Applied Statistics and Probability for Engineers

106 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

EXAMPLE 4-7 In Example 4-1, Xis the current measured in milliamperes. What is the expected value of the

squared current? Now, Therefore,

In the previous example, the expected value of X^2 does not equal E(X) squared. However, in

the special case that for any constants aand b, This

can be shown from the properties of integrals.

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

Integration by parts can be used to show that

The variance of Xis

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

EXERCISES FOR SECTION 4-4

V 1 X 2  



12.5

1 x12.55 22 f 1 x 2 dx

E 1 X 2 xe^201 x12.5^2 

e^201 x12.5^2

20

`



12.5

12.50.0512.55

E 1 X 2  



12.5

xf 1 x 2 dx 



12.5

x 20 e^201 x12.5^2 dx

h 1 X 2 aXb E 3 h 1 X 24 aE 1 X 2 b.

E 3 h 1 X 24  





x^2 f 1 x 2 dx

20

0

0.05x^2 dx0.05

x^3

3

`

20

0

133.33

h 1 X 2 X^2.

If Xis a continuous random variable with probability density function f(x),

E 3 h 1 X 24  (4-5)





h 1 x 2 f 1 x 2 dx

Expected Value

of a Function of

a Continuous

Random

Variable

4-22. Suppose for Determine the

mean and variance of X.

4-23. Suppose for Determine the

mean and variance of X.

4-24. Suppose for Determine

the mean and variance of X.

f 1 x 2 1.5x^2  1 x1.

f 1 x 2 0.125x 0 x4.

f 1 x 2 0.25 0 x4. 4-25. Suppose that for Determine

the mean and variance for x.

4-26. Determine the mean and variance of the weight of

packages in Exercise 4.7.

4-27. The thickness of a conductive coating in micrometers

has a density function of 600x^2 for 100 m x 120 m.

f 1 x 2 x 8 3 x5.

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

a function of a discrete random variable.

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