Applied Statistics and Probability for Engineers

70 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Suppose Xis a discrete uniform random variable on the consecutive integers

for The mean of Xis

The variance of Xis

2  (3-6)

1 ba   122  1

12

E 1 X 2 

b a

2

a, a 1, a 2,p, b, ab.

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 Xthat assumes each of the values

x 1 , x 2 ,p, xn,with equal probability (^1) n,is frequently of interest.
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 Xis the first digit of the serial number, Xhas a dis-
crete uniform distribution with probability 0.1 for each value in. That is,
for each value in R. The probability mass function of Xis shown in Fig. 3-7.
Suppose the range of the discrete random variable Xis the consecutive integers a,
for The range of Xcontains b a 1 values each with proba-
bility. Now,
The algebraic identity can be used to simplify the result to
 1 b a (^2)  (^2) .The derivation of the variance is left as an exercise.
a
b
ka
k
b 1 b 12  1 a 12 a
2
a
b
ka
k^ a
1
ba 1
b
(^1)  1 ba 12
a 1, a 2,p, b, ab.
f 1 x 2 0.1
R 5 0, 1, 2,p, 96
A random variable Xhas a discrete uniform distributionif each of the nvalues in
its range, say, has equal probability. Then,
f 1 xi 2  (^1) n (3-5)
x 1 , x 2 ,p, xn,
Definition
f(x)
0 1 2 3 4 5 6 7 89 x
0.1
Figure 3-7 Probability
mass function for a
discrete uniform ran-
dom variable.
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