Page 172 - Applied statistics and probability for engineers

150 Chapter 4/Continuous Random Variables and Probability Distributions

Example 4-28 The time to complete a task in a large project is modeled as a generalized beta distribution with
minimum and maximum times a = 8 and b = 20 days, respectively, along with mode of m = 16
/
days. Also, assume that the mean completion time is μ = (a + m + ) 6. Determine the parameters
b
4
α and β of the generalized beta distribution with these properties.
,
,
The values (a m b ) specify the minimum, mode, and maximum times, but the mode value alone does not uniquely deter-
/
mine the two parameters α and β. Consequently, the mean completion time, μ, is assumed to equal μ = (a + m + ) 6.
b
4
−
a
)
Here the generalized beta random variable is W = + ( b a X, where X is a beta random variable. Because the
minimum and maximum values for W are 8 and 20, respectively, a = 8 and b = 20. The mean of W is
α
−
−
a
a
(
(
μ = + b a E X) = + b a)
(
)
+
( α β)
+
16
4
The assumed mean is μ = (8 + ( ) 20 ) / = 15 .333. The mode of W is
6
α −1
−
a
m = + ( b a)
α β
+ − 2
with m = 16. These equations can be solved for α and β to obtain
a
2
α = ( μ − )( m − − )a b
(m − μ)(b − )a
β = α(b − μ)
μ − a
Therefore,
(15 .333 − )( ( )8 2 16 − − 20 )
8
α = = . 3 665
(16 −15 .333 )(20 − )8
.
β = . 3 665 (20 −115 333) = .
2 333
.
−
15 333 8
Practical Interpretation: The program evaluation and review technique (PERT) widely uses the distribution of W to
model the duration of tasks. Therefore, W is said to have a PERT distribution. Notice that we need only specify the
minimum, maximum, and mode (most likely time) for a task to specify the distribution. The model assumes that the
mean is the function of these three values and allows the α and β parameters to be computed.
Exercises FOR SECTION 4-12
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-184. Suppose that X has a beta distribution with parameters (a) Calculate the mode, mean, and variance of the distribution
α = 2.5 and β = 2.5. Sketch an approximate graph of the prob- for α = 3 and β = 1 4. .
ability density function. Is the density symmetric? (b) Calculate the mode, mean, and variance of the distribution
.
4-185. Suppose that x has a beta distribution with parameters for α = 10 and β = 6 25.
α = 2.5 and β = 1. Determine the following: (c) Comment on the difference in dispersion in the distribution
. (
(
(a) P X < 0 25) (b) P 0 25 < X < 0.75) from parts (a) and (b).
.
(c) Mean and variance 4-188. The length of stay at a hospital emergency department
is the sum of the waiting and service times. Let X denote the
4-186. Suppose that X has a beta distribution with parameters proportion of time spent waiting and assume a beta distribution
α = 1 and β = 4 2. Determine the following: with α = 10 and β = 1. Determine the following:
.
(
. (
(a) P X < 0 25) (b) P 0 5 < ) (c) Mean and variance (a) P X > 0 9) (b) P X < 0 5) (c) Mean and variance
(
(
.
X
.
.
4-187. A European standard value for a low-emission win- 4-189. The maximum time to complete a task in a project is 2.5
dow glazing uses 0.59 as the proportion of solar energy that days. Suppose that the completion time as a proportion of this max-
enters a room. Suppose that the distribution of the proportion imum is a beta random variable with α = 2 and β = 3. What is the
of solar energy that enters a room is a beta random variable. probability that the task requires more than two days to complete?