Page 170 - Applied statistics and probability for engineers

148 Chapter 4/Continuous Random Variables and Probability Distributions

0
4-175. The lifetime of a semiconductor laser has a log- (a) P X( < .0 02 ) (b) Value for x such that P X( ≤ x) = .95
normal distribution, and it is known that the mean and standard (c) Mean and variance of X
deviation of lifetime are 10,000 and 20,000, respectively. 4-181. An article in Applied Mathematics and Computa-
(a) Calculate the parameters of the lognormal distribution. tion [“Conidence Intervals for Steady State Availability of
(b) Determine the probability that a lifetime exceeds 10,000 a System with Exponential Operating Time and Lognormal
hours. Repair Time” (2003, Vol.137(2), pp. 499-509)] considered the
(c) Determine the lifetime that is exceeded by 90% of lasers. long-run availability of a system with an assumed lognormal
4-176. An article in Health and Population: Perspectives distribution for repair time. In a given example, repair time
and Issues (2000, Vol. 23, pp. 28–36) used the lognormal distri- follows a lognormal distribution with θ = ω = 1. Determine the
bution to model blood pressure in humans. The mean systolic following:
blood pressure (SBP) in males age 17 was 120.87 mm Hg. If the (a) Probability that repair time is more than ive time units
co-eficient of variation (100% × Standard deviation/mean) (b) Conditional probability that a repair time is less than eight
is 9%, what are the parameter values of the lognormal time units given that it is more than ive time units
distribution? (c) Mean and variance of repair time
4-177. Derive the probability density function of a lognormal 4-182. An article in Chemosphere [“Statistical Evaluations
random variable from the derivative of the cumulative distribu- Relecting the Skewness in the Distribution of TCDD Lev-
tion function. els in Human Adipose Tissue” (1987, Vol.16(8), pp. 2135-
4-178. Suppose that X has a lognormal distribution with 2140)] concluded that the levels of 2,3,7,8-TCDD (colorless
2
parameters θ = 10 and ω = 16. Determine the following: persistent environmental contaminants with no distinguish-
(
(
(a) P X < 2000) (b) P X > 1500) able odor at room temperature) in human adipose tissue
(c) Value exceeded with probability 0.7 has a lognormal distribution (based on empirical evidence
4-179. Suppose that the length of stay (in hours) at a hospital from North America). The mean and variance of this log-
emergency department is modeled with a lognormal random normal distribution in the USA are 8 and 21, respectively.
variable X with θ = 1 5. and ω = 0 4. . Determine the following Let X denote this lognormal random variable. Determine the
in parts (a) and (b): following:
(
(a) Mean and variance (b) P X < 8) (a) P(2000 < X < 2500 )
(c) Comment on the difference between the probability (b) Value exceeded with probability 10%
P X < 0) calculated from this lognormal distribution and (c) Mean and variance of X
(
a normal distribution with the same mean and variance. 4-183. Consider the lifetime of a laser in Example 4-26.
4-180. An article in Journal of Hydrology [“Use of a Lognormal Determine the following in parts (a) and (b):
Distribution Model for Estimating Soil Water Retention Curves (a) Probability the lifetime is less than 1000 hours
from Particle-Size Distribution Data” (2006, Vol. 323(1), pp. (b) Probability the lifetime is less than 11,000 hours given that
325–334)] considered a lognormal distribution model to estimate it is more than 10,000 hours
water retention curves for a range of soil textures. The particle-size (c) Compare the answers to parts (a) and (b) and comment on
distribution (in centimeters) was modeled as a lognormal random any differences between the lognormal and exponential
variable X with θ = − .3 8 and ω = .0 7. Determine the following: distributions.

4-12 Beta Distribution

A continuous distribution that is lexble but bounded over a inite range is useful for probabil-
ity models. The proportion of solar radiation absorbed by a material or the proportion (of the
maximum time) required to complete a task in a project are examples of continuous random
variables over the interval [0, 1].

The random variable X with probability density function
Γ α + β)
(
α−
1
f x ( ) = x (1 − x) β−1 , for x [0 1in , ]
Γ α ( ) Γ β ( )

is a beta random variable with parameters a > 0 and b > 0.
The shape parameters α and β allow the probability density function to assume many differ-
ent shapes. Figure 4-28 provides some examples. If α = β, the distribution is symmetric about
.
x = 0 5, and if α = β = 1, the beta distribution equals a continuous uniform distribution. Figure
4-28 illustrates that other parameter choices generate nonsymmetric distributions.