Finding Probability with Mean and Poisson Distribution

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Discussion Overview

The discussion revolves around determining the probability of an event occurring given only the mean of a distribution, specifically in the context of using the Poisson distribution. Participants explore the implications of having limited information about the distribution, including the absence of standard deviation.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to find the probability of an event with only the mean provided, suggesting a Poisson distribution may be applicable.
  • Another participant emphasizes that knowing only the mean does not provide sufficient information about the distribution, arguing that the standard deviation is necessary if assuming a normal distribution.
  • A participant asserts that if the context involves counting events like car accidents, a Poisson distribution is appropriate since it cannot yield negative values.
  • Some participants discuss the characteristics of Poisson processes and their relevance to the problem, noting that certain processes are inherently Poisson distributed.
  • There is a suggestion that the definition of a Poisson process is not clearly understood among participants, leading to confusion about its application.
  • One participant acknowledges the potential for misunderstanding in the phrasing of the problem, indicating a need for clarity in the question being addressed.
  • Another participant points out that the standard deviation of a Poisson distribution is equal to the mean, indicating that the mean alone provides more information than initially assumed.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of the standard deviation and the appropriateness of the Poisson distribution for the scenario described. The discussion remains unresolved regarding the best approach to calculate the probability given the constraints of the information available.

Contextual Notes

Participants acknowledge limitations in their understanding of the definitions and applications of probability distributions, particularly regarding the Poisson and Gaussian distributions. There is uncertainty about the assumptions underlying their reasoning.

Poop-Loops
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Is there a way to figure out the probability of getting a number if all you have is the mean? Everything I can find tells me I need the STD for that, but I don't have it given.
 
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Er, what? What does "getting a number" mean?

- Warren
 
Sorry. You have a mean number of an event occurring and then you want to find the probability of another event occurring (i.e. getting a number).

So for example you have that the mean is 5 and you want to find the probability of getting a 6.

And actually I'm fairly certain this is a Poisson distribution, so the STD is just the root of the mean. But, if you know of any way to find the probability using only the mean for a Gaussian distribution, please tell me, since I'm not 100% certain.
 
Yeah, you need the entire probability distribution. If all you know is the mean, you know nothing about the distribution. Even the standard deviation is useless, unless you actually know a priori that the distribution is a normal distribution.

- Warren
 
I got it now.

You had the number of car accidents per a night, so what was the probability of getting a number the next night. Since you can't have less than 0, it HAD to be a Poisson distribution and not a normal one, right? Then I don't need the STD to calculate the probability.
 
Sounds reasonable. I cannot say for sure that you're right, of course.

- Warren
 
Yeah, me neither, but it's the best thing I can think of.
 
Often, you would infer the distribution to use from what you're modelling. In particular, if you're modelling a Poisson process, then you'd use the Poisson distribution. If you aren't modelling a Poisson process, then you probably wouldn't use a Poisson distribution.
 
I know, I thought of that. I mean, Poisson stuff is used for things like radioactive decay, right? Not much to do with cars, which you would think would be a Gaussian distribution, but I can't find any formula for finding the probability of an event occurring if I don't have the STD, and also you can't have less than 0 events occurring, so you wouldn't be able to have the left tail of the distribution. We've only learned about two distributions in class, so I have to conclude that this is what I have to do. Or do you have any ideas?

EDIT: Actually, since we are given an interval (one night), and the accidents could occur at any time, it could follow a Poisson distribution... right?
 
Last edited:
  • #10
Poop-Loops said:
I know, I thought of that. I mean, Poisson stuff is used for things like radioactive decay, right? Not much to do with cars,
There's a definition of a Poisson process. Anything that is a Poisson process would be Poisson distributed. There's no reason to think different Poisson processes should have much to do with each other.

which you would think would be a Gaussian distribution
Why would you think that?
 
  • #11
...ummm... because most things are... and I have nothing else to go by really hahaha

But no, I think it's best described as a Poisson distribution from what I read in my book, since it can't be less than 0 to even out the mean and it has to do with intervals.
 
  • #12
Is that how your book defines "Poisson process"?
 
  • #13
It doesn't. It only gives explains when Poisson distributions are used.
 
  • #14
Poop-Loops said:
I got it now.

You had the number of car accidents per a night, so what was the probability of getting a number the next night. Since you can't have less than 0, it HAD to be a Poisson distribution and not a normal one, right? Then I don't need the STD to calculate the probability.

You are still talking very "sloppy". What could the "number of car accidents per night" possibly have to do with "getting a number the next night" (i.e. phone number of the girl you just met in a bar!).:smile:

Oh, wait a minute! Possibly you mean "given the mean number of car accidents a night, find the probability that there will be a given number of car accidents the next night". That's not at all what you said!:rolleyes:
You say "it HAD to be a Poisson distribution". And then say "I don't need the STD". Of course you do- you just assumed the STD: the Poisson distribution with the given mean. (One nice thing about the Poisson distribution is that it's standard deviation and other moments is the same as the mean- you are using a lot more information than you think!)
 

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