Introduction to Probability and Statistics for Engineers and Scientists

Chapter 5 Special Random Variables

Certain types of random variables occur over and over again in applications. In this chapter,
we will study a variety of them.

5.1The Bernoulli and Binomial Random Variables

Suppose that a trial, or an experiment, whose outcome can be classified as either a “success”
or as a “failure” is performed. If we letX=1 when the outcome is a success andX= 0
when it is a failure, then the probability mass function ofXis given by

P{X= 0 }= 1 −p (5.1.1)

P{X= 1 }=p

wherep,0≤p≤1, is the probability that the trial is a “success.”
A random variableXis said to be a Bernoulli random variable (after the Swiss mathe-
matician James Bernoulli) if its probability mass function is given by Equations 5.1.1 for
somep∈(0, 1). Its expected value is

E[X]= 1 ·P{X= 1 }+ 0 ·P{X= 0 }=p

That is, the expectation of a Bernoulli random variable is the probability that the random
variable equals 1.
Suppose now thatnindependent trials, each of which results in a “success” with prob-
abilitypand in a “failure” with probability 1−p, are to be performed. IfXrepresents
the number of successes that occur in thentrials, thenXis said to be abinomialrandom
variable with parameters (n,p).

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