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

In 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.[1] 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 or multivariate random variables whose 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 integrated over an interval to yield a probability.[2]

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, and provides the possible values and their associated probabilities. It is the function defined by

for ,[2] where is a probability measure. can also be simplified as .[3]

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

and

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

Measure theoretic formulation

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

Suppose that is a probability space and that is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of . In this setting, a random variable is discrete provided its image is countable. The pushforward measure —called the distribution of in this context—is a probability measure on whose restriction to singleton sets induces the probability mass function (as mentioned in the previous section) since for each .

Now suppose that is a measure space equipped with the counting measure μ. The probability density function of with respect to the counting measure, if it exists, is the Radon–Nikodym derivative of the pushforward measure of (with respect to the counting measure), so and is a function from to the non-negative reals. As a consequence, for any we have

demonstrating that is in fact a probability mass function.

When there is a natural order among the potential outcomes , 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 of . That is, may be defined for all real numbers and for all as shown in the figure.

The image of has a countable subset on which the probability mass function is one. Consequently, the probability mass function is zero for all but a countable number of values of .

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

Examples

Main articles: Bernoulli distribution, Binomial distribution, and Geometric distribution

Finite

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

Infinite

The following exponentially declining distribution is an example of a distribution with an infinite number of possible outcomes—all the positive integers:

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

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

  1. ^ 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.
  2. ^ a b A 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)
  3. ^ Rao, Singiresu S. (1996). Engineering optimization : theory and practice (3rd ed.). New York: Wiley. ISBN 0-471-55034-5. OCLC 62080932.

Further reading