Page 178 - Fundamentals of Probability and Statistics for Engineers

6

Some Important Discrete

Distributions

This chapter deals with some distributions of discrete random variables that are
important as models of scientific phenomena. The nature and applications of
these distributions are discussed. An understanding of the situations in which
these random variables arise enables us to choose an appropriate distribution
for a scientific phenomenon under consideration. Thus, this chapter is also
concerned with the induction step discussed in Chapter 1, by which a model
is chosen on the basis of factual understanding of the physical phenomenon
under study (step B to C in Figure 1.1).
Some important distributions of continuous random variables will be studied
in Chapter 7.

6.1 BERNOULLI TRIALS

A large number of practical situations can be described by the repeated per-
formance of a random experiment of the following basic nature: a sequence
of trials is performed so that (a) for each trial, there are only two possible
outcomes, say, success and failure; (b) the probabilities of the occurrence of
these outcomes remain the same throughout the trials; and (c) the trials are
carried out independently. Trials performed under these conditions are called
Bernoulli trials. Despite of the simplicity of the situation, mathematical models
arising from this basic random experiment have wide applicability. In fact,
we have encountered Bernoulli trials in the random walk problems described
in Examples 3.5 (page 52) and 4.17 (page 106) and also in the traffic problem
examined in Example 3.9 (page 64). More examples will be given in the
sections to follow.
ˆ
Let us denote event ‘success’ by S, and event ‘failure’ by F. Also, let P(S) p,
and P(F) ˆ q, where p ‡ q ˆ 1. Possible outcomes resulting from performing
Fundamentals of Probability and Statistics for Engineers T.T. Soong  2004 John Wiley & Sons, Ltd
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)

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