Probability mass function: Difference between revisions

Revision as of 00:15, 31 October 2021

Revision as of 23:58, 5 January 2022

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{{Short description|Discrete-variable probability distribution}}

[[Image:Discrete probability distrib.svg|right|thumb|The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.]]

In [[probability theory|probability]] and [[statistics]], a '''probability mass function''' is a function that gives the probability that a [[discrete random variable]] is exactly equal to some value.<ref>{{cite book|author=Stewart, William J.| title=Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling|publisher=Princeton University Press|year=2011|isbn=978-1-4008-3281-1|page=105|url=https://books.google.com/books?id=ZfRyBS1WbAQC&pg=PT105}}</ref> Sometimes it is also known as the discrete density function. The probability mass function is often the primary means of defining a [[discrete probability distribution]], and such functions exist for either [[Scalar variable|scalar]] or [[multivariate random variable]]s whose [[Domain of a function|___domain]] is discrete.

A probability mass function differs from a [[probability density function]] (PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be [[integration (mathematics)|integrated]] over an interval to yield a probability.<ref name=":0">{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|___location=London|oclc=262680588}}</ref>

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==Formal definition==

Probability mass function is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities. It is the function <math>p:\mathbb{ \R}</math> <math>\rightarrowto [0,1]</math> defined by

{{Equation box 1

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|equation = <math>p_X(x) = P(X = x) </math>

</math>

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for <math>-\infin < x < \infin</math>,<ref name=":0" /> where <math>P</math> is a [[probability measure]]. <math>p_X(x)</math> can also be simplified as <math>p(x)</math>.<ref>{{Cite book|title=Engineering optimization : theory and practice| last=Rao, | first = Singiresu S., 1944-|date=1996|publisher=Wiley|isbn=0-471-55034-5|edition=3rd|___location=New York|oclc=62080932}}</ref>

The probabilities associated with all (hypothetical) values must be non-negative and sum up to 1,

:<math display="block">\sum_x p_X(x) = 1\quad </math> and <math display="block">\quad p_X(x)\geq 0.</math>.

Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes <math>x</math>.

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Now suppose that <math>(B, \mathcal B, \mu)</math> is a [[measure space]] equipped with the counting measure μ. The probability density function <math>f</math> of <math>X</math> with respect to the counting measure, if it exists, is the [[Radon–Nikodym derivative]] of the pushforward measure of <math>X</math> (with respect to the counting measure), so <math> f = d X_*P / d \mu</math> and <math>f</math> is a function from <math>B</math> to the non-negative reals. As a consequence, for any <math>b \in B</math> we have

:<math display="block">P(X=b)=P( X^{-1} b ) = (X_*P)b =</math><math> \int_{ b } f d \mu = f(b),</math>

demonstrating that <math>f</math> is in fact a probability mass function.

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0, &x \notin \{0, 1\}.

\end{cases}</math>

* [[Binomial distribution]], models the number of successes when someone draws n times with replacement. Each draw or experiment is independent, with two possible outcomes. The associated probability mass function is <math display="inline">\binom{n}{k} p^k (1-p)^{n-k}</math>. [[Image:Fair dice probability distribution.svg|right|thumb|The probability mass function of a [[Dice|fair die]]. All the numbers on the {{dice}} have an equal chance of appearing on top when the die stops rolling.]]{{pb}}An example of the binomial distribution is the probability of getting exactly one 6 when someone rolls a fair die three times.

* Geometric distribution describes the number of trials needed to get one success. Its probability mass function is <math display="inline">p_X(k) = (1-p)^{k-1} p</math>.{{pb}}An example is tossing a coin until the first "heads" appears. <math>p</math> denotes the probability of the outcome "heads", and <math>k</math> denotes the number of necessary coin tosses. {{pb}}Other distributions that can be modeled using a probability mass function are the [[categorical distribution]] (also known as the generalized Bernoulli distribution) and the [[multinomial distribution]].

* If the discrete distribution has two or more categories one of which may occur, whether or not these categories have a natural ordering, when there is only a single trial (draw) this is a categorical distribution.

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===Infinite===

*The following exponentially declining distribution is an example of a distribution with an infinite number of possible outcomes—all the positive integers: <math display="block">\text{Pr}(X=i)= \frac{1}{2^i}\qquad \text{for } i=1, 2, 3, \dots </math> Despite the infinite number of possible outcomes, the total probability mass is 1/2 + 1/4 + 1/8 + ⋯ = 1, satisfying the unit total probability requirement for a probability distribution.

*The following exponentially declining distribution is an example of a distribution with an infinite number of possible outcomes—all the positive integers: <math display="block">\text{Pr}(X=i)= \frac{1}{2^i}\qquad \text{for } i=1, 2, 3, \dots </math> Despite the infinite number of possible outcomes, the total probability mass is 1/2 + 1/4 + 1/8 + ⋯ = 1, satisfying the unit total probability requirement for a probability distribution.

==Multivariate case==