Weibull Distribution (Optional)

6.10 Weibull Distribution (Optional)

Modern technology has enabled engineers to design many complicated systems whose operation and safety depend on the reliability of the various components making up the systems. For example, a fuse may burn out, a steel column may buckle, or a heat-sensing device may fail. Identical components subjected to iden- tical environmental conditions will fail at different and unpredictable times. We have seen the role that the gamma and exponential distributions play in these types of problems. Another distribution that has been used extensively in recent years to deal with such problems is the Weibull distribution, introduced by the Swedish physicist Waloddi Weibull in 1939.

Weibull The continuous random variable X has a Weibull distribution, with param- Distribution eters α and β, if its density function is given by

where α > 0 and β > 0. The graphs of the Weibull distribution for α = 1 and various values of the param-

eter β are illustrated in Figure 6.30. We see that the curves change considerably in shape for different values of the parameter β. If we let β = 1, the Weibull dis- tribution reduces to the exponential distribution. For values of β > 1, the curves become somewhat bell shaped and resemble the normal curve but display some skewness.

The mean and variance of the Weibull distribution are stated in the following theorem. The reader is asked to provide the proof in Exercise 6.52 on page 206.

Theorem 6.8: The mean and variance of the Weibull distribution are

β Like the gamma and exponential distributions, the Weibull distribution is also

applied to reliability and life-testing problems such as the time to failure or

204 Chapter 6 Some Continuous Probability Distributions

Figure 6.30: Weibull distributions (α = 1).

life length of a component, measured from some specified time until it fails. Let us represent this time to failure by the continuous random variable T , with probability density function f (t), where f (t) is the Weibull distribution. The Weibull distribution has inherent flexibility in that it does not require the lack

of memory property of the exponential distribution. The cumulative distribution function (cdf) for the Weibull can be written in closed form and certainly is useful in computing probabilities.

cdf for Weibull The cumulative distribution function for the Weibull distribution is Distribution given by

F (x) = 1 − e −αx β ,

for x ≥ 0,

for α > 0 and β > 0. Example 6.24: The length of life X, in hours, of an item in a machine shop has a Weibull distri-

bution with α = 0.01 and β = 2. What is the probability that it fails before eight hours of usage?

Solution : 2 P (X < 8) = F (8) = 1 − e −(0.01)8 = 1 − 0.527 = 0.473.

The Failure Rate for the Weibull Distribution

When the Weibull distribution applies, it is often helpful to determine the fail- ure rate (sometimes called the hazard rate) in order to get a sense of wear or deterioration of the component. Let us first define the reliability of a component or product as the probability that it will function properly for at least a specified time under specified experimental conditions . Therefore, if R(t) is defined to be

6.10 Weibull Distribution (Optional) 205

the reliability of the given component at time t, we may write ∞

R(t) = P (T > t) =

f (t) dt = 1 − F (t),

where F (t) is the cumulative distribution function of T . The conditional probability that a component will fail in the interval from T = t to T = t + ∆t, given that it survived to time t, is

F (t + ∆t) − F (t) . R(t)

Dividing this ratio by ∆t and taking the limit as ∆t → 0, we get the failure rate, denoted by Z(t). Hence,

Z(t) = lim

1 − F (t) which expresses the failure rate in terms of the distribution of the time to failure.

Since Z(t) = f (t)/[1 − F (t)], the failure rate is given as follows: Failure Rate for The failure rate at time t for the Weibull distribution is given by

Weibull Distribution

Z(t) = αβt β−1 ,

t > 0.

Interpretation of the Failure Rate

The quantity Z(t) is aptly named as a failure rate since it does quantify the rate of change over time of the conditional probability that the component lasts an additional ∆t given that it has lasted to time t. The rate of decrease (or increase) with time is important. The following are crucial points.

(a) If β = 1, the failure rate = α, a constant. This, as indicated earlier, is the special case of the exponential distribution in which lack of memory prevails.

(b) If β > 1, Z(t) is an increasing function of time t, which indicates that the

component wears over time. (c) If β < 1, Z(t) is a decreasing function of time t and hence the component

strengthens or hardens over time. For example, the item in the machine shop in Example 6.24 has β = 2, and

hence it wears over time. In fact, the failure rate function is given by Z(t) = 0.02t. On the other hand, suppose the parameters were β = 3/4 and α = 2. In that case, Z(t) = 1.5/t 1/4 and hence the component gets stronger over time.

206 Chapter 6 Some Continuous Probability Distributions

Exercises

6.39 Use the gamma function with y = √ 2x to show (a) How long can such a battery be expected to last? that Γ(1/2) = π.

(b) What is the probability that such a battery will be

operating after 2 years?

