Continuous distributions introduction

Continuous distributions can take any number of values over a certain range for x. This range may be infinite (e.g. for the Normal distribution) in which case we speak of an unbounded distribution or finite (e.g. the Uniform distribution) in which case we speak of a bounded distribution.

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

Note that in some cases the moment formulae 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 continuous distributions

The table below gives an overview of various continuous 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.

Distribution	Example use
Beta	Models uncertainty or variation of a probability, fraction or prevalence.
Bradford	Description of population variation.
Cauchy	Models the points of impact of a fixed straight line of particles emitted from a point source. Ratio of two Normal distributions.
Chi Squared	The sum of unit Normal distributions squared. Used widely in classical statistics where sample measures can be transformed to be approximately a sum of unit Normal distributions squared too.
Cumulative ascending	Used to create an empirically-based distribution. Useful in creating a non-parametric fitted to data.
Cumulative descending	Another form of the cumulative distribution. Uses the probability of being greater than or equal to their corresponding x-values.
Dirichlet	Used to describe uncertainty about the probabilities of a Multinomial distribution: a multi-dimensional version of a Beta distribution.
Error	Another format for the Normal distribution with a zero mean.
Erlang	A special case of the Gamma distribution where the first parameter is discrete.
Exponential	Models the time until an event occurs in a Poisson process.
Extreme value	Models the distribution of the extreme values that a variable can take.
F	Used in statistics to compare the variance between two (assumed Normally distributed) populations.
Frechet	Models the distribution of the extreme values that a variable can take.
Gamma	Models the time until a number of events occurs in a Poisson process.
General	Used to create an empirically-based distribution from relative frequency data.
Generalized Extreme Value	A distribution that incorporates the Fréchet, Weibull and Gumbel extreme value families of distributions.
Histogram	Useful for replicating the distribution shape of a set of data.
Inverse Gaussian	Models the time to cover a distance in Brownian motion.
JohnsonB	Fitting a bounded distribution to known moments.
JohnsonU	Fitting an unbounded distribution to known moments.
KernelCU	A kernel estimated distribution based on a set of data, assuming the variable is continuous (C) and unbounded (U).
Laplace	A symmetric distribution, useful for having longer tails than a Normal distribution.
Lifetime2	The lifetime of a device with a linear hazard rate.
Lifetime3	The lifetime of a device with a quadratic hazard rate.
LifetimeExp	The lifetime of a device with an exponentially increasing hazard rate.
LogLaplace	An asymmetric distribution which offers a greater variety of shapes.
Logistic	Popular in demographic and economic modeling, mostly as a growth equation. Similar to a Normal distribution, but more peaked.
LogLogistic	The log of the logistic distribution, so if X is loglogistically distributed, log X is logistically distributed.
Lognormal (format 1)	Useful for modeling naturally occurring variables that are the product of a number of other naturally occurring variables. If log X is Normally distributed, then X is lognormally distributed.
Lognormal (format 2)	An alternative way of defining a Lognormally distributed variable, using the mean and standard deviation of the corresponding Normal.
LogTriangle	Used when estimating a variable with a range covering orders of magnitude.
LogUniform	Used when estimating a variable with a range covering orders of magnitude.
Maxwell	A distribution often used in physics and engineering.
Normal	Models variations of naturally occurring variables. Also an approximate distribution to many other distributions in certain circumstances.
NormalAlt	Used for modeling expert estimates of some continuous quantity with equal tails, like time or cost to complete a task.
NormalMix	Used where a Normal distribution gives insufficient peakedness (kurtosis).
Pareto (1st kind)	Used to model any variable that has a minimum, and also its most likely, value and for which the probability density decreases geometrically towards zero. Often used because it has a very long right tail.
Pareto (2nd kind)	Is a shifted Pareto distribution.
Pearson V	A member of the Pearson system of distributions, and little used.
Pearson VI	Another member of the Pearson system of distributions.
PERT	A smoothed, triangular-like distribution, based on the Beta distribution.
Modified PERT	A more controllable version of the PERT distribution.
Ogive	A distribution based on the cumulative distribution of a data set.
Rayleigh	A special case of the Weibull distribution. Models distance to nearest neighbour where they are Poisson distributed in space.
Skew Normal	Used where a Normal distribution gives insufficient lopsidedness (skewness).
Slash	Developed as a deviation to the Normal distribution to allow for fatter tails.
Student	Used in statistical estimation.
Uniform	Used as a very approximate model where there are very little or no available data.
Triangle (various)	Used as a rough modeling tool where the range and the most likely value within the range can be estimated.
Weibull	Used to model the time until occurrence of an event where the momentary probability of occurrence changes over time.
Weibull3	The lifetime of a device where there is no possibility of failure for some limited time.

A continuous distribution is used to represent a variable that can take any value within a defined range (domain). For example, the height of an adult English male picked at random will have a continuous distribution because the height of a person is essentially infinitely divisible. We could measure his height to the nearest centimetre, millimetre, tenth of a millimetre, etc. The scale can be repeatedly divided up generating more and more possible values.

Properties like time, mass and distance, that are infinitely divisible, are modelled using continuous distributions. In practice, we also use continuous distributions to model variables that are, in truth, discrete but where the gap between allowable values is insignificant: for example, project cost (which is discrete with steps of one penny, one cent, etc.), exchange rate (which is only quoted to a few significant figures), number of employees in a large organization, etc.

The vertical scale of a relative frequency plot of an input continuous probability distribution is the probability density. It does not represent the actual probability of the corresponding x-axis value since that probability is zero. Instead, it represents the probability per x-axis unit of generating a value within a very small range around the x-axis value.

In a continuous relative frequency distribution, the area under the curve must equal one. This means that the vertical scale must change according to the units used for the horizontal scale. For example, the figure below shows a theoretical distribution of the cost of a project using Normal(Ј4 200 000, Ј350 000).

Since this is a continuous distribution, the cost of the project being precisely Ј4M is zero. The vertical scale reads a value of 9.7x10-7 (about one in a million). The x-axis units are Ј1, so this y-axis reading means that there is a one in a million chance that the project cost will be Ј4M plus or minus 50p (a range of Ј1). By comparison, the figure below shows the same distribution but using million pounds as the scale i.e. Normal(4.2, 0.35). The y-axis value at x = Ј4M is 0.97, one million times the above value.

This does not however mean that there is a 97% chance of being between Ј3.5M and Ј4.5M, because the probability density varies very considerably over that range. The logic used in interpreting the 9. 7x10-7 value for the first figure is an approximation that is valid there because the probability density is essentially constant over that range (Ј4M +/- 50p).

The links below discuss different ways of categorizing distributions that may help in your selection of the most appropriate distribution to use: