Page 15 - Elements of Distribution Theory

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Properties of Probability Distributions

1.1 Introduction
Distribution theory is concerned with probability distributions of random variables, with
the emphasis on the types of random variables frequently used in the theory and application
of statistical methods. For instance, in a statistical estimation problem we may need to
determine the probability distribution of a proposed estimator or to calculate probabilities
in order to construct a confidence interval.
Clearly, there is a close relationship between distribution theory and probability theory; in
some sense, distribution theory consists of those aspects of probability theory that are often
used in the development of statistical theory and methodology. In particular, the problem
of deriving properties of probability distributions of statistics, such as the sample mean
or sample standard deviation, based on assumptions on the distributions of the underlying
random variables, receives much emphasis in distribution theory.
In this chapter, we consider the basic properties of probability distributions. Although
these concepts most likely are familiar to anyone who has studied elementary probability
theory, they play such a central role in the subsequent chapters that they are presented here
for completeness.

1.2 Basic Framework

The starting point for probability theory and, hence, distribution theory is the concept of
an experiment. The term experiment may actually refer to a physical experiment in the
usual sense, but more generally we will refer to something as an experiment when it has
the following properties: there is a well-defined set of possible outcomes of the experiment,
each time the experiment is performed exactly one of the possible outcomes occurs, and
the outcome that occurs is governed by some chance mechanism.
Let denote the sample space of the experiment, the set of possible outcomes of the
experiment; a subset A of is called an event. Associated with each event A is a probability
P(A). Hence, P is a function defined on subsets of and taking values in the interval [0, 1].
The function P is required to have certain properties:

(P1) P( ) = 1
(P2) If A and B are disjoint subsets of , then P(A ∪ B) = P(A) + P(B).

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