Probability mass function
In probability and statistics, a probability mass function (sometimes called probability function or frequency function[1]) is a function that gives the probability that a discrete random variable is exactly equal to some value.
[2] 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 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.
[3] The value of the random variable having the largest probability mass is called the mode.
[4] 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 as is the total probability for all hypothetical outcomes
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
with respect to the counting measure.
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.
whose restriction to singleton sets induces the probability mass function (as mentioned in the previous section)
(with respect to the counting measure), so
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
may be defined for all real numbers and
has a countable subset on which the probability mass function
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.
means that the casual event
is certain (it is true in 100% of the occurrences); on the contrary,
means that the casual event
This statement isn't true for a continuous random variable
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.
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.
