Probability mass function
Encyclopedia
In probability theory
and statistics
, a probability mass function (pmf) is a function that gives the probability that a discrete random variable
is exactly equal to some value. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variable
s, given that the distribution is discrete.
A probability mass function differs from a probability density function
(p.d.f.) in that the latter is associated with continuous rather than discrete random variables; the values of the latter are not probabilities as such: a p.d.f. must be integrated over an interval to yield a probability.
Suppose that X: S → A (A 
R) is a discrete random variable defined on a sample space S. Then the probability mass function fX: A → [0, 1] for X is defined as
Note that fX is defined for all real number
s, including those not in the image
of X; indeed, fX(x) = 0 for all x
X(S). Essentially the same definition applies for a discrete multivariate random variable
X: S → An, with scalar values being replaced by vector values.
The total probability for all X must equal 1
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 is related to the fact that the cumulative distribution function
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.
This is a special case of the binomial distribution.
An example of a multivariate discrete distribution, and of its pmf, is provided by the multinomial distribution.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a probability mass function (pmf) is a function that gives the probability that a discrete random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
is exactly equal to some value. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variable
Multivariate random variable
In mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose values is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.More formally, a multivariate random...
s, given that the distribution is discrete.
A probability mass function differs from a probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
(p.d.f.) in that the latter is associated with continuous rather than discrete random variables; the values of the latter are not probabilities as such: a p.d.f. must be integrated over an interval to yield a probability.
Formal definition
R) is a discrete random variable defined on a sample space S. Then the probability mass function fX: A → [0, 1] for X is defined asNote that fX is defined for all real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, including those not in the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
of X; indeed, fX(x) = 0 for all x
X(S). Essentially the same definition applies for a discrete multivariate random variableMultivariate random variable
In mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose values is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.More formally, a multivariate random...
X: S → An, with scalar values being replaced by vector values.
The total probability for all X must equal 1
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 is related to the fact that the cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
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.
Examples
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 isThis is a special case of the binomial distribution.
An example of a multivariate discrete distribution, and of its pmf, is provided by the multinomial distribution.