1. In a certain city, the daily consumption of water (in millions of liters) follows approximately a gamma distribution with ? = 2 and ? = 3. If the daily capacity of that city is 9 million liters of water, what is the probability that on any given day the water supply is inadequate?
2. 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 ? = 6 and variance ? = 12.
a) Find the values of ? and ?.
b) Find the probability that on any given day the daily power consumption will exceed 12 million kilowatt-hours.
3. The life, in years, of a certain type of electrical switch has an exponential distribution with an average life ? = 2. If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year?
4. The lifetime, in weeks, of a certain type of transistor is known to follow a gamma distribution with mean 10 weeks and standard deviation ï?-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?
5. 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?
6. 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 ? = 6, we know that the time, in hours, between successive calls has an exponential distribution with parameter ? = 1/6. What is the probability of waiting more than 15 minutes between any two successive calls?
7. A manufacturer of a certain type of large machine wishes to buy rivets form 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 man 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?
8. Consider an electrical component failure rate of once every 5 hours. It is important to consider the time that it takes for 2 components to fail.
a) Assuming that the gamma distribution applies, what is the mean time that it takes for failure of 2 components?
b) What is the probability that 12 hours will elapse before 2 components fail?
9. A controlled satellite is known to have an error (distance from target) that is normally distributed with mean zero and standard deviation 4 feet. The manufacturer of the satellite defines a ?success? as a firing in which the satellite comes within 10 feet of the target. Compute the probability that the satellite fails?
See attached file for full problem description.
The solution gives complete solution to problems related to Poisson distribution, exponential distribution and gamma distribution. Probabilities of different events are calculated in step by step approach.