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In probability theory
Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
, a probability mass function (abbreviated pmf) is a function that gives the probability that a discrete random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
 is exactly equal to some value. A pmf differs from a probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 (abbreviated pdf) in that the values of a pdf, defined only for continuous random variables, are not probabilities as such. Instead, the integral of a pdf over a range of possible values (a, b] gives the probability of the random variable falling within that range.






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In probability theory
Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
, a probability mass function (abbreviated pmf) is a function that gives the probability that a discrete random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
 is exactly equal to some value. A pmf differs from a probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 (abbreviated pdf) in that the values of a pdf, defined only for continuous random variables, are not probabilities as such. Instead, the integral of a pdf over a range of possible values (a, b] gives the probability of the random variable falling within that range. See notation
Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
 for the meaning of (a, b].

Mathematical description

Suppose that X: S ? R is a discrete random variable defined on a sample space
Sample space

In probability theory, the sample space or universal sample space, often denoted S, O, or U , of an experiment or random trial and error is the set of all possible outcomes....
 S. Then the probability mass function fX: R ? [0, 1] for X is defined as Note that fX is defined for all real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, including those not in the image of X; indeed, fX(x) = 0 for all x ∉ X(S).

Since the image of X is countable, the probability mass function fX(x) is zero for all but a countable number of values of x. The discontinuity of probability mass functions reflects the fact that the cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 of a discrete random variable is also discontinuous. Where it is differentiable, the derivative is zero, just as the probability mass function is zero at all such points.

Example

Suppose that S is the sample space of all outcomes of a single toss of a fair coin, and X is the random variable defined on S assigning 0 to "tails" and 1 to "heads". Since the coin is fair, the probability mass function is

See also

  • Discrete probability distribution
    Discrete probability distribution

    Discrete probability distributions arise in the mathematical description of probability theory and statistical analysis in which the values that might be observed are restricted to being within a pre-defined list of possible values....