Introduction to Probability and Statistics for Engineers and Scientists

148 Chapter 5: Special Random Variables

Binomial Distribution

Enter Value For p:

Enter Value For n:

Enter Value For i:

.75

100

70

Probability (Number of Successes = i)

Probability (Number of Successes <= i)

.04575381

.14954105

Start

Quit

FIGURE 5.2

EXAMPLE 5.1f IfXis a binomial random variable with parametersn=100 andp=.75,
findP{X= 70 }andP{X≤ 70 }.

SOLUTION The text disk gives the answers shown in Figure 5.2. ■

5.2The Poisson Random Variable

A random variableX, taking on one of the values 0, 1, 2,..., is said to be a Poisson
random variable with parameterλ,λ>0, if its probability mass function is given by

P{X=i}=e−λ

λi

i!

, i=0, 1,... (5.2.1)

The symbolestands for a constant approximately equal to 2.7183. It is a famous constant in
mathematics, named after the Swiss mathematician L. Euler, and it is also the base of the
so-called natural logarithm.
Equation 5.2.1 defines a probability mass function, since

∑∞

i= 0

p(i)=e−λ

∑∞

i= 0

λi/i!=e−λeλ= 1

A graph of this mass function whenλ=4 is given in Figure 5.3.
The Poisson probability distribution was introduced by S. D. Poisson in a book he wrote
dealing with the application of probability theory to lawsuits, criminal trials, and the like.
This book, published in 1837, was entitledRecherches sur la probabilité des jugements en
matière criminelle et en matière civile.