Editing Probability theory (section)

==Treatment==
Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more.

===Motivation===
Consider an [[Experiment (probability theory)|experiment]] that can produce a number of outcomes. The set of all outcomes is called the ''[[sample space]]'' of the experiment. The ''[[power set]]'' of the sample space (or equivalently, the event space) is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called ''events''. In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, that event is said to have occurred.

Probability is a [[Function (mathematics)|way of assigning]] every "event" a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) be assigned a value of one. To qualify as a [[probability distribution]], the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that any of these events occurs is given by the sum of the probabilities of the events.<ref>{{cite book |last=Ross |first=Sheldon |title=A First Course in Probability |publisher=Pearson Prentice Hall |edition=8th |year=2010 |isbn=978-0-13-603313-4 |pages=26–27 |url=https://books.google.com/books?id=Bc1FAQAAIAAJ&pg=PA26 |access-date=2016-02-28 }}</ref>

The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that the probability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} has a probability of 1, that is, absolute certainty.

When doing calculations using the outcomes of an experiment, it is necessary that all those [[elementary event]]s have a number assigned to them. This is done using a [[random variable]]. A random variable is a function that assigns to each elementary event in the sample space a [[real number]]. This function is usually denoted by a capital letter.<ref>{{Cite book |title =Introduction to Probability and Mathematical Statistics |last1 =Bain |first1 =Lee J. |last2 =Engelhardt |first2 =Max |publisher =Brooks/Cole |location =[[Belmont, California]] |page =53 |isbn =978-0-534-38020-5 |edition =2nd |date =1992 }}</ref> In the case of a die, the assignment of a number to certain elementary events can be done using the [[identity function]]. This does not always work. For example, when [[coin flipping|flipping a coin]] the two possible outcomes are "heads" and "tails". In this example, the random variable ''X'' could assign to the outcome "heads" the number "0" (<math display="inline">X(\text{heads})=0</math>) and to the outcome "tails" the number "1" (<math>X(\text{tails})=1</math>).

===Discrete probability distributions===
{{Main|Discrete probability distribution}}

[[File:NYW-DK-Poisson(5).svg|thumb|300px|The [[Poisson distribution]], a discrete probability distribution]]

{{em|Discrete probability theory}} deals with events that occur in [[countable]] sample spaces.

Examples: Throwing [[dice]], experiments with [[deck of cards|decks of cards]], [[random walk]], and tossing [[coin]]s.

{{em|Classical definition}}:
Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see [[Classical definition of probability]].

For example, if the event is "occurrence of an even number when a dice is rolled", the probability is given by <math>\tfrac{3}{6}=\tfrac{1}{2}</math>, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.

{{em|Modern definition}}:
The modern definition starts with a [[Countable set|finite or countable set]] called the [[sample space]], which relates to the set of all ''possible outcomes'' in classical sense, denoted by <math>\Omega</math>. It is then assumed that for each element <math>x \in \Omega\,</math>, an intrinsic "probability" value <math>f(x)\,</math> is attached, which satisfies the following properties:
# <math>f(x)\in[0,1]\mbox{ for all }x\in \Omega\,;</math>
# <math>\sum_{x\in \Omega} f(x) = 1\,.</math>

That is, the probability function ''f''(''x'') lies between zero and one for every value of ''x'' in the sample space ''Ω'', and the sum of ''f''(''x'') over all values ''x'' in the sample space ''Ω'' is equal to 1. An {{em|[[Event (probability theory)|event]]}} is defined as any [[subset]] <math>E\,</math> of the sample space <math>\Omega\,</math>. The {{em|probability}} of the event <math>E\,</math> is defined as
:<math>P(E)=\sum_{x\in E} f(x)\,.</math>

So, the probability of the entire sample space is 1, and the probability of the null event is 0.

The function <math>f(x)\,</math> mapping a point in the sample space to the "probability" value is called a {{em|probability mass function}} abbreviated as {{em|pmf}}. 

===Continuous probability distributions===
{{Main|Continuous probability distribution}}

[[File:Gaussian distribution 2.jpg|thumb|300px|The [[normal distribution]], a continuous probability distribution]]

{{em|Continuous probability theory}} deals with events that occur in a continuous sample space.

