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Probability mass function

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Discrete-variable probability distribution

The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.

Inprobability andstatistics, aprobability mass function (sometimes calledprobability function orfrequency function^[1]) is a function that gives the probability that adiscrete random variable is exactly equal to some value.^[2] Sometimes it is also known as thediscrete probability density function. The probability mass function is often the primary means of defining adiscrete probability distribution, and such functions exist for eitherscalar ormultivariate random variables whosedomain is discrete.

A probability mass function differs from acontinuous probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A continuous PDF must beintegrated over an interval to yield a probability.^[3]

The value of the random variable having the largest probability mass is called themode.

Formal definition

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Probability mass function is the probability distribution of adiscrete random variable, and provides the possible values and their associated probabilities. It is the function $p:\mathbb {R} \to [0,1]$ defined by

$p_{X}(x)=P(X=x)$

for $-\infty <x<\infty$ ,^[3] where $P {\displaystyle P}$ is aprobability measure. $p_{X}(x)$ can also be simplified as $p(x)$ .^[4]

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

$\sum _{x}p_{X}(x)=1$ and $p_{X}(x)\geq 0.$

Thinking of probability as mass helps to avoid mistakes since the physical mass isconserved as is the total probability for all hypothetical outcomes $x {\displaystyle x}$ .

Measure theoretic formulation

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A probability mass function of a discrete random variable $X {\displaystyle X}$ can be seen as a special case of two more general measure theoretic constructions: thedistribution of $X {\displaystyle X}$ and theprobability density function of $X {\displaystyle X}$ with respect to thecounting measure. We make this more precise below.

Suppose that $(A,{\mathcal {A}},P)$ is aprobability spaceand that $(B,{\mathcal {B}})$ is a measurable space whose underlyingσ-algebra is discrete, so in particular contains singleton sets of $B {\displaystyle B}$ . In this setting, a random variable $X\colon A\to B$ is discrete provided its image is countable.Thepushforward measure $X_{*}(P)$ —called the distribution of $X {\displaystyle X}$ in this context—is a probability measure on $B {\displaystyle B}$ whose restriction to singleton sets induces the probability mass function (as mentioned in the previous section) $f_{X}\colon B\to \mathbb {R}$ since $f_{X}(b)=P(X^{-1}(b))=P(X=b)$ for each $b\in B$ .

Now suppose that $(B,{\mathcal {B}},\mu )$ is ameasure space equipped with the counting measure $\mu$ . The probability density function $f {\displaystyle f}$ of $X {\displaystyle X}$ with respect to the counting measure, if it exists, is theRadon–Nikodym derivative of the pushforward measure of $X {\displaystyle X}$ (with respect to the counting measure), so $f=dX_{*}P/d\mu$ and $f {\displaystyle f}$ is a function from $B {\displaystyle B}$ to the non-negative reals. As a consequence, for any $b\in B$ we have $P(X=b)=P(X^{-1}(b))=X_{*}(P)(b)=\int _{b}fd\mu =f(b),$

demonstrating that $f {\displaystyle f}$ is in fact a probability mass function.

When there is a natural order among the potential outcomes $x {\displaystyle x}$ , it may be convenient to assign numerical values to them (orn-tuples in case of a discretemultivariate random variable) and to consider also values not in theimage of $X {\displaystyle X}$ . That is, $f_{X}$ may be defined for allreal numbers and $f_{X}(x)=0$ for all $x\notin X(S)$ as shown in the figure.

The image of $X {\displaystyle X}$ has acountable subset on which the probability mass function $f_{X}(x)$ is one. Consequently, the probability mass function is zero for all but a countable number of values of $x {\displaystyle x}$ .

The discontinuity of probability mass functions is related to the fact that thecumulative distribution function of a discrete random variable is also discontinuous. If $X {\displaystyle X}$ is a discrete random variable, then $P(X=x)=1$ means that the casual event $(X=x)$ is certain (it is true in 100% of the occurrences); on the contrary, $P(X=x)=0$ means that the casual event $(X=x)$ is always impossible. This statement isn't true for acontinuous random variable $X {\displaystyle X}$ , for which $P(X=x)=0$ for any possible $x {\displaystyle x}$ .Discretization is the process of converting a continuous random variable into a discrete one.

Examples

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Main articles:Bernoulli distribution,Binomial distribution, andGeometric distribution

Finite

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There are three major distributions associated, theBernoulli distribution, thebinomial distribution and thegeometric distribution.

Bernoulli distribution:ber(p), is used to model an experiment with only two possible outcomes. The two outcomes are often encoded as 1 and 0. $p_{X}(x)={\begin{cases}p,&{\text{if }}x{\text{ is 1}}\\1-p,&{\text{if }}x{\text{ is 0}}\end{cases}}$ An example of the Bernoulli distribution is tossing a coin. Suppose that $S {\displaystyle S}$ is the sample space of all outcomes of a single toss of afair coin, and $X {\displaystyle X}$ is the random variable defined on $S {\displaystyle S}$ assigning 0 to the category "tails" and 1 to the category "heads". Since the coin is fair, the probability mass function is $p_{X}(x)={\begin{cases}{\frac {1}{2}},&x=0,\\{\frac {1}{2}},&x=1,\\0,&x\notin \{0,1\}.\end{cases}}$
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 ${\textstyle {\binom {n}{k}}p^{k}(1-p)^{n-k}}$ .
The probability mass function of afair die. All the numbers on the die have an equal chance of appearing on top when the die stops rolling.
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 ${\textstyle p_{X}(k)=(1-p)^{k-1}p}$ .
An example is tossing a coin until the first "heads" appears. $p {\displaystyle p}$ denotes the probability of the outcome "heads", and $k {\displaystyle k}$ denotes the number of necessary coin tosses.
Other distributions that can be modeled using a probability mass function are thecategorical distribution (also known as the generalized Bernoulli distribution) and themultinomial 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.
An example of amultivariate discrete distribution, and of its probability mass function, is provided by themultinomial distribution. Here the multiple random variables are the numbers of successes in each of the categories after a given number of trials, and each non-zero probability mass gives the probability of a certain combination of numbers of successes in the various categories.

Infinite

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The following exponentially declining distribution is an example of a distribution with an infinite number of possible outcomes—all the positive integers: ${\text{Pr}}(X=i)={\frac {1}{2^{i}}}\qquad {\text{for }}i=1,2,3,\dots$ 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

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Main article:Joint probability distribution

Two or more discrete random variables have a joint probability mass function, which gives the probability of each possible combination of realizations for the random variables.

References

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^7.2 - Probability Mass Functions | STAT 414 - PennState - Eberly College of Science
^Stewart, William J. (2011).Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press. p. 105.ISBN 978-1-4008-3281-1.
^^a ^bA modern introduction to probability and statistics : understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005.ISBN 978-1-85233-896-1.OCLC 262680588.{{cite book}}: CS1 maint: others (link)
^Rao, Singiresu S. (1996).Engineering optimization : theory and practice (3rd ed.). New York: Wiley.ISBN 0-471-55034-5.OCLC 62080932.

v t e Theory ofprobability distributions
probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function
raw moment central moment mean variance standard deviation skewness kurtosis L-moment
moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant

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Formal definition

Measure theoretic formulation

Examples

Finite

Infinite

Multivariate case

References

Further reading