\[X=\left( \right)={\left( X_1…X_n \right)}'\]

  • as an \(n×1\) random vector.

Distributions

  • If \(X\) is a continuous random vector , then the probability density function of \(X\) is written as

\[f\left( x \right)=f(x_1,…,x_n)\]

  • If, in addition, \(Y\) is an \(m×1\) continuous random vector defined on the same probability space then

\[f\left( x,y \right)=f(x_1,…,x_n,y_1,…,y_m)\]

  • denotes the joint probability density function of \(X\) and \(Y\) .
  • The marginal densities are given by

\[f_X\left( x \right)=\int_{y}{ f\left( x,y \right)dy }\]

\[f_Y\left( y \right)=\int_{x}{ f\left( x,y \right)dx }\]

  • We say that the random vectors \(X,Y\) are independent if , for all \(x,y\)

\[f\left( x,y \right)=f_X\left( x \right)f_Y\left( y \right)\]

  • If \(X,Y\) are independent random vectors then \(X_i\) is independent of \(Y_j\) for each \(i=1,…,n\) and \(j=1,…,m\) .
  • The conditional densities are defined by

\[f_{X|Y}\left( y \right)= \frac{f\left( x,y \right)}{f_Y\left( y \right)}\]

  • and

\[f_{Y|X}\left( x \right)= \frac{f\left( x,y \right)}{f_X\left( x \right)}\]