Page 243 - Probability and Statistical Inference

220 4. Functions of Random Variables and Sampling Distribution

independent random variables X , X , ..., X where X is distributed as
1
0
i
p
, i = 0, 1, ..., p. This is often the situation in the case of analysis of
variance, where these X’s customarily stand for the sums of squares due to
the error and treatments. We denote the new random variables

The marginal distribution of the random variable U is F , . This follows
i
νi ν0
from the definition of a F random variable. Jointly, however, (U , ..., U ) is
p
1
said to have the p-dimensional F distribution, denoted by MF (ν , ν , ..., ν ).
p
0
p
1
One may refer to Johnson and Kotz (1972, p. 240). The joint pdf of U , ...,
1
U , is distributed as MF (ν , ν , ..., ν ), is given by
p p 0 1 p
with u’ = (u , ..., u ) and . The derivation of this joint pdf is left
1
p
as the Exercise 4.6.13.
Fundamental works in this area include references to Finney (1941) and
Kimbal (1951), among others. One may also refer to Johnson and Kotz (1972)
and Tong (1990) to find other sources.

4.7 Importance of Independence in Sampling
Distributions

In the reproductive property of the normal distributions and Chi-square distri-
butions (Theorem 4.3.2, parts (i) and (iii)) or in the definitions of the Student’s
t and F distributions, we assumed independence among various random vari-
ables. Now, we address the following important question. If those indepen-
dence assumptions are violated, would one have similar standard distributions
in the end? In general, the answer is “no” in many circumstances. In this
section, some of the ramifications are discussed in a very simple way.

4.7.1 Reproductivity of Normal Distributions

In the Theorem 4.3.2, part (i), we learned that independent normal variables
add up to another normal random variable. According to the Definition
4.6.1, as long as (X , ..., X ) is distributed as the n-dimensional normal N ,
n
n
1
this reproductive property continues to hold. But, as soon as we stray