Page 166 - Applied statistics and probability for engineers

144 Chapter 4/Continuous Random Variables and Probability Distributions

3.0
d b
2.5 1 0.5
1 1
1 2
2.0 4.5 6.2

1.5
f (x)
1.0

0.5

0.0

–0.5
0 1 2 3 4 5 6 7 8
x
FIGURE 4-26 Weibull probability density functions for selected values of δ and β.

Example 4-25 Bearing Wear The time to failure (in hours) of a bearing in a mechanical shaft is satisfactorily
/
modeled as a Weibull random variable with β = 1 2 and δ = 5000 hours . Determine the mean time
until failure.
From the expression for the mean,
/
E X ( ) = 5000 Γ ⎡1 + (1 2 )⎤ = 5000 Γ . [ ] = 5000 × . 0 5 π = 4431 hours
1
.1
5
⎦
⎣
Determine the probability that a bearing lasts at least 6000 hours. Now,
⎤
2
P X > 6000) = − ( ⎡ − ⎢ ⎛ ⎜ 6000⎞ ⎟ ⎥ = e − . = 0 2 . 337
(
1 44
F 6000) = exp
1
⎢ ⎣ ⎝ 5000⎠ ⎥ ⎦
Practical Interpretation: Consequently, only 23.7% of all bearings last at least 6000 hours.
EXERCISES FOR SECTION 4-10

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-153. Suppose that X has a Weibull distribution with β = .2 (a) Determine the probability that a bearing lasts at least 8000
0
and δ = 100 hours. Determine the mean and variance of X. hours.
4-154. Suppose that X has a Weibull distribution with (b) Determine the mean time until failure of a bearing.
β = .2 and δ = 100 hours. Determine the following: (c) If 10 bearings are in use and failures occur independently, what
0
(
(
(a) P X <10 ,000) (b) P X > 5000) is the probability that all 10 bearings last at least 8000 hours?
4-155. If X is a Weibull random variable with β = 1 and 4-157. The life (in hours) of a computer processing unit
δ = 1000, what is another name for the distribution of X, and (CPU) is modeled by a Weibull distribution with parameters
what is the mean of X? β = 3 and δ = 900 hours. Determine (a) and (b):
4-156. Assume that the life of a roller bearing follows a (a) Mean life of the CPU. (b) Variance of the life of the CPU.
Weibull distribution with parameters β = 2 and δ = 10 000, hours. (c) What is the probability that the CPU fails before 500 hours?