# CHAPTER 2

# PRELIMINARIES

## Outline

**2.5** Multivariate Normal Distribution

**2.6** Multivariate t-Distribution

In this chapter, we discuss some basic results on various distributions, particularly the normal, chi-square, Student’s t-, multivariate t-distributions.

# 2.1 Normal Distribution

The most basic distribution in statistical theory is the normal distribution, (*θ*, σ^{2}), with pdf

(2.1.1)

where *θ* is the mean and σ^{2} is the variance of this distribution.

It is well known that

**(i)**If

*Y*is

*N*(

*θ*, σ

^{2}), then .

**(ii)**If

*Y*is (

*θ*, σ

^{2}) and φ(

*Y*) is a differentiable function satisfying |φ′(

*Y*)| < ∞, then

(2.1.2)

**(iii)**If

*Z*is

*N*(θ, 1), then

(2.1.3)

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