Page 98 - Applied statistics and probability for engineers

76 Chapter 3/Discrete Random Variables and Probability Distributions

Because the units of the variables in this example are millions of dollars and because the variance of a random vari-
2
able squares the deviations from the mean, the units of σ are millions of dollars squared. These units make interpreta-
tion difi cult.
Because the units of standard deviation are the same as the units of the random variable, the standard deviation σ is
.
easier to interpret. Here σ = 5 25 = 2 29 millions of dollars and σ is large relative to μ.
.

Example 3-11 Messages The number of e-mail messages received per hour has the following distribution:

x = number of messages 10 11 12 13 14 15
f x ( ) 0.08 0.15 0.30 0.20 0.20 0.07

Determine the mean and standard deviation of the number of messages received per hour.
(
E X) = 10 ( . ) +11 ( . ) + … +15(0.07) = 12.5
15
0
08
0
2
2
2
1
( .
V X ( ) = 10 2 . (0 08 ) +11 ( . 0 15) + … + 15 0 07) − 12 5 = .
1 85
.
( ) =
.
σ = V X 1 85 = .
1 36
The variance of a random variable X can be considered to be the expected value of
a speciic function of X, namely, h X ( ) = ( X − μ) . In general, the expected value of any
2

function h X ( ) of a discrete random variable is deined in a similar manner.
Expected Value of a
(
Function of a Discrete If X is a discrete random variable with probability mass function f x , )
Random Variable
( )
E h X ( )⎤ = ∑ h x f x ( ) (3-4)
⎡
⎦
⎣
x
Example 3-12 Digital Channel In Example 3-9, X is the number of bits in error in the next four bits transmitted.
What is the expected value of the square of the number of bits in error? Now, h X ( ) = X . Therefore,
2
⎡
2
2
2
2
2
0 0486
0 00011
0 0036
E h X ( )⎤ = 0 × . +1 × . + 2 × . + 3 × . + 4 × .
0 6561
0 2916
⎣
⎦
= 0 52
.
Practical Interpretation: The expected value of a function of a random variable is simply a weighted average of the
function evaluated at the values of the random variable.
In Example 3-12, the expected value of h X) = X does not equal h E X[ ( ) ]. However, in the
2
(
special case that h X ( ) = aX b (for any constants a and b), the following can be shown from
+
the properties of sums in the deinition in Equation 3-4.

(
E aX b) = aE X ( ) + b
+
and
+
V aX b) = a V X)
2
(
(