CHAPTER 10
Statistical Distributions
In this chapter, we shall present some probability distributions that play a central
role in econometric theory.
First, we shall present the distributions of some
discrete random variables that have either a finite set of values or that take
values that can be indexed by the entire set of positive integers. We shall also
present the multivariate generalisations of one of these distributions.
Next, we shall present the distributions of some continuous random variables
that take values in intervals of the real line or over the entirety of the real line.
Amongst these is the normal distribution, which is of prime importance and for
which we shall consider, in detail, the multivariate extensions.
Associated with the multivariate normal distribution are the socalled sam
pling distributions that are important in the theory of statistical inference. We
shall consider these distributions in the final section of the chapter, where it will
transpire that they are special cases of the univariate distributions described in
the preceding section.
Discrete Distributions
Suppose that there is a population of
N
elements,
Np
of which belong to
class
A
and
N
(1
−
p
) to class
A
c
. When we select
n
elements at random from the
population in
n
successive trials, we wish to know the probability of the event
that
x
of them will be in
A
and that
n
−
x
of them will be in
A
c
.
The probability will be a
ff
ected by the way in which the
n
elements are
selected; and there are two ways of doing this. Either they can be put aside after
they have been sampled, or else they can be restored to the population. Therefore
we talk of sampling without replacement and of sampling with replacement.
If we sample with replacement, then the probabilities of selecting an element
from either class will the same in every trial, and the size
N
of the population will
have no relevance. In that case, the probabilities are governed by the binomial
law. If we sample without replacement, then, in each trial, the probabilities of
selecting elements from either class will depend on the outcomes of the previous
trials and upon the size of the population; and the probabilities of the outcomes
from
n
successive trials will be governed by the hypergeometric law.
Binomial Distribution
When there is sampling with replacement, the probability is
p
that an ele
ment selected at random will be in class
A
, and the probability is 1
−
p
that it will
be in class
A
c
. Moreover, the outcomes of successive trials will be statistically
independent. Therefore, if a particular sequence has
x
elements in
A
in
n
−
x
elements
A
c
, then, as a statistical outcome, its probability will be
p
x
(1
−
p
)
n
−
x
.
1
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D.S.G. POLLOCK: ECONOMETRICS
There are altogether
nC
x
=
n
!
/
{
(
n
−
x
)!
x
!
}
such sequences, with
x
elements
in
A
in
n
−
x
elements in
A
c
.
These sequences represent a set of mutually
exclusive ways in which the event in question can occur; and their probabilities
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 Spring '12
 D.S.G.Pollock
 Normal Distribution, Probability theory, density function, RHS

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