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I am learning about discrete probability distributions and found 2 definitions for Geometric Distributions from wikipedia: 1. The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...} 2. The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }

It seems like a subtle difference, but i'm having trouble wrapping my head around when i would use which? Any insights appreciated.

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I always thought about it like this: if the geometric random variable can assume $0$, then use $(1-p)^{n}p$. Else use $(1-p)^{n-1}p$. E.g. is it possible to have zero failures? –  PEV Nov 25 '10 at 14:51
    
Geometric always means "the number of trials before the state changes", i.e. from success to failure or vise-versa, and you have to interpret it according to what is being described. Given the simplicity of the distribution, the one to which an author is referring is almost always obvious after a second of thought. –  Josh Guffin Nov 26 '10 at 17:17

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When you would use which really depends on the random variable you're interested in. If you're interested in the number of trials needed to obtain the first success, use the first kind of geometric distribution. If you're interested in the number of successes you are able to achieve before the first failure occurs then use the second kind of geometric distribution. (Inverting the interpretation of the second version like this also requires you to redefine the success probability as $1-p$ and the failure probability as $p$.)

As the Wikipedia page says, "Which of these one calls 'the' geometric distribution is a matter of convention and convenience."

I find the two different versions confusing myself. When I read somewhere that $X$ has a "geometric distribution," the writer isn't always careful to specify which one is meant, and then I have to spend time figuring out which one it is. This confusion also spreads to the negative binomial distribution, which is a generalization of the geometric distribution. The exponential distribution, which is a continuous version of the geometric distribution, and the gamma distribution (a generalization of the exponential), have more than one definition, too. It's too bad that these weren't standardized with one definition for each, but part of the reason they weren't is that the different versions are useful in different scenarios.

In response to Josh Guffin's comment: Yes, in many contexts it is easy to figure out which geometric distribution a writer is referring to. However, in some it is not. Wikipedia's page on moment-generating functions is a classic example. The table there gives the mgf's for the negative binomial (and thus geometric), exponential, and gamma distributions, but it doesn't specify which convention for each one is being used.

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Thanks for the explanation. My current simple-minded thinking is to choose based on the answer to: "can success occur after 1 experiment?" if yes, i'd use the second form since it works for {0, 1, ...}. –  PPC-Coder Nov 26 '10 at 17:46

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