# Thread: Difference between binomial and negative binomial distribution

1. ## Difference between binomial and negative binomial distribution

If I were given a word problem, how would I know which distribution to use such as if the material were to cover bernoulii, binomial, and negative binomial distribution?

2. All involve indep trials with p=P(success on one trial).
Now the Bernoulli is just a Binomial with n=1.
Hence P(X=1)=p and P(X=0)=1-p.
Now the Binomial is a sum of n Bernoullis
Here you are count how many successes in n trial.

$\displaystyle P(X=x)={n\choose x}p^x(1-p)^{n-x}$ where $\displaystyle x=0,1\ldots ,n$.

The next distribution is really the Geometric. In this case the random variable, X,
is the trial on which the first success occurs. Now, some books count the trial on which
that success (say Godot appears) and some don't. Hence

$\displaystyle P(X=x)=p(1-p)^{x-1}$ where $\displaystyle x=1,2\ldots$.

Finally, we have the Neg Binomial, which should probably be called the multigeometric instead.
Here we are waiting for the $\displaystyle r^{th}$ success. So

$\displaystyle P(X=x)={x-1\choose r-1}p^r(1-p)^{x-r}$ where $\displaystyle x=r,1+1\ldots$.

Note, that if r=1, we have our geo once again.

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# distinguish between binomaial and negative binomaial distribution

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