In probability and statistics, a probability mass function (sometimes called probability function or frequency function) is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete probability density function. 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 variables whose domain is discrete.

A probability mass function differs from a continuous probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A continuous PDF must be integrated over an interval to yield a probability.

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

Formal definition

Probability mass function is the probability distribution of a discrete random variable defined by

, is a probability measure.

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

Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved.

Measure theoretic formulation

can be seen as a special case of two more general measure theoretic constructions: the distribution and the probability density function with respect to the counting measure. We make this more precise below.

is a probability space is a measurable space whose underlying σ-algebra is discrete provided its image is countable. The pushforward measure.

is a measure space with respect to the counting measure, if it exists, is the Radon–Nikodym derivative

is in fact a probability mass function.

, it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable) and to consider also values not in the image may be defined for all real numbers as shown in the figure.

has a countable.

The discontinuity of probability mass functions is related to the fact that the cumulative distribution function is always impossible. This statement isn't true for a continuous random variable. Discretization is the process of converting a continuous random variable into a discrete one.

Examples

Finite

There are three major distributions associated, the Bernoulli distribution, the binomial distribution and the geometric distribution.

• Bernoulli distribution: ber(p) is the sample space of all outcomes of a single toss of a fair coin

• . An example of the binomial distribution is the probability of getting exactly one 6 when someone rolls a fair die three times.

• . denotes the number of necessary coin tosses. 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.

• An example of a multivariate discrete distribution, and of its probability mass function, is provided by the multinomial 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

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

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

Further reading