Fundamentals of Probability and Statistics for Engineers

Definition 3.2.The function

is defined as the (pmf) of. Again, the subscript is
used to identify the associated random variable.

For the random variable defined in Example 3.1, the pmf is zero everywhere
except at 1, 2,... , and has the appearance shown in Figure 3.3.
This is a typical shape of pmf for a discrete random variable. Since
0 for any for continuous random variables, it does not exist in
the case of the continuous random variable. We also observe that, like
the specification of completely characterizes random variable ; further-
more, these two functions are simply related by:

(assuming ..).
The upper limit for the sum in Equation (3.7) means that the sum is taken
over all satisfying Hence, we see that the PDF and pmf of a discrete
random variable contain the same information; each one is recoverable from
the other.

–2 –1 0123 4

1

1

2

8

pX(x)

x

Figure 3.3 Probability mass function of for the random variable defined

in Example 3.1

42 Fundamentals of Probability and Statistics for Engineers

X,pX 9 x),

pX…x†ˆP…Xˆx†: … 3 : 5 †

probability mass function X X

xi,iˆ

P 9 Xˆx)ˆ x
FX 9 x),
pX 9 x) X

pX…xi†ˆFX…xi†FX…xi 1 †; … 3 : 6 †

FX…x†ˆ

iX:xix

iˆ 1

pX…xi†; … 3 : 7 †

x 1 <x 2 <.

i xix.