By definition a discrete probability distribution exists if the following two requirements are met.
 It exists a set of random variables and each of these variables has a corresponding probability with
 All probabilities always sum up to 1
Special cases
Arbitrary finite number of random variables
Any arbitrary function that maps a probability to an arbitrary set of random variables is a probability distribution if the above mentioned requirements are met. For instance for the set of random variables the set of probabilities are a discrete probability distribution as all these probabilities sum up to 1.
Two random variables
In this case, only one probability is needed as the other can be calculated by the sum which is always one.
This distribution is called Bernulli distribution.
One random variable
This is an exceptional case which is 100% certain. For instance the sun rises every morning. It might not be useful to consider such a case, as it has no practical consequences in statistics. It is however important to develop the fauceir abstraction of control.
Infinite number of random variables
The requirements are met if the sum of probabilities converges to 1 as in covergent series. Conclusively every series converging to 1 can be considered a random distribution whatever its practical implication.
Named discrete probability distributions
 Bernulli distribution
 Binomial distribution
 Poisson distribution
 Geometric distribution
 Zipf distribution
Tags: Statistics
Categories: Mathematics
