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.

where .

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

where .

Finally, we have the Neg Binomial, which should probably be called the multigeometric instead.

Here we are waiting for the success. So

where .

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