Editing Probability distribution (section)

==Discrete probability distribution==
{{Main|Probability mass function}}
[[File:Dice Distribution (bar).svg|thumb|250px|right|Figure 3: The [[probability mass function]] (pmf) <math>p(S)</math> specifies the probability distribution for the sum <math>S</math> of counts from two [[dice]]. For example, the figure shows that <math>p(11) = 2/36 = 1/18</math>. The pmf allows the computation of probabilities of events such as <math>P(X > 9) = 1/12 + 1/18 + 1/36 = 1/6</math>, and all other probabilities in the distribution.]]

[[File:Discrete probability distrib.svg|right|thumb|Figure 4: The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]
[[File:Discrete probability distribution.svg|right|thumb|Figure 5: The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...]]
[[File:Normal probability distribution.svg|right|thumb|Figure 6: ... of a continuous probability distribution, ...]]
[[File:Mixed probability distribution.svg|right|thumb|Figure 7: ... of a distribution which has both a continuous part and a discrete part]]

A '''discrete probability distribution''' is the probability distribution of a random variable that can take on only a countable number of values<ref>{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer| isbn=9780387878591| location=New York|pages=51|oclc=710149819}}</ref> ([[almost surely]])<ref>{{Cite book|title=Measure theory| last=Cohn|first=Donald L.|date=1993|publisher=Birkhäuser}}</ref> which means that the probability of any event <math>E</math> can be expressed as a (finite or [[Series (mathematics)|countably infinite]]) sum:
<math display="block">P(X\in E) = \sum_{\omega\in A \cap E} P(X = \omega),</math>
where <math>A</math> is a countable set with <math>P(X \in A) = 1</math>. Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a [[probability mass function]] <math>p(x) = P(X=x)</math>. In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if <math>p(n) = \tfrac{1}{2^n}</math> for <math>n = 1, 2, ...</math>, the sum of probabilities would be <math>1/2 + 1/4 + 1/8 + \dots = 1</math>.

Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], the [[negative binomial distribution]] and [[categorical distribution]].<ref name=":1" /> When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete, and which provides information about the population distribution. Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.

===Cumulative distribution function===
A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take.
Thus the cumulative distribution function has the form
<math display="block">F(x) = P(X \leq x) = \sum_{\omega \leq x} p(\omega).</math>
The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.

===Dirac delta representation===
A discrete probability distribution is often represented with [[Dirac measure]]s, also called one-point distributions (see below), the probability distributions of [[Degenerate distribution|deterministic random variable]]s. For any outcome <math>\omega</math>, let <math>\delta_\omega</math> be the Dirac measure concentrated at <math>\omega</math>. Given a discrete probability distribution, there is a countable set <math>A</math> with <math>P(X \in A) = 1</math> and a probability mass function <math>p</math>. If <math>E</math> is any event, then
<math display="block">P(X \in E) = \sum_{\omega \in A} p(\omega) \delta_\omega(E),</math>
or in short,
<math display="block">P_X = \sum_{\omega \in A} p(\omega) \delta_\omega.</math>

Similarly, discrete distributions can be represented with the [[Dirac delta function]] as a [[Generalized function|generalized]] [[probability density function]] <math>f</math>, where
<math display="block">f(x) = \sum_{\omega \in A} p(\omega) \delta(x - \omega),</math>
which means
<math display="block">P(X \in E) = \int_E f(x) \, dx = \sum_{\omega \in A} p(\omega) \int_E \delta(x - \omega) = \sum_{\omega \in A \cap E} p(\omega)</math>
for any event <math>E.</math><ref>{{Cite journal|last=Khuri|first=André I.|date=March 2004| title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology| language=en|volume=35|issue=2|pages=185–195| doi=10.1080/00207390310001638313|s2cid=122501973|issn=0020-739X}}</ref>

===Indicator-function representation===
For a discrete random variable <math>X</math>, let <math>u_0, u_1, \dots</math> be the values it can take with non-zero probability. Denote
<math display="block">\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots</math>
These are [[disjoint set]]s, and for such sets
<math display="block">P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.</math>
It follows that the probability that <math>X</math> takes any value except for <math>u_0, u_1, \dots</math> is zero, and thus one can write <math>X</math> as
<math display="block">X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)</math>
except on a set of probability zero, where <math>1_A</math> is the indicator function of <math>A</math>. This may serve as an alternative definition of discrete random variables.

===One-point distribution===
A special case is the discrete distribution of a random variable that can take on only one fixed value, in other words, a Dirac measure. Expressed formally, the random variable <math>X</math> has a one-point distribution if it has a possible outcome <math>x</math> such that <math>P(X{=}x)=1.</math><ref>{{cite book |title=Probability Theory and Mathematical Statistics |first=Marek |last=Fisz |edition=3rd |publisher=John Wiley & Sons |year=1963 |isbn=0-471-26250-1 |page=129}}</ref> All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 before <math>x</math> to 1 at <math>x</math>. It is closely related to a deterministic distribution, which cannot take on any other value, while a one-point distribution can take other values, though only with probability 0. For most practical purposes the two notions are equivalent.