This post presents exercises on gamma distribution and Poisson distribution, reinforcing the concepts discussed in this blog post in a companion blog and blog posts in another blog. Because the shape parameter of the gamma distribution in the following problems is a positive integer, the calculation of probabilities for the gamma distribution is based on Poisson distribution.
Practice Problem 2A 
Suppose that is the useful working life (in years) of a brand new industrial machine. The following is the probability density function of .
A manufacturing plant has just purchased such a new machine. Determine the probability that the machine will be in operation for the next 20 years. 
Practice Problem 2B 
The annual rainfall (in inches) in Western Colorado is modeled by a distribution with the following cumulative distribution function. In a year in which the annual rainfall is above 20 inches, determine the probability that the annual rainfall is above 30 inches. 
Practice Problem 2C 
The annual rainfall (in inches) in Western Colorado is modeled by a distribution with the following cumulative distribution function. Determine the mean and the variance of the annual rainfall in this region. 
Practice Problem 2D 
The repair time (in hours) for an industrial machine has a gamma distribution with mean 1.5 and variance 0.75.

Practice Problem 2E 
Customers arriving at a jewelry store according to a Poisson process with an average rate of 2.5 per hours. The store opens its door at 9 AM.

Practice Problem 2F 
In a certain city, telephone calls to 911 emergency response system arrive on the average of two every 3 minutes. Suppose that the arrivals of 911 calls are modeled by a Poisson process.

Practice Problem 2G 
Customers arrive at a store at an average rate of 30 per hour according to a Poisson process.

Practice Problem 2H 
Cars arrive at a highway tollbooth at an average rate of 6 cars every 10 minutes according to a Poisson process.

Practice Problem 2I 
The number of claims in a year for an insured from a large group of insureds is modeled by the following model.
The parameter varies from insured to insured. However, it is known that is modeled by the following density function. An insured is randomly selected from the large group of insureds. Determine the mean and the variance of the number of claims for this insured in the next year. 
Practice Problem 2J 
Suppose that the number of accidents per year per driver in a large group of insured drivers follows a Poisson distribution with mean . The parameter follows a gamma distribution with mean 0.9 and variance 0.27.
Given that a randomly selected insured has at least one claim, determine the probability that the insured has more than one claim. 
Practice Problem 2K 
Customers arrive at a shop according to a Poisson process. The waiting time (in minutes) until the 5th customer is modeled by the following density function.

Problem  ………..Answer 

2A 

2B 

2C 

2D 

2E 

2F 

2G 

2H 

2I 

2J 

2K 

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Tagged: Gamma Distribution, Poisson Distribution
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