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78 Chapter 3/Discrete Random Variables and Probability Distributions

3-5 Discrete Uniform Distribution

The simplest discrete random variable is one that assumes only a inite number of possible

values, each with equal probability. A random variable X that assumes each of the values
x , x , … , x n with equal probability 1/ n is frequently of interest.
2
1
Discrete Uniform
Distribution A random variable X has a discrete uniform distribution if each of the n values in
its range, x , x ,… , x , has equal probability. Then
1
2
n
f x i ( ) = 1/ n (3-5)

Example 3-13 Serial Number The irst 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 irst digit of the serial number, X

has a discrete uniform distribution with probability 0.1 for each value in R = {0 , , ,… , } 9 . That is,
2
1
.
f x ( ) = 0 1
for each value in R. The probability mass function of X is shown in Fig. 3-7.
f(x)
0.1

0 1 2 3 4 5 6 7 8 9 x
FIGURE 3-7 Probability mass function for a discrete uniform random variable.

Suppose that the range of the discrete random variable X equals the consecutive integers
a, a +1, a + 2,..., b, for a ≤ b. The range of X contains b a− +1 values each with probability
(
1/ b a + 1). Now
−
b ⎛ 1 ⎞
μ = ∑ k ⎜ ⎟
− + ⎠ 1
k =a ⎝ b a
(
b b b + ) −( a − ) 1 a
1
The algebraic identity ∑ k = 2 can be used to simplify the result
=

k a
+ )
to μ = (b a / 2. The derivation of the variance is left as an exercise.
Mean and Variance
Suppose that X is a discrete uniform random variable on the consecutive integers
+
+
a,a 1 ,a 2 ,… , b, for a ≤ b. The mean of X is
μ = ( ) = b + a
X
E
2
The variance of X is
2
(b − + ) −1a 1
2
σ = (3-6)
12
Example 3-14 Number of Voice Lines As in Example 3-1, let the random variable X denote the number of 48
voice lines that are used at a particular time. Assume that X is a discrete uniform random variable
with a range of 0 to 48. Then,
+ ) 2
E X ( ) = (48 0 / = 24
and