The assigning of probabilties to events generates a function referred to as a probability function. A probabilty function takes as an input any event in the sample space and outputs a real number from 0 to 1.

• A Brief Review of Functions - from Cut-the-Knot
  
  

    There are two distinct types of probability functions, discrete and continuous as determined by the sample space. Sample spaces that contain a finite or countably infinite number of outcomes are called discrete, while sample spaces having an uncountably infinite number of outcomes are called continuous. This is important since discrete and continuous probability functions have fundamentally different interpretations and they are treated mathematically in very different ways. To better get a handle on the difference we need to review the concept of the Power Set of a set.

    Let S bet a set, in our case the sample space for an experiment. The power set of S, denoted
Ã(S), is the set of all the possible subsets of S. A probability function P is defined as

    Discrete Probability Functions

    In the discrete case, the assignment of probabilities to events, or probability measure, is a discrete probability function between the collection of subsets of the sample space and the real numbers that satisfies the three Kolmogorov axioms:

        Axiom 1: The probability of an event is a nonnegative real number; that is, P(A) ³ 0 for any subset A of S.

        Axiom 2: P(S) = 1

        Axiom 3: If A₁,A₂,A₃ ... is a finite or infinite sequence of mutually exclusive events of S, then

    Continuous Probability Functions

    Continuous sample spaces arise in practice whenever the outcomes of experiments are measuremnts of physical properties, such as temperture, times, speed, pressure,..., that are measured on continuous scales. The interpretation of a continuous probability function is complicated by the fact that the Kolmogorov axioms need to be modified. (give an example) As our author says, "It is a curious mathematical fact that the Kolmogorov axioms are inconsistent with the notion that every subset of every sample space has a probability." In order to handle this mathematicians have created what is called measurable spaces composed of collections of subsets referred to as σ-algebras.

    σ-algebras from Wikipedia.

A σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory.

Formally, X is a σ-algebra if and only if it has the following properties:

1. The empty set is in X,
2. If E is in X then so is the complement of E.
3. If E₁, E₂, E₃, ... is a sequence in X then their (countable) union is also in X.

From 1 and 2 it follows that S is in X; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections.

An ordered pair (S, X), where S is a set and X is a σ-algebra over S, is called a measurable space.

    If S is a sample space with an uncountable number of outcomes and if f is a real valued function defined on S, then f is said to be a continuous probability function if:

        Axiom 1:If y is an outcome         

        Axiom 2:

        Axiom 3: If A is any event defined on S,

    If f is a continuous probability function, f(y) is not the probability that the outcome of the experiment is y. Rather, f is that particular function that has the property that for any event A, the probability of A, P(A), is the integral of f over A. (example of area under a curve)

    Probabilities 0 and 1

    In the case of the continuous probability function...

    Random Selection of Points from Intervals

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