{{em|Classical definition}}:
The classical definition breaks down when confronted with the continuous case. See [[Bertrand's paradox (probability)|Bertrand's paradox]].

{{em|Modern definition}}:
If the sample space of a random variable ''X'' is the set of [[real numbers]] (<math>\mathbb{R}</math>) or a subset thereof, then a function called the {{em|[[cumulative distribution function]]}} ({{em|CDF}}) <math>F\,</math> exists, defined by <math>F(x) = P(X\le x) \,</math>. That is, ''F''(''x'') returns the probability that ''X'' will be less than or equal to ''x''.

The CDF necessarily satisfies the following properties.
# <math>F\,</math> is a [[Monotonic function|monotonically non-decreasing]], [[right-continuous]] function;
# <math>\lim_{x\rightarrow -\infty} F(x)=0\,;</math>
# <math>\lim_{x\rightarrow \infty} F(x)=1\,.</math>

The random variable <math>X</math> is said to have a continuous probability distribution if the corresponding CDF <math>F</math> is continuous. If <math>F\,</math> is [[absolutely continuous]], then its derivative exists almost everywhere and integrating the derivative gives us the CDF back again. In this case, the random variable ''X'' is said to have a {{em|[[probability density function]]}} ({{em|PDF}}) or simply {{em|density}} <math>f(x)=\frac{dF(x)}{dx}\,.</math>

For a set <math>E \subseteq \mathbb{R}</math>, the probability of the random variable ''X'' being in <math>E\,</math> is
:<math>P(X\in E) = \int_{x\in E} dF(x)\,.</math>

In case the PDF exists, this can be written as
:<math>P(X\in E) = \int_{x\in E} f(x)\,dx\,.</math>

Whereas the ''PDF'' exists only for continuous random variables, the ''CDF'' exists for all random variables (including discrete random variables) that take values in <math>\mathbb{R}\,.</math>

These concepts can be generalized for [[Dimension|multidimensional]] cases on <math>\mathbb{R}^n</math> and other continuous sample spaces.

===Measure-theoretic probability theory===
The utility of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two.

An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a PDF of <math>(\delta[x] + \varphi(x))/2</math>, where <math>\delta[x]</math> is the [[Dirac delta function]].

Other distributions may not even be a mix, for example, the [[Cantor distribution]] has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using [[measure theory]] to define the [[probability space]]:

Given any set <math>\Omega\,</math> (also called {{em|sample space}}) and a [[sigma-algebra|σ-algebra]] <math>\mathcal{F}\,</math> on it, a [[measure (mathematics)|measure]] <math>P\,</math> defined on <math>\mathcal{F}\,</math> is called a {{em|probability measure}} if <math>P(\Omega)=1.\,</math>

If <math>\mathcal{F}\,</math> is the [[Borel algebra|Borel σ-algebra]] on the set of real numbers, then there is a unique probability measure on <math>\mathcal{F}\,</math> for any CDF, and vice versa. The measure corresponding to a CDF is said to be {{em|induced}} by the CDF. This measure coincides with the pmf for discrete variables and PDF for continuous variables, making the measure-theoretic approach free of fallacies.

The ''probability'' of a set <math>E\,</math> in the σ-algebra <math>\mathcal{F}\,</math> is defined as
<!--the correct formulation; X has nothing to do with it-->
:<math>P(E) = \int_{\omega\in E} \mu_F(d\omega)\,</math>
where the integration is with respect to the measure <math>\mu_F\,</math> induced by <math>F\,.</math>

Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside <math>\mathbb{R}^n</math>, as in the theory of [[stochastic process]]es. For example, to study [[Brownian motion]], probability is defined on a space of functions.

When it is convenient to work with a dominating measure, the [[Radon-Nikodym theorem]] is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure.  Discrete densities are usually defined as this derivative with respect to a [[counting measure]] over the set of all possible outcomes.  Densities for [[absolutely continuous]] distributions are usually defined as this derivative with respect to the [[Lebesgue measure]].  If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others;  separate proofs are not required for discrete and continuous distributions.