Message-ID: <357077908.2918.1604105072194.JavaMail.confluence@modelassist.epixanalytics.com> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_2917_143641587.1604105072193" ------=_Part_2917_143641587.1604105072193 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Distributions

# Distributions

### Use of Distributions

Distributions are used in risk analysis to model three conceptually diff= erent things:

1. The variability of individuals in a population (frequenc= y distribution);

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

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

=20 =20 =20 =20
=20 =20 =20 =20 =20 =20 =20

This section is a resource to allow you to look up specific distribution= s by name, to see how they are used, learn how to generate their values whe= n they are not directly available in your preferred software, and review th= e most useful equations associated with each distribution. We have also des= cribed the inter-relationships between various distributions and provided l= inks to related topics.

The distributions are split into two categories: discrete and continuous. We explain what you need to think about when= selecting a distribution to go in your model. We also provide a se= ction that describes how you can approximate one distribution with another, or= use = recursive formulas, so that you can avoid the parameter restrictions se= t by your simulation software.

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.

Once you have reviewed the material in this section, you might like to t= est how much you have learned by taking the self-test quizzes:

A quiz on Distributions:

A quiz on Probability distribution identities (intermediate level): = ;

A quiz on Probability distribution identities (advanced level):

### Useful formulas and fun= ctions

When reviewing the formulae associated with each distribution you will f= requently come across a number of unusual (though not very complicated!) &n= bsp;mathematical functions that are worth explaining here:

#### Binomial coefficient

The binomial coefficient is defined as follows:

=20 =20 =20 where:  n! =3D 1 * 2 * 3 * .. * (n<= /em>-1) * n

In Excel use =3DCOMBIN(n,r). The Excel function FACT(n) also returns n!

#### The Gamma function

=CE=93(n) is the gamma function and has the follow= ing properties:

=20 =20 =20 =CE=93(n+1) =3D n=CE=93(n) =3D n!     &= nbsp; where n is an integer

=CE=93(0.5) =3D =E2=88=9A=CF=80

=CE=93(0) =3D 1

So, for example, =CE=93 (1.5) =3D 1/2 * =CE=93 (0.5) =3D 1/2 =E2=88=9A= =CF=80 . Excel offers the function GAMMALN( ) which returns the natural log of the= Gamma function, so to get =CE=93(n) you write =3DEXP= (GAMMALN(n)).

Other functions

=20 =20
tan-1(x) The arc tan of x in radians. =3DATAN(x) in Ex= cel The absolute value of x, i.e. |-x| =3D x, |x|= =3D x. Use =3DABS(x) in Excel The cosecant of x, =3D 1/sin(x). Use =3D1/SIN= (x) in Excel The nearest integer at or below x. Use =3DROU= NDDOWN(x,0) in Excel The nearest integer at or above x. Use =3DROU= NDUP(x,0) in Excel The natural log of x, so that x =3D exp(ln(x)= ). Use =3DLN(x) in Excel The natural exponent of x, =3D 2.718281828..<= span>x. Use =3DEXP(x) in Excel

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