Summary of probability distributions

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Bernoulli process

An experiment whose dichotomous outcomes are random (coin toss, rolling a die, polls).

Beta Distribution

models events that are constrained to take place within an interval defined by a minimum and maximum value.The Beta probability density function is given by

\[\frac {x^{\alpha-1}(1-x)^{\beta-1}} {B(\alpha,\beta)}\]

where

  • α is a positive shape parameter
  • β is a positive shape parameter

The Beta distribution is used in a range of disciplines including rule of succession, Bayesian statistics, and task duration modeling. Examples of events that may be modeled by Beta distribution include:

  • The time it takes to complete a task
  • The proportion of defective items in a shipment

Geometric distribution

The probability distribution of the number of Bernoulli trials needed to get one success.

Hyper Geometric

Discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement.

Poisson distribution

A discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event, that is, it predicts the degree of spread around a known average rate of occurrence.

\[\frac {\lambda^k e^{-\lambda}} {k!}\]

where

  • e is the base of the natural logarithm (e = 2.71828…)
  • k! is the factorial of k
  • λ is a positive real number, equal to the expected number of occurrences during the given interval. As a function of k, this is the probability mass function. The parameter λ is not only the mean number of occurrences , but also its variance.

Gamma distribution

Gamma distribution is a distribution that arises naturally in processes for which the waiting times between events are relevant. It can be thought of as a waiting time between Poisson distributed events.
The waiting time until the hth Poisson event with a rate of change λ is

\[P(x)=\lambda(\lambda x)^{h-1}\]

The gamma distribution can be used a range of disciplines including queuing models, climatology, and financial services. Examples of events that may be modeled by gamma distribution include:

  • The amount of rainfall accumulated in a reservoir
  • The size of loan defaults or aggregate insurance claims
  • The flow of items through manufacturing and distribution processes
  • The load on web servers
  • The many and varied forms of telecom exchange

The gamma distribution is also used to model errors in a multi-level Poisson regression model because the combination of a Poisson distribution and a gamma distribution is a negative binomial distribution.

Exponential Distribution

A special case of the gamma distribution. The gamma distribution is the waiting time for more than one event, the exponential distribution describes the time between a single Poisson event. The exponential probability density function is given by

\[e^{-\lambda x}\]

where:

  • e is the natural number (e = 2.71828…)
  • λ is the mean time between events
  • x is a random variable

The exponential distribution occurs naturally when describing the waiting time in a homogeneous Poisson process. It can be used in a range of disciplines including queuing theory, physics, reliability theory, and hydrology. Examples of events that may be modeled by exponential distribution include:

  • The time until a radioactive particle decays
  • The time between clicks of a Geiger counter
  • The time until default on payment to company debt holders
  • The distance between roadkills on a given road
  • The distance between mutations on a DNA strand
  • The time it takes for a bank teller to serve a customer
  • The height of various molecules in a gas at a fixed temperature and pressure in a uniform gravitational field
  • The monthly and annual maximum values of daily rainfall and river discharge volumes

Pareto Distribution

A skewed, heavy-tailed distribution that is sometimes used to model that distribution of incomes. The basis of the distribution is that a high proportion of a population has low income while only a few people have very high incomes. The Pareto probability density function is given by

\[\frac {ax^a_m} {x^{a+1}}\]

where

  • $x_m$ is the minimum possible value of X
  • α is a positive parameter which determines the concentration of data towards the mode
  • x is a random variable ( $x > x_m$ )

The Pareto distribution is sometimes expressed more simply as the “80-20 rule”, which describes a range of situations. In customer support, it means that 80% of problems come from 20% of customers. In economics, it means 80% of the wealth is controlled by 20% of the population. Examples of events that may be modeled by Pareto distribution include:

  • The sizes of human settlements (few cities, many villages)
  • The file size distribution of Internet traffic which uses the TCP protocol (few larger files, many smaller files)
  • Hard disk drive error rates
  • The values of oil reserves in oil fields (few large fields, many small fields)
  • The length distribution in jobs assigned supercomputers (few large ones, many small ones)
  • The standardized price returns on individual stocks
  • The sizes of sand particles
  • The sizes of meteorites
  • The number of species per genus
  • The areas burned in forest fires
  • The severity of large casualty losses for certain businesses, such as general liability, commercial auto, and workers compensation