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PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 70

70 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

3-5 DISCRETE UNIFORM DISTRIBUTION

The simplest discrete random variable is one that assumes only a finite number of possible
values, each with equal probability. A random variable X that assumes each of the values
x 1 , x 2 , p , x n , with equal probability 1 n, is frequently of interest.

Definition
A random variable X has a discrete uniform distribution if each of the n values in
its range, say, x , x , p , x , has equal probability. Then,
1
2
n
f 1x 2 1 n (3-5)
i

EXAMPLE 3-13 The first digit of a part’s serial number is equally likely to be any one of the digits 0 through 9.
If one part is selected from a large batch and X is the first digit of the serial number, X has a dis-
crete uniform distribution with probability 0.1 for each value in R 50, 1, 2, p , 96 . That is,
f 1x2 0.1

for each value in R. The probability mass function of X is shown in Fig. 3-7.
Suppose the range of the discrete random variable X is the consecutive integers a,
a 1, a 2, p , b, for a b. The range of X contains b a 1 values each with proba-
bility 1 1b a 12 . Now,

b 1
a k a b
k a b a 1

b b1b 12 1a 12a
The algebraic identity a k can be used to simplify the result to
k a 2
1b a2 2 . The derivation of the variance is left as an exercise.

Suppose X is a discrete uniform random variable on the consecutive integers
a, a 1, a 2, p , b, for a b. The mean of X is

b a
E1X2
2
The variance of X is
2
1b a 12 1
2
(3-6)
12

Figure 3-7 Probability f(x)
mass function for a 0.1
discrete uniform ran-
dom variable. 0 1 2 3 4 5 6 7 8 9 x