Applied Statistics and Probability for Engineers

3-4 MEAN AND VARIANCE OF A DISCRETE RANDOM VARIABLE 67

The mean of a discrete random variable Xis a weighted average of the possible values of

X, with weights equal to the probabilities. If is the probability mass function of a loading

on a long, thin beam, is the point at which the beam balances. Consequently,

describes the “center’’ of the distribution of Xin a manner similar to the balance point of a

loading. See Fig. 3-5.

The variance of a random variable Xis a measure of dispersion or scatter in the possible

values for X. The variance of Xuses weight as the multiplier of each possible squared

deviation. Figure 3-5 illustrates probability distributions with equal means but dif-

ferent variances. Properties of summations and the definition of can be used to show the

equality of the formulas for variance.

Either formula for can be used. Figure 3-6 illustrates that two probability distributions

can differ even though they have identical means and variances.

EXAMPLE 3-9 In Example 3-4, there is a chance that a bit transmitted through a digital transmission channel

is received in error. Let Xequal the number of bits in error in the next four bits transmitted.

The possible values for Xare. Based on a model for the errors that is presented

in the following section, probabilities for these values will be determined. Suppose that the

probabilities are

P 1 X 12 0.2916 P 1 X 32 0.0036

P 1 X 02 0.6561 P 1 X 22 0.0486 P 1 X 42 0.0001

5 0, 1, 2, 3, 46

V 1 x 2

 a

x

x^2 f 1 x 2  22   2  a

x

x^2 f 1 x 2 ^2

V 1 X 2 a

x

1 x 22 f 1 x 2 a

x

x^2 f 1 x 2  (^2) a
x
xf 1 x (^2 2) a
x
f 1 x 2
1 x 22
f 1 x 2
E 1 X 2 E 1 X 2
f 1 x 2
0 2 4 6 8 10
(a)
0 2 4 6 8 10
(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
(a)
0 2 4 6 8 10
(b)
Figure 3-6 The probability distributions illustrated in Parts (a) and (b) differ even
though they have equal means and equal variances.
PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 67