Fundamentals of Probability and Statistics for Engineers

6Some Important DiscreteDistributionsThis chapter deals with some distributions of discrete random variables that areimportant as models of scientific phenomena. The nature and applications ofthese distributions are discussed. An understanding of the situations in whichthese random variables arise enables us to choose an appropriate distributionfor a scientific phenomenon under consideration. Thus, this chapter is alsoconcerned with the induction step discussed in Chapter 1, by which a modelis chosen on the basis of factual understanding of the physical phenomenonunder study (step B to C in Figure 1.1).Some important distributions of continuous random variables will be studiedin Chapter 7.6.1 BERNOULLI TRIALSA large number of practical situations can be described by the repeated performanceof a random experiment of the following basic nature: a sequenceof trials is performed so that (a) for each trial, there are only two possibleoutcomes, say, success and failure; (b) the probabilities of the occurrence ofthese outcomes remain the same throughout the trials; and (c) the trials arecarried out independently. Trials performed under these conditions are calledBernoulli trials. Despite of the simplicity of the situation, mathematical modelsarising from this basic random experiment have wide applicability. In fact,we have encountered Bernoulli trials in the random walk problems describedin Examples 3.5 (page 52) and 4.17 (page 106) and also in the traffic problemexamined in Example 3.9 (page 64). More examples will be given in thesections to follow.Let us denote event ‘success’by S, and event ‘failure’by F. Also, let P(S) ˆ p,and P(F) ˆ q, where p ‡ q ˆ 1. Possible outcomes resulting from performingFundamentals of Probability and Statistics for Engineers T.T. Soong© 2004 John Wiley & Sons, LtdISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)TLFeBOOK