Probability theory and statistics

Chapter 5 : Random vectors

By Lund University

In the last chapter we considered the case of defining several random variables although most of it was done using only two random variables. If we want to look at more than two random variables, then it is much more useful to define a k×1 random vector containing k random variables. We will see that many expressions, such as a linear function of several random variables can be expressed more compactly using vector in matrix notation.

Random vectors and functions of random vectors

The first section is basically a generalization of much of chapter four using vector notation. It generalizes the various distribution functions and functions of random variables using vector notation.

Random vectors

Function of a random vector

Moments of random vectors

In this section we will begin by defining the expected value of a random vector. Basically, it will be a new vector containing the expected value of each random variable making up our rental vector. We then move onto the definition of the variance of a random vector which will turn out to be a matrix. We end this section by looking at the conditional expectations of a random vector.

The expected value of a random vector

Problem: Expected value of X’Y

The variance of a random vector

Problem: Variance of c’X

Conditional expectations and random vectors

Multivariate normal distribution

The normal distribution is the most for a single random variable. In this last section we look at the generalization for random vectors following a multivariate normal distribution.

Multivariate normal