### Terminology

A*probability space*consists of a triple

*(Ω,A,P)*where

*Ω*is the sample space (the points of which are called sample points or elementary events),

*A*is the σ-algebra of events (subsets of

*Ω*which are assigned a probability), and

*P*is the probability measure on

*A*(a nonnegative σ-additive set function such that

*P(Ω) = 1).*

A

*real-valued random variable*on the probability space

*(Ω,A,P)*is a Borel measureable function

*X:Ω→R*, i.e.

*X*for any Borel set

^{ -1}(B) ∈ A*B*.

Given a random variable

*X:Ω→R*, the

*probability distribution*of

*X*is

*P*for every Borel set

_{X}(B) = P({ω∈Ω | X(ω) ∈ B}) = P(X ∈ B) = P(X^{ -1}(B))*B*.

Two definitions of the distribution function,

*F:R→[0,1]*, of a real-valued random variable

*X:Ω→R*are in use:

*F(x) = P({ω∈Ω | X(ω) < x}) = P(X < x)*and

*F(x) = P({ω∈Ω | X(ω) ≤ x}) = P(X ≤ x)*

*F*is a monotonic nondecreasing function for which

*lim*and

_{x→-∞}F(x) = 0*lim*.

_{x→∞}F(x) = 1For the first definition,

*F*is continuous from below and for the second definition

*F*is continuous from above.

Concerning the names of the

*C*functions, the term

*distribution function*refers to the first definition and the qualified term

*cumulative distribution function*refers to the second definition.

Note that the only discontinuities of a distribution function

*F*are jumps and if

*F*is continuous, then

*P(X=x) = 0*so that both definitions agree.

Given the distribution function

*F*of a random variable

*X*, if there exists a function

*f:R→R*such that

*F(x) = ∫*then

_{-∞}^{x}f(t) dt*f*is called the

*probability density function*of

*F*. Note that if the probability density function exists, then the distribution function

*F(x)*is necessarily continuous.

If the random variable

*X*assumes values in a finite subset or possibly in a countably infinite nowhere dense subset of

*R*, then if

*X(ω) = x*, then there exists an open interval

_{i}*I*containing

*x*and not containing any other point assumed by

_{i}*X*such that

*P(X = {x*, i.e. the probability distribution of

_{i}}) = P(X ∈ I)*X*is concentrated at discrete points. In this case the random variable

*X*is said to be discrete and the probability distribution of

*X*, the distribution function of

*X*and the cumulative distribution function of

*X*can be calculated from the individual probabilities

*P*.

_{X}(x_{i}) = P(X = {x_{i}})For discrete real-valued random variables the names of the

*C*functions return the probabilities

*P*are called

_{X}(x_{i})*point distributions*.

Currently all the distribution functions of real-valued random variables for which a

*C*function is available at this website are either continuous or discrete. In the advent that a distribution function which is neither continuous nor discrete is programmed both forms of the distribution function will be available.

### Distribution Functions, Cumulative Distribution Functions, Densities, Point Distributions, Tables of Various Probability Distributions

#### Continuous Probability Distributions

- Beta Distribution
- Cauchy Distribution
- Chi Square Distribution
- Exponential Distribution
- F Distribution
- Gamma Distribution
- Gaussian Distribution
- Gumbel's Extreme Value Maximum Distribution
- Gumbel's Extreme Value Minimum Distribution
- Kolmogorov's Asymptotic Distribution
- Kumaraswamy's Distribution
- Laplace's Distribution
- Logistic Distribution
- Pareto's Distribution
- Student's T Distribution
- Student's T Distribution with 2 Degrees of Freedom
- Uniform (0,1) Distribution
- Weibull Distribution

#### Discrete Probability Distributions

- Binomial Distribution
- Geometric Distribution
- Hypergeometric Distribution
- Log Series Distribution
- Negative Binomial Distribution
- Poisson Distribution

## Random Variates

Monte Carlo simulation requires that random variables be generated according to some preassigned distribution function. The classical methods for generating such random variates are: the inversion method, the rejection methods, Marsalia's ziggurat method, table look-up methods, polar methods etc. The fastest method is usually a table look-up method, but such a method requires significant setup time and is probably better suited to be programmed using an object-oriented language. The methods which are presented below do not use a table look-up method, but rather include the other methods mentioned.### Random Variates

- Bernoulli Random Variate
- Beta Random Variate
- Binomial Random Variate
- Cauchy Random Variate
- Exponential Random Variate
- Gamma Random Variate
- Gaussian Random Variate
- Geometric Random Variate
- Gumbel's Extreme Value Maximum Random Variate
- Gumbel's Extreme Value Minimum Random Variate
- Kumaraswamy's Random Variate
- Laplace's Random Variate
- Logistic Random Variate
- Log Series Random Variate
- Negative Binomial Random Variate
- Pareto's Random Variate
- Poisson Random Variate
- Student's T with 2 degrees of freedom Random Variate
- Uniform (0,1) Random Variate
- Weibull Random Variate