6.40 In a certain city, the daily consumption of water (in millions of liters) follows approximately a gamma 6.48 Derive the mean and variance of the beta distri- distribution with α = 2 and β = 3. If the daily capac- bution. ity of that city is 9 million liters of water, what is the

probability that on any given day the water supply is 6.49 Suppose the random variable X follows a beta inadequate?

distribution with α = 1 and β = 3. 6.41 If a random variable X has the gamma distribu- (a) Determine the mean and median of X.

tion with α = 2 and β = 1, find P (1.8 < X < 2.4). (b) Determine the variance of X. (c) Find the probability that X > 1/3.

6.42 Suppose that the time, in hours, required to repair a heat pump is a random variable X having 6.50 If the proportion of a brand of television set re- a gamma distribution with parameters α = 2 and quiring service during the first year of operation is a β = 1/2. What is the probability that on the next random variable having a beta distribution with α = 3 service call

and β = 2, what is the probability that at least 80% of (a) at most 1 hour will be required to repair the heat the new models of this brand sold this year will require pump?

service during their first year of operation? (b) at least 2 hours will be required to repair the heat

pump? 6.51 The lives of a certain automobile seal have the √ Weibull distribution with failure rate Z(t) =1/ t. 6.43 (a) Find the mean and variance of the daily wa- Find the probability that such a seal is still intact after ter consumption in Exercise 6.40.

4 years.

(b) According to Chebyshev’s theorem, there is a prob- 6.52 Derive the mean and variance of the Weibull dis- ability of at least 3/4 that the water consumption tribution. on any given day will fall within what interval?

6.53 In a biomedical research study, it was deter- 6.44 In a certain city, the daily consumption of elec- mined that the survival time, in weeks, of an animal tric power, in millions of kilowatt-hours, is a random subjected to a certain exposure of gamma radiation has variable X having a gamma distribution with mean 2 a gamma distribution with α = 5 and β = 10. µ = 6 and variance σ = 12.

(a) What is the mean survival time of a randomly se- (a) Find the values of α and β.

lected animal of the type used in the experiment? (b) Find the probability that on any given day the daily (b) What is the standard deviation of survival time? power consumption will exceed 12 million kilowatt- (c) What is the probability that an animal survives hours.

more than 30 weeks?

6.45 The length of time for one individual to be 6.54 The lifetime, in weeks, of a certain type of tran- served at a cafeteria is a random variable having an ex- sistor is known to follow a gamma distribution with ponential distribution with a mean of 4 minutes. What

√ is the probability that a person is served in less than 3 mean 10 weeks and standard deviation

50 weeks. minutes on at least 4 of the next 6 days?

(a) What is the probability that a transistor of this type will last at most 50 weeks?

6.46 The life, in years, of a certain type of electrical (b) What is the probability that a transistor of this switch has an exponential distribution with an average

type will not survive the first 10 weeks? life β = 2. If 100 of these switches are installed in dif- ferent systems, what is the probability that at most 30 6.55 Computer response time is an important appli- fail during the first year?

cation of the gamma and exponential distributions. Suppose that a study of a certain computer system

6.47 Suppose that the service life, in years, of a hear- reveals that the response time, in seconds, has an ex- ing aid battery is a random variable having a Weibull ponential distribution with a mean of 3 seconds. distribution with α = 1/2 and β = 2.

Review Exercises 207 (a) What is the probability that response time exceeds (a) What is the probability that more than 10 auto-

5 seconds? mobiles appear at the intersection during any given (b) What is the probability that response time exceeds

minute of time?

10 seconds? (b) What is the probability that more than 2 minutes elapse before 10 cars arrive? 6.56 Rate data often follow a lognormal distribution. Average power usage (dB per hour) for a particular 6.59 Consider the information in Exercise 6.58. company is studied and is known to have a lognormal (a) What is the probability that more than 1 minute distribution with parameters µ = 4 and σ = 2. What

elapses between arrivals?

is the probability that the company uses more than 270 dB during any particular hour?

(b) What is the mean number of minutes that elapse

between arrivals?

6.57 For Exercise 6.56, what is the mean power usage (average dB per hour)? What is the variance?

6.60 Show that the failure-rate function is given by 6.58 The number of automobiles that arrive at a cer-

Z(t) = αβt β−1 , t > 0, tain intersection per minute has a Poisson distribution

with a mean of 5. Interest centers around the time that if and only if the time to failure distribution is the elapses before 10 automobiles appear at the intersec- Weibull distribution tion.

f (t) = αβt β−1 e −αt β , t > 0.

