# How do I determine which Single Variable Discrete Model is appropriate to use?

I am taking a statistics course and we are studying Random Variables and Probability Models right now. I am learning a lot about the probability functions for different variables, but I am having a really hard time determining which model to use for any given question that is assigned. We have covered:

• Discrete Uniform
• Hypegeometric
• Binomial
• Negative Binomial
• Geometric
• Poisson

Does any one have any tips on knowing which one is appropriate to use in which situations?

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Welcome to MSE @TopGunCpp. It will help us answer your questions if you narrow it down a bit - this is pretty broad. Do you have examples of different situations for which you are confused about which probability function to use? –  Alexander Gruber Nov 3 '12 at 17:05
Unfortunately you are right, the question is quite broad. As I have been studying the content I notice that I deduce to use one method but it always seems like I make the wrong assumptions. I thought it might be beneficial to ask just in general, although I see my error now in asking such a broad question. Sorry about that :\ –  TopGunCpp Nov 4 '12 at 5:22
No problem @TopGunCpp. Feel free to edit the question if you think of something. –  Alexander Gruber Nov 4 '12 at 5:37
Dont apologize. You asked a question about concept. Abstract. They are broad by default. I respect you a hell of a lot more than the person who asks a specific question that they drew from their evening homework. And I lose respect for anyone who mocks you for asking a question that demands conceptual explanation, because if doing that is too difficult for them then they probably dont understand the material as well as they want to get credit for. –  CogitoErgoCogitoSum Feb 24 '13 at 21:21

I think the best thing to do, if you havent already, is to prove each distribution function from their assumptions and basic principles in probability. That is what I did.

The Binomial asks for $k$ discrete successes out of $n$ discrete chances.

The Poisson is essentially the Binomial. Except that the Poisson is taken over infinitely many infinitesimally small $n$ chances. Basically, we are asking for $k$ discrete successes over a continuous variable, such as time or distance.

They are easy to distinguish by asking what kind of variable are our successes (discrete or continuous) and what kind of variable is the domain it occurs in (discrete or continuous).

The hypergeometric is, oddly enough, also similar to the Binomial. Except instead of two discrete possibilities (success or fail), there are more than two discrete possibilities (such as sets A,B,C). When one of these sets contains no elements at all, you are left with the binomial.

The negative binomials is... based on the binomial. Coincidence? The only difference is that we are not asking for the probability of $k$ events in $n$ chances. Rather, we are asking for the probability that it will take $n$ chances to achieve $k$ successes.

I really dont know how to answer your question except to compel you to learn the basis for each. How each was proved from simpler probability concepts. All the ones I explained are fundamentally based on the binomial. But the equations are simplified from certain assumptions made about the problems. If you understand these assumptions you can run through the proofs yourself, and you will surely understand how to use each distribution.

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