# Probability Distributions, Densities, Tables, and Random Variates

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### 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 -1(B) ∈ A for any Borel set B.
Given a random variable X:Ω→R, the probability distribution of X is PX(B) = P({ω∈Ω | X(ω) ∈ B}) = P(X ∈ B) = P(X -1(B)) for every Borel set 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)
For both definitions F is a monotonic nondecreasing function for which limx→-∞F(x) = 0 and limx→∞F(x) = 1.
For 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) = ∫-∞xf(t) dt then 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) = xi , then there exists an open interval I containing xi and not containing any other point assumed by X such that P(X = {xi}) = P(X ∈ I), i.e. the probability distribution of 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 PX(xi) = P(X = {xi}).
For discrete real-valued random variables the names of the C functions return the probabilities PX(xi) are called 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.

## 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.