Practice Problem Set 2 – Poisson and Gamma

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.

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Practice Problem 2-A
Suppose that X is the useful working life (in years) of a brand new industrial machine. The following is the probability density function of X.

    \displaystyle f(t)=\frac{1}{24} \ \biggl(\frac{1}{5}\biggr)^5 \ t^4 \ e^{-\frac{1}{5} \ t} \ \ \ \ \ \ t>0

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.

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Practice Problem 2-B

The annual rainfall (in inches) in Western Colorado is modeled by a distribution with the following cumulative distribution function.

    \displaystyle F(x)=1-e^{-0.2 x}-0.2 \ x \ e^{-0.2 x}-0.02 \ x^2 \ e^{-0.2 x} \ \ \ \ \ \ \ 0<x<\infty

In a year in which the annual rainfall is above 20 inches, determine the probability that the annual rainfall is above 30 inches.

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Practice Problem 2-C

The annual rainfall (in inches) in Western Colorado is modeled by a distribution with the following cumulative distribution function.

    \displaystyle F(x)=1-e^{-0.2 x}-0.2 \ x \ e^{-0.2 x}-0.02 \ x^2 \ e^{-0.2 x} \ \ \ \ \ \ \ 0<x<\infty

Determine the mean and the variance of the annual rainfall in this region.

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Practice Problem 2-D

The repair time (in hours) for an industrial machine has a gamma distribution with mean 1.5 and variance 0.75.

  • Determine the probability that a repair time exceeds 2 hours.
  • Determine the probability that a repair time is at least 5 hours given that it already exceeds 2 hours.

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Practice Problem 2-E

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.

  • What is the probability that the first customer arrives at the store before 11 AM?
  • What is the probability that the first two customers arrive at the store before 11 AM?
  • What is the probability that the first three customers arrive at the store before 11 AM?
  • What is the probability that the first five customers arrive at the store before 11 AM?

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Practice Problem 2-F
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.

  • What is the probability of four or more calls arriving in a 5-minute period?
  • A call to the 911 system just ended. What is the probability that the wait time for the next 6 calls is more than 10 minutes?

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Practice Problem 2-G

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

  • Determine the probability that at least 5 customers arrive at the store in the first 10 minutes after opening on a given day.
  • Determine the probability that, after opening, it will take more than 15 minutes for the 6th customer to arrive at the store.

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

  • Determine the probability that the toll collector will have to wait longer than 20 minutes before collecting the seventh toll.
  • A toll collector just starts his shift. Determine the median time (in minutes) until he collects the first toll.

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Practice Problem 2-I
The number of claims in a year for an insured from a large group of insureds is modeled by the following model.

    \displaystyle P(X=x \lvert \lambda)=\frac{e^{-\lambda} \lambda^x}{x!} \ \ \ \ \ x=0,1,2,3,\cdots

The parameter \lambda varies from insured to insured. However, it is known that \lambda is modeled by the following density function.

    \displaystyle g(\lambda)=32 \ \lambda^2 \ e^{-4 \lambda} \ \ \ \ \ \ \lambda>0

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.

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Practice Problem 2-J
Suppose that the number of accidents per year per driver in a large group of insured drivers follows a Poisson distribution with mean \lambda. The parameter \lambda 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.

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Practice Problem 2-K

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.

    \displaystyle f(t)=324 \ t^4 \ e^{-6 \ t} \ \ \ \ \ \ t>0

  • Determine mean and variance of the time until the 6th customer after the opening of the shop on a given day.
  • Determine the probability that the wait for the 7th customer is longer than 2 minutes.

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Problem ………..Answer
2-A
  • \displaystyle \frac{103}{3} e^{-4}=0.6288
2-B
  • \displaystyle \frac{25}{13} e^{-2}=0.2603
2-C
  • mean = 15
  • variance = 75
2-D
  • \displaystyle P(X>2)=13 e^{-4}
  • \displaystyle P(X>5 \lvert X>2)=\frac{61}{13} \ e^{-6}=0.01163
2-E
  • \displaystyle 1-e^{-5}=0.99326
  • \displaystyle 1-6 e^{-5}=0.95957
  • \displaystyle 1-18.5 e^{-5}=0.87535
  • \displaystyle 1-\frac{1569}{24} e^{-5}=0.55951
2-F
  • \displaystyle 1-\frac{1301}{81} e^{-10/3}=0.4270
  • \displaystyle 1-\frac{197789}{729} e^{-20/3}=0.6547
2-G
  • \displaystyle 1-\frac{1569}{24} e^{-5}=0.5595
  • \displaystyle \frac{52383.28125}{120} e^{-7.5}=0.241436
2-H
  • \displaystyle 7457.8 e^{-12}=0.04582
  • \displaystyle \frac{\text{ln}(0.5)}{-0.6}=1.15525 min
2-I
  • mean = 0.75
  • variance = 0.9375
2-J
  • \displaystyle \frac{0.6561}{1.3^4}=0.22972
2-K
  • mean = 1, variance = 1/6
  • \displaystyle 7457.8 e^{-12}=0.04582

Daniel Ma mathematics

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3 thoughts on “Practice Problem Set 2 – Poisson and Gamma

  1. […] problems can be found here in a companion blog for practice […]

  2. […] Two sets of practice problems are found here and here. […]

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