A discrete random variable is one that can only take on a countable number of discrete values, e.g. the score on a die. Discrete random variables are often counts, e.g. the number of goals scored in a football match.

## Probability Function

The probability function for a discrete random variable gives the probability of each possible value.

E.g. The following table gives the probability function for the score on a fair die:

x_{i} |
1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

P(X=x)_{i} |

Clearly for any distribution.

## Binomial Distribution

The binomial distribution gives the probability of obtaining `X` successes from `n` trials. The trials must be independent of each other, and the probability of success, `p`, must be constant. `n` and `p` are known as the parameters of the distribution.

indicates that `X` is binomially distributed with parameters `n` and `p`.

The probability function is given by:

where is the binomial coefficient (also written as ^{n}C_{r}).

The mean is given by:

and the variance by:

## Poisson Distribution

The Poisson distribution gives the probability of `X` events occurring in a given time interval, e.g. the number of suicides per year in a given population. The events are independent and cannot occur simultaneously (although they can occur very close together). The mean number of events occurring in the given time interval is constant and known as the Poisson parameter.

indicates that `X` has a Poisson distribution with parameter `λ`.

The probability function is given by:

The mean is given by:

and the variance by:

The Poisson distribution can be used as an approximation to the binomial distribution if `n` is large and `p` is small.

Then: