Discrete distributions

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Discrete distributions can only take a discrete number of values. This number may be infinite (e.g. for the Poisson distribution) or finite (e.g. the Bernoulli distribution).

For every discrete distribution available in ModelRisk, we have also included the following equations:

Probability Mass Function
Cumulative Distribution Function
Parameter restriction
Domain (i.e. x-range)
Mean
Mode
Variance
Skewness
Kurtosis

Note that in some cases the moment formulas can grow extremely complicated, in which case it does not make a lot of sense anymore to display it entirely. In those cases you will see complicated instead of the exact moment formula.

List of discrete distributions

The table below gives an overview of various discrete distributions commonly used in risk analysis modeling, so that you can most easily focus on which ones might be most appropriate for your modeling needs. Follow the links for an in-depth explanation of each. We have used the most common name for each distribution.

Distributions	Example use
Bernoulli	Returns a 1 with probability p and a zero otherwise.
Binomial	Shows the number of successes from n independent trials where there is a probability p of success in each trial.
Beta-binomial	A binomial variable where p is also a Beta-distributed random variable.
Burnt Finger Poisson	Like a Poisson, except that the risk of an event decreases after it has happened one time because one becomes more ‘cautious’.
Discrete	Describes a variable that can take one of several explicit discrete values with different probabilities. Returns the difference between two independent Poisson distributions.
Discrete uniform	Describes a variable that can take one of several explicit discrete values with equal probabilities.
Geometric	Models the number of failures that will occur before a success, given that p is the probability of succeeding.
Hypergeometric	Models the number of items of a particular type there will be in a sample of size n where that sample is drawn from a population of size M of which D are also of that particular type.
HypergeoD	Used to estimate the size of a sub-population in a Hypergeometric process.
HypergeoM	Used to estimate the size of a population in a Hypergeometric process.
Inverse Hypergeometric	Models the number of failures one would have before achieving the s-th success in a hypergeometric sampling.
Logarithmic	A one parameter, positive distribution sometimes used to model frequency of insurance claims. Also used for insect species abundance
Multinomial	An extension of the Binomial distribution where more than two different states of a trial exist.
Multivariate Hypergeometric	An extension of the Hypergeometric distribution where more than two sub-populations of interest exist.
Negative Binomial	Models the number of failures there will be before s successes are achieved where there is a probability p of success with each trial. Also models a Poisson random variable whose mean is a (Gamma) random variable.
Poisson	Models the number of occurrences of an event in a time t when the time between successive events follows a Poisson process
PoissonUniform	Events occur randomly with a randomly (Uniformly) varying risk level.
Skellam	The difference in number between two Poisson distributed events.
Poisson	Models the number of occurrences of an event in a time t when the time between successive events follows a Poisson process

A discrete distribution may take one of a set of identifiable values, each of which has a calculable probability of occurrence. Discrete distributions are used to model parameters like the number of bridges a roading scheme may need, the number of key personnel to be employed or the number of customers that will arrive at a service station in a hour. Clearly, variables such as these can only take specific values: one cannot build half a bridge, employ 2.7 people or serve 13.6 customers.

The vertical scale of a relative frequency plot of a discrete distribution is the actual probability of occurrence, sometimes called the probability mass. These probabilities must sum to one.

The most common examples of discrete distributions are: Binomial, Geometric, Hypergeometric, Negative Binomial, Poisson and, of course, the generalised Discrete distribution.