Distributions | Vose Software

# Distributions

Distributions by category

Discrete Univariate Distributions

Zero-modified counting distributions

Distributions are used in risk analysis to model three conceptually different things

1. The variability of individuals in a population (frequency distribution).

2. The value of a random variable (probability distribution).

3. The uncertainty we have about a fixed, but imprecisely known, parameter in nature.

### Organization of this section

Click on an image
to read the explanation associated with it.

This section is a resource to allow you to look up specific distributions by name, to see how they are used, learn how to generate their values when they are not directly available in ModelRisk, and review the most useful equations associated with each distribution. We have also described the inter-relationships between various distributions and provided links to related topics.

The distributions are split into two categories: discrete and continuous. We explain how what you need to think about when selecting a distribution to go in your model. We also provide a section that describes how you can approximate one distribution with another, or use recursive formulae, so that you can avoid possible parameter restrictions..

Finally, if none of the distributions we describe suit your purpose, you can always create your own! We show you a variety of ways to do that.

### Useful formulae and functions

When reviewing the formulae associated with each distribution you will frequently come across a number of unusual (though not very complicated!) mathematical functions that it is worth explaining here:

##### Binomial coefficient

The binomial coefficient is defined as follows:

where:  n! = 1 * 2 * 3 * .. * (n-1) * n

In Excel use =COMBIN(n,r). The Excel function FACT(n) also returns n!

##### The Gamma function

Γ(n) is the gamma function and has the following properties:

Γ(n+1) = nΓ(n) = n!       where n is an integer

Γ(0.5) = √π

Γ(0) = 1

So, for example, Γ (1.5) = 1/2 * Γ (0.5) = 1/2 √π . Excel offers the function GAMMALN( ) which returns the natural log of the Gamma function, so to get Γ(n) you write =EXP(GAMMALN(n)).

##### Other functions
 tan-1(x) The arc tan of x in radians. =ATAN(x) in Excel |x| The absolute value of x, i.e. |-x| = x, |x| = x. Use =ABS(x) in Excel csc(x) The cosecant of x, = 1/sin(x). Use =1/SIN(x) in Excel The nearest integer at or below x. Use =ROUNDDOWN(x,0) in Excel The nearest integer at or above x. Use =ROUNDUP(x,0) in Excel ln(x) The natural log of x, so that x = exp(ln(x)). Use =LN(x) in Excel exp(x), ex The natural exponent of x, = 2.718281828..x. Use =EXP(x) in Excel