91Ó°ÊÓ

In a certain city, the daily consumption of electric power, in millions of kilowatt-hours, is a random variable \(X\) having a gamma distribution with mean \(\mu=6\) and variance \(a^{2}=12\) (a) Find the values of \(\alpha\) and \(\beta\). (b) Find the probability that on any given day the daily power consumption will exceed 12 million kilowatthours.

The life, in years, of a certain type of electrical switch has an exponential distribution with an average life \(\beta=2\). If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year?

Suppose that the service life, in years, of a hearing aid battery is a random variable having a Weibull distribution with \(\alpha=1 / 2\) and \(\beta=2\) (a) How long can such a battery be expected to last? (b) What is the probability that such a battery will be operating after 2 years?

Derive the mean and variance of the Weibull distribution.

The lifetime, in weeks, of a certain type of transistor is known to follow a gamma distribution with mean 10 weeks and standard deviation \(\sqrt{50}\) weeks. (a) What is the probability that the transistor will last at most 50 weeks? (b) What is the probability that the transistor will not survive the first 10 weeks?

The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5 . Interest centers around the time that elapses before 10 automobiles appear at the intersection. (a) What is the probability that more than 10 automobiles appear at the intersection during any given minute of time? (b) What is the probability that more than 2 minutes are required before 10 cars arrive?

The exponential distribution is frequently applied to the waiting times between successes in a Poisson process. If the number of calls received per hour by a telephone answering service is a Poisson random variable with parameter \(X=6,\) we know that the time, in hours, between successive calls has an exponential distribution with parameter \(\beta=1 / 6\). What is the probability of waiting more than 15 minutes between any two successive calls?

A manufacturer of a certain type of large machine wishes to buy rivets from one of two manufacturers. It is important that the breaking strength of each rivet exceed 10,000 psi. Two manufacturers (A and \(B\) ) offer this type of rivet and both have rivets whose breaking strength is normally distributed. The mean breaking strengths for manufacturers \(A\) and \(B\) are 14,000 psi and 13,000 psi, respectively. The standard deviations are 2000 psi and 1000 psi, respectively. Which manufacturer will produce, on the average, the fewest number of defective rivets?

The life of a certain type of device has an advertised failure rate of 0.01 per hour. The failure rate is constant and the exponential distribution applies. (a) What is the mean time to failure? (b) What is the probability that 200 hours will pass before a failure is observed?

The average rate of water usage (thousands of gallons per hour) by a certain community is known to involve the lognormal distribution with parameters \(\mu=5\) and \(a=2 .\) It is important for planning purposes to get a sense of periods of high usage. What is the probability that, for any given hour, 50,000 gallons of water are used?