Review Exercises

6.61 According to a study published by a group of so- is the probability that the next 3 calls will be received ciologists at the University of Massachusetts, approx- within the next 30 minutes? imately 49% of the Valium users in the state of Mas-

sachusetts are white-collar workers. What is the prob- 6.64 A manufacturer of a certain type of large ma- ability that between 482 and 510, inclusive, of the next chine wishes to buy rivets from one of two manufac- 1000 randomly selected Valium users from this state turers. It is important that the breaking strength of are white-collar workers?

each rivet exceed 10,000 psi. Two manufacturers (A and B) offer this type of rivet and both have rivets

6.62 The exponential distribution is frequently ap- whose breaking strength is normally distributed. The plied to the waiting times between successes in a Pois- mean breaking strengths for manufacturers A and B son process. If the number of calls received per hour are 14,000 psi and 13,000 psi, respectively. The stan- by a telephone answering service is a Poisson random dard deviations are 2000 psi and 1000 psi, respectively. variable with parameter λ = 6, we know that the time, Which manufacturer will produce, on the average, the in hours, between successive calls has an exponential fewest number of defective rivets?

distribution with parameter β =1/6. What is the prob- 6.65 According to a recent census, almost 65% of all ability of waiting more than 15 minutes between any households in the United States were composed of only two successive calls?

one or two persons. Assuming that this percentage is 6.63 When α is a positive integer n, the gamma dis- still valid today, what is the probability that between

tribution is also known as the Erlang distribution. 590 and 625, inclusive, of the next 1000 randomly se- Setting α = n in the gamma distribution on page 195, lected households in America consist of either one or two persons? the Erlang distribution is

6.66 A certain type of device has an advertised fail-

f (x) = β n (n−1)! , x > 0, ure rate of 0.01 per hour. The failure rate is constant

n−1 e −x/β

elsewhere.

and the exponential distribution applies. (a) What is the mean time to failure?

It can be shown that if the times between successive (b) What is the probability that 200 hours will pass events are independent, each having an exponential

before a failure is observed? distribution with parameter β, then the total elapsed waiting time X until all n events occur has the Erlang 6.67 In a chemical processing plant, it is important distribution. Referring to Review Exercise 6.62, what that the yield of a certain type of batch product stay

208 Chapter 6 Some Continuous Probability Distributions

above 80%. If it stays below 80% for an extended pe- 6.74 The average rate of water usage (thousands of riod of time, the company loses money. Occasional gallons per hour) by a certain community is known defective batches are of little concern. But if several to involve the lognormal distribution with parameters batches per day are defective, the plant shuts down µ = 5 and σ = 2. It is important for planning purposes and adjustments are made. It is known that the yield to get a sense of periods of high usage. What is the is normally distributed with standard deviation 4%.

probability that, for any given hour, 50,000 gallons of (a) What is the probability of a “false alarm” (yield water are used? below 80%) when the mean yield is 85%?

6.75 For Review Exercise 6.74, what is the mean of (b) What is the probability that a batch will have a the average water usage per hour in thousands of gal-

yield that exceeds 80% when in fact the mean yield lons? is 79%?

6.76 In Exercise 6.54 on page 206, the lifetime of a 6.68 For an electrical component with a failure rate transistor is assumed to have a gamma distribution of once every 5 hours, it is important to consider the

√ with mean 10 weeks and standard deviation

50 weeks. time that it takes for 2 components to fail.

Suppose that the gamma distribution assumption is in- (a) Assuming that the gamma distribution applies, correct. Assume that the distribution is normal. what is the mean time that it takes for 2 compo- (a) What is the probability that a transistor will last nents to fail?

at most 50 weeks?

(b) What is the probability that 12 hours will elapse (b) What is the probability that a transistor will not before 2 components fail?

survive for the first 10 weeks? (c) Comment on the difference between your results

6.69 The elongation of a steel bar under a particular here and those found in Exercise 6.54 on page 206. load has been established to be normally distributed

with a mean of 0.05 inch and σ = 0.01 inch. Find the 6.77 The beta distribution has considerable applica- probability that the elongation is

tion in reliability problems in which the basic random (a) above 0.1 inch;

variable is a proportion, as in the practical scenario il- (b) below 0.04 inch;

lustrated in Exercise 6.50 on page 206. In that regard, (c) between 0.025 and 0.065 inch.

consider Review Exercise 3.73 on page 108. Impurities in batches of product of a chemical process reflect a serious problem. It is known that the proportion of

6.70 A controlled satellite is known to have an error impurities Y in a batch has the density function (distance from target) that is normally distributed with mean zero and standard deviation 4 feet. The manu-

0 ≤ y ≤ 1, facturer of the satellite defines a success as a firing in

f (y) =

elsewhere. which the satellite comes within 10 feet of the target. Compute the probability that the satellite fails.

