binomial, geo and neg binomial should be considered together.
ALL of these are discrete by the way.
In those three, we do an experiment repeatedly
The chance of success is always p and it stays p from trial to trial
IN the bio case, we are doing this experiment n times.
n is fixed in advance
The random variable, X is the number of successes out of those n.
There are two outcomes and a success is one of the two you decide on in advance.
Then with the geo and the neg binomial, you still do this success/failure experiment repeatedly
HERE Y is a geometric, IT is the trial on which we observe the FIRST success.
Now some people say that's including that trial. some count all the previous trial and not that last one.
For example if we are waiting for a HEAD, when tossing a coin and we see TTH, then Y=3 or 2 depending on how you define that geo.
The neg bio is the same as a geo, but here you are waiing for the second or third.... success.
It can be viewed as the sum of r independent geometrics.
That is the easiest way to obtain its mean and variance