Probability Poisson Process and Gamma Distribution

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SUMMARY

The discussion centers on the relationship between the Poisson process and the Gamma distribution in probability theory. Specifically, the user Casey successfully calculates P(X>10) using cumulative Poisson values but struggles to understand the application of the Gamma distribution in part (b) of the problem. The key takeaway is that the Gamma distribution is used to model the waiting times between events in a Poisson process, as it generalizes the exponential distribution for multiple events. Understanding this relationship is crucial for solving problems involving event occurrences over time.

PREREQUISITES
  • Understanding of Poisson processes and their properties.
  • Familiarity with the Gamma distribution and its applications.
  • Knowledge of cumulative distribution functions (CDFs) for discrete distributions.
  • Basic probability theory, including concepts of event occurrence and waiting times.
NEXT STEPS
  • Study the derivation and properties of the Gamma distribution in relation to Poisson processes.
  • Learn how to apply the exponential distribution in modeling waiting times for a single event.
  • Explore the relationship between Poisson processes and other distributions, such as the negative binomial distribution.
  • Practice solving problems involving both Poisson and Gamma distributions to reinforce understanding.
USEFUL FOR

Students and professionals in statistics, data science, and operations research who are looking to deepen their understanding of probability distributions, particularly in the context of event-driven processes.

Saladsamurai
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Homework Statement



Screenshot2010-06-15at114822PM.png



The Attempt at a Solution



Part (a) is no problem, it is simply P(X>10) = 1 - P(X<=10) which requires the use of tabulated cumulative poisson values.

Part (b) is throwing for a loop. I know that I need to invoke the Gamma distribution since that is what the solution is doing. But I don't really understand why. I think that it is because there is some relationship between a Poisson process and the Gamma distribution, but I am not exactly sure what it is.

What I do know, is that for Poisson processes, the probability that an event occurs X = x number of times in 't' time depends on the average number of times the Poisson event occurs per unit time. But how can I use this knowledge to solve part (b)?

Any thoughts?
Casey



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Here is the solution if it helps to generate any ideas. I am just not sure why they are using the Gamma distribution.

Screenshot2010-06-16at120204AM.png
 
Last edited:
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Any thoughts on this one? For some reason, I feel like it would make more sense to use the exponential distribution, but again, I am not really sure why. I still don't see why the fact that it is a poisson process allows me to infer that I can or should be using a gamma distribution?
 

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