(a) Verify that the above is a valid density function. (b) What is the probability that a batch is considered

6.71 A technician plans to test a certain type of resin not acceptable (i.e., Y > 0.6)? developed in the laboratory to determine the nature (c) What are the parameters α and β of the beta dis- of the time required before bonding takes place. It

tribution illustrated here? is known that the mean time to bonding is 3 hours

and the standard deviation is 0.5 hour. It will be con- (d) The mean of the beta distribution is α α+β . What is sidered an undesirable product if the bonding time is

the mean proportion of impurities in the batch? either less than 1 hour or more than 4 hours. Com- (e) The variance of a beta distributed random variable

ment on the utility of the resin. How often would its

is

performance be considered undesirable? Assume that 2 αβ time to bonding is normally distributed.

2 (α + β) . (α + β + 1) 6.72 Consider the information in Review Exercise

What is the variance of Y in this problem? 6.66. What is the probability that less than 200 hours

will elapse before 2 failures occur? 6.78 Consider now Review Exercise 3.74 on page 108. The density function of the time Z in minutes between

6.73 For Review Exercise 6.72, what are the mean calls to an electrical supply store is given by and variance of the time that elapses before 2 failures

occur? 1

f (z) = 10 e −z/10 ,

0 < z < ∞,

elsewhere.

6.11 Potential Misconceptions and Hazards 209 (a) What is the mean time between calls?

6.85 From the relationship between the chi-squared (b) What is the variance in the time between calls?

random variable and the gamma random variable, (c) What is the probability that the time between calls prove that the mean of the chi-squared random variable exceeds the mean?

is v and the variance is 2v.

6.86 The length of time, in seconds, that a computer 6.79 Consider Review Exercise 6.78. Given the as- user takes to read his or her e-mail is distributed as a

sumption of the exponential distribution, what is the lognormal random variable with µ = 1.8 and σ 2 = 4.0. mean number of calls per hour? What is the variance in the number of calls per hour?

(a) What is the probability that a user reads e-mail for more than 20 seconds? More than a minute?

6.80 In a human factor experimental project, it has (b) What is the probability that a user reads e-mail for been determined that the reaction time of a pilot to a

a length of time that is equal to the mean of the visual stimulus is normally distributed with a mean of

underlying lognormal distribution? 1/2 second and standard deviation of 2/5 second. (a) What is the probability that a reaction from the

6.87 Group Project : Have groups of students ob- pilot takes more than 0.3 second?

serve the number of people who enter a specific coffee (b) What reaction time is that which is exceeded 95% shop or fast food restaurant over the course of an hour, of the time?

beginning at the same time every day, for two weeks. The hour should be a time of peak traffic at the shop

6.81 or restaurant. The data collected will be the number

The length of time between breakdowns of an es- of customers who enter the shop in each half hour of sential piece of equipment is important in the decision of the use of auxiliary equipment. An engineer thinks time. Thus, two data points will be collected each day. Let us assume that the random variable X, the num- that the best model for time between breakdowns of a ber of people entering each half hour, follows a Poisson generator is the exponential distribution with a mean distribution. The students should calculate the sam- of 15 days.

ple mean and variance of X using the 28 data points (a) If the generator has just broken down, what is the collected.

probability that it will break down in the next 21 (a) What evidence indicates that the Poisson distribu- days?

tion assumption may or may not be correct? (b) What is the probability that the generator will op- (b) Given that X is Poisson, what is the distribution of

erate for 30 days without a breakdown? T , the time between arrivals into the shop during 6.82 The length of life, in hours, of a drill bit in a

a half hour period? Give a numerical estimate of mechanical operation has a Weibull distribution with

the parameter of that distribution. α = 2 and β = 50. Find the probability that the bit (c) Give an estimate of the probability that the time will fail before 10 hours of usage.

between two arrivals is less than 15 minutes. (d) What is the estimated probability that the time

6.83 Derive the cdf for the Weibull distribution. between two arrivals is more than 10 minutes? [Hint: In the definition of a cdf, make the transfor-

mation z = y β .] (e) What is the estimated probability that 20 minutes after the start of data collection not one customer

6.84 Explain why the nature of the scenario in Re-

has appeared?

view Exercise 6.82 would likely not lend itself to the exponential distribution.