# CHAPTER 2

# PRELIMINARIES

## Outline

**2.1** Normal Distribution

**2.2** Chi-Square Distribution

**2.3** Student’s t-Distribution

**2.4** F-Distribution

**2.5** Multivariate Normal Distribution

**2.6** Multivariate t-Distribution

**2.7** Problems

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)

For more information see Stein (1981).

**(iii)** If *Z* is *N*(θ, 1), then

(2.1.3)

where Φ(.) is the cumulative distribution function (cdf) of the standard normal (zero mean and unit variance) distribution. ...