Continuous distributions | Vose Software

# Continuous distributions

Continuous distributions can take any value within a defined range. 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.

### 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.

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:

Bounded and unbounded distributions

Parametric and non-parametric distributions

Discrete distributions