Discrete distributions | Vose Software

# Discrete distributions

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. Beta-binomial A binomial variable where p is also a Beta-distributed random variable. BetaGeometric Models the number of failures that will occur before a success, given that p is also a Beta-distributed random variable. BetaNegBin Models the number of failures that will occur before a success, given that p is also a Beta-distributed random variable. Binomial Shows the number of successes from n independent trials where there is a probability p of success in each trial. 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’. Delaporte Models the number of occurrences of an event in a specific period where the expected rate of occurrence in that period is also a shifted -Gamma-distributed random variable. 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 Fitted An empirical discrete distribution constructed based on a set of data where the possible outcomes all lie a fixed distance Increment from each other. 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 D in a Hypergeometric process. HypergeoM Used to estimate the size of a population M 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. Polya Models the number of occurrences of an event in a specific period where the expected rate of occurrence in that period is also a Gamma-distributed random variable. Skellam The difference in number between two Poisson distributed events. Step Uniform Models a variable that can take, with equal probability, a value from a set of numbers equally spaced between the minimum and maximum.

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.