Fundamentals of Probability and Statistics for Engineers

One can also give PDF and pmf a useful physical interpretation. In terms of
the distribution of one unit of mass over the real line the PDF of
a random variable at can be interpreted as the total mass associated
with point and all points lying to the left of. The pmf, in contrast, shows
the distribution of this unit of mass over the real line; it is distributed at discrete
points with the amount of mass equal to 1,2,....

Example 3.2.A discrete distribution arising in a large number of physical
models is the M uch more will be said of this important
distribution in Chapter 6 but, at present, let us use it as an illustration for
graphing the PDF and pmf of a discrete random variable.
A discrete random variable has a binomial distribution when

where and are two parameters of the distribution, being a positive integer,
and 0 1. The binomial coefficient

is defined by

The pmf and PDF of for 10 and 0 2 are plotted in Figure 3.4.

02 46810

0.1

0.2

0.3

0.4

pX(x)

x

0246810

1.0

0.8

0.6

0.4

0.2

FX(x)

x

(a) (b)

Figure 3.4 (a) Probability mass function, ( ), and (b) probability distribution

function, ( ), for the discrete random variable described in Example 3.2

Random Variables and Probability D istributions 43

1<x< 1 ,

x,FX 9 x),

x x

xi pX 9 xi)atxi,iˆ

binomial distribution.

X

pX…k†ˆ

n

k



pk… 1 p†nk; kˆ 0 ; 1 ; 2 ;...;n; … 3 : 8 †

n p n

<p<

n

k



n

k



ˆ

n!

k!…nk†!

: … 3 : 9 †

X nˆ pˆ :

pXx

FXx X