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?

- Mar 11th 2009, 10:33 PMMathRules!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?

- Mar 11th 2009, 11:09 PMmatheagle
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.