Page 165 - Applied statistics and probability for engineers

Section 4-10/Weibull Distribution 143

(a) What is the mean time until the 10th arrival? intensity, the number of electrons arriving at an APD follows a
(b) What is the probability that more than 20 minutes is Poisson distribution with a mean of 1.74 particles per detection
required for the third arrival? window of 200 nanoseconds.
4-150. The total service time of a multistep manufacturing (a) What is the mean and variance of the time for 100 arrivals?
operation has a gamma distribution with mean 18 minutes and (b) What is the probability that the time until the ifth particle
standard deviation 6. arrives is greater than 1.0 nanosecond?
(a) Determine the parameters λ and r of the distribution. 4-152. An article in Mathematical Biosciences [“Inluence of
(b) Assume that each step has the same distribution for service Delayed Viral Production on Viral Dynamics in HIV-1 Infected
time. What distribution for each step and how many steps Patients” (1998, Vol.152(2), pp. 143–163)] considered the time
produce this gamma distribution of total service time? delay between the initial infection by immunodeiciency virus
4-151. An article in Sensors and Actuators A: Physical type 1 (HIV-1) and the formation of productively infected cells.
[“Characterization and Simulation of Avalanche PhotoDiodes In the simulation model, the time delay is approximated by a
.
for Next-Generation Colliders” (2011, Vol.172(1), pp.181– gamma distribution with parameters r = 4 and 1/ =λ 0 25 days.
188)] considered an avalanche photodiode (APD) to detect Determine the following:
charged particles in a photo. The number of arrivals in each (a) Mean and variance of time delay
detection window was modeled with a Poisson distribution (b) Probability that a time delay is more than half a day
with a mean depending on the intensity of beam. For one beam (c) Probability that a time delay is between one-half and one day

4-10 Weibull Distribution

As mentioned previously, the Weibull distribution is often used to model the time until failure
of many different physical systems. The parameters in the distribution provide a great deal
of lexibility to model systems in which the number of failures increases with time (bearing
wear), decreases with time (some semiconductors), or remains constant with time (failures
caused by external shocks to the system).
Weibull Distribution
The random variable X with probability density function
β ⎛ x⎞ β−1 ⎡ ⎛ x⎞ β ⎤
f x ( ) = ⎜ ⎟ exp − ⎢ ⎜ ⎟ , ⎥ for x > 0 (4-20)
⎝
δ δ ⎠ ⎢ ⎣ ⎝ δ ⎠ ⎥ ⎦
is a Weibull random variable with scale parameter δ > 0 and shape parameter β > 0 .

The graphs of selected probability density functions in Fig. 4-26 illustrate the lexibility of
the Weibull distribution. By inspecting the probability density function, we can see that when
β = 1, the Weibull distribution is identical to the exponential distribution. Also, the Raleigh
distribution is a special case when the shape parameter is 2.
The cumulative distribution function is often used to compute probabilities. The following
result can be obtained.

Cumulative Distribu-
tion Function If X has a Weibull distribution with parameters δ and β, then the cumulative distribu-
tion function of X is
β
⎛ x⎞
− ⎜ ⎟
⎝ ⎠
δ
1
F x ( ) = − e
Also, the following results can be obtained.
Mean and Variance
If X has a Weibull distribution with parameters δ and β,
⎛ 1 ⎞ ⎛ 2 ⎞ ⎡ ⎛ 1 ⎞ ⎤ 2
2
2
2
μ = ( ) = δΓE X ⎜ ⎝ 1 + β⎠ ⎟ and σ = ( ) = δ Γ ⎜ ⎝ 1 + β⎠ ⎟ − δ ⎢ ⎣ ⎣ Γ 1 + β⎠ ⎟ ⎥ (4-21)
V
X
⎜
⎝
⎦