By definition a **discrete probability distribution** exists if the following two requirements are met.

- It exists a set of random variables [Error: Macro 'me' doesn't exist]
and each of these variables has a corresponding probability [Error: Macro 'me' doesn't exist]
with

[Error: Macro 'me' doesn't exist]

- All probabilities always sum up to 1

[Error: Macro 'me' doesn't exist]

### 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 [Error: Macro 'me' doesn't exist]
the set of probabilities [Error: Macro 'me' doesn't exist]
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.

[Error: Macro 'me' doesn't exist]

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