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106 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

The expected value of a function h(X) of a continuous random variable is defined similarly to
a function of a discrete random variable.

Expected Value
of a Function of If X is a continuous random variable with probability density function f(x),
a Continuous

Random
Variable E3h1X24 h1x2 f 1x2 dx (4-5)

EXAMPLE 4-7 In Example 4-1, X is the current measured in milliamperes. What is the expected value of the
2
squared current? Now, h1X2 X . Therefore,
20 3 20
2 2 x
E3h1X24 x f 1x2 dx 0.05x dx 0.05 ` 133.33
3 0
0
2
In the previous example, the expected value of X does not equal E(X) squared. However, in
the special case that h1X2 aX b for any constants a and b, E3h1X24 aE1X2 b. This
can be shown from the properties of integrals.

EXAMPLE 4-8 For the drilling operation in Example 4-2, the mean of X is

201x 12.52
E1X2 xf 1x2 dx x 20e dx
12.5 12.5
Integration by parts can be used to show that

e 201x 12.52
E1X2 xe 201x 12.52 ` 12.5 0.05 12.55
20 12.5
The variance of X is

2
V1X2 1x 12.552 f 1x2 dx

12.5
Although more difficult, integration by parts can be used two times to show that V(X) 0.0025.

EXERCISES FOR SECTION 4-4

4-22. Suppose f 1x2 0.25 for 0 x 4. Determine the 4-25. Suppose that f 1x2 x 8 for 3 x 5. Determine
mean and variance of X. the mean and variance for x.
4-23. Suppose f 1x2 0.125x for 0 x 4. Determine the 4-26. Determine the mean and variance of the weight of
mean and variance of X. packages in Exercise 4.7.
4-24. Suppose f 1x2 1.5x 2 for 1 x 1. Determine 4-27. The thickness of a conductive coating in micrometers
the mean and variance of X. has a density function of 600x 2 for 100
m x 120
m.