Page 235 - Probability and Statistical Inference

212 4. Functions of Random Variables and Sampling Distribution

–2
du
z) , 0 < u < ∞, 0 < z < 1. One may also check that (ν /ν ) - = (1 – z) . Thus,
–1
dz
2
1
combining (4.4.1) and (4.5.13) in a straightforward fashion, we can write
down the pdf of Z as follows: For 0 < z < 1, we have
which simplifies to

It coincides with the beta pdf defined in (1.7.35) where α = 1/2ν and β = 1/
1
2ν . That is,
2
[(ν /ν )F , ν ]/[1 + (ν /ν ) F , ] has Beta (1/2ν , 1/2ν ) distribution.
1
2
ν1 ν2
2
1
2
ν1
2
1
4.6 Special Continuous Multivariate Distributions
We now include some interesting aspects of the multivariate normal, t, and F
distributions. It will become clear shortly that both the multivariate t and F
distributions are close associates of the multivariate normal distribution. One
may review the Section 4.8 for some of the details about matrices.
Tong’s (1990) book is devoted to the multivariate normal distributions and
includes valuable tables. It briefly discusses the multivariate t and F distribu-
tions too. The references to the tables and other features for the multivariate
normal, t and F distributions can be found in Johnson and Kotz (1972).
We included important properties of a bivariate normal distribution in the
Section 3.6. The sampling distributions in the context of a bivariate normal
population, however, is included in the present section.

4.6.1 The Normal Distribution
The bivariate normal density was given in (3.6.1). The general multivariate
normal density is more involved. But, without explicitly referring to the pdf,
one can derive many interesting and useful properties. The following broad
definition of the p-dimensional normality can be found in Rao (1973, p. 518.)
The advantage of adopting this definition over another relying explicitly on the
multivariate pdf will be clear from the Examples 4.6.1-4.6.2.
Definition 4.6.1 A p(≥ 1) random vector X = (X , ..., X ) is said to
p
1
have a p-dimensional normal distribution, denoted by N , if and only if
p