Page 96 - Applied statistics and probability for engineers

74 Chapter 3/Discrete Random Variables and Probability Distributions

3-4 Mean and Variance of a Discrete Random Variable

Two numbers are often used to summarize a probability distribution for a random variable X. The
mean is a measure of the center or middle of the probability distribution, and the variance is a
measure of the dispersion, or variability in the distribution. These two measures do not uniquely
identify a probability distribution. That is, two different distributions can have the same mean and
variance. Still, these measures are simple, useful summaries of the probability distribution of X.

Mean, Variance, and (
Standard Deviation The mean or expected value of the discrete random variable X, denoted as μ or E X , ) is
( )
E X
μ = ( ) = ∑ xf x (3-3)
x
(
2
The variance of X, denoted as σ or V X , ) is
(
)
(
(
(
(
2
2
2
σ =V X) = E X −μ) 2 = ∑ x −μ f x) = ∑ x f x) − μ 2
x x
2
The standard deviation of X is σ = σ .
The mean of a discrete random variable X is a weighted average of the possible values of X
with weights equal to the probabilities. If f x ( ) is the probability mass function of a loading on a
long, thin beam, E X ( ) is the point at which the beam balances. Consequently, E X ( ) describes the
“center” of the distribution of X in a manner similar to the balance point of a loading. See Fig. 3-5.
The variance of a random variable X is a measure of dispersion or scatter in the pos-
sible values for X. The variance of X uses weight f x ( ) as the multiplier of each possi-
(
2
ble squared deviation x − μ) . Figure 3-5 illustrates probability distributions with equal
means but different variances. Properties of summations and the deinition of μ can be
used to show the equality of the formulas for variance.
V X ( ) = ( x − μ) ( x f x ( ) − μ∑ xf x ( ) + μ ∑ f x ( )
f x) = ∑
∑
2
2
2
2
x x x x
= ∑ x f x ( ) − μ + μ = ∑ x f ( ) − μx 2
+
2
2
2
2
2
x x
Either formula for V x ( ) can be used. Figure 3-6 illustrates that two probability distributions
can differ even though they have identical means and variances.
0 2 4 6 8 10 0 2 4 6 8 10
(a) (b)
FIGURE 3-5 A probability distribution can be viewed as a loading with the mean equal to the
balance point. Parts (a) and (b) illustrate equal means, but part (a) illustrates a larger variance.

0 2 4 6 8 10 0 2 4 6 8 10
(a) (b)
FIGURE 3-6 The probability distributions illustrated in parts (a) and (b) differ even
though they have equal means and equal variances.