Distributions are used in risk analysis to model three conceptually diff= erent things:
The variability of individuals in a population (frequenc= y distribution);
The value of a random variable (probability distribution= ); and
The uncertainty we have about a fixed, but= imprecisely known, parameter in nature (uncertainty distribution).
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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 interrelationships 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 selftest quizzes:
A quiz on Distributions:
A quiz on Probability distribution identities (intermediate level): = ;
A quiz on Probability distribution identities (advanced level): Take quiz
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:
The binomial coefficient is defined as follows:
where: n! =3D 1 * 2 * 3 * .. * (n<= /em>1) * n
In Excel use =3DCOMBIN(n,r). The Excel function FACT(n) also returns n!
=CE=93(n) is the gamma function and has the follow= ing properties:
=CE=93(n+1) =3D n=CE=93(
=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
tan1(x) 
The arc tan of x in radians. =3DATAN(x) in Ex= cel 

x 
The absolute value of x, i.e. x =3D x, x= =3D x. Use =3DABS(x) in Excel 
csc(x) 
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 
ln(x) 
The natural log of x, so that x =3D exp(ln(x)= ). Use =3DLN(x) in Excel 
exp(x), ex 
The natural exponent of x, =3D 2.718281828..<= span>x. Use =3DEXP(x) in Excel 