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Concept
Probability mass function
Summary
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. 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.
The value of the random variable having the largest probability mass is called the mode.
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 , where is a probability measure. can also be simplified as .
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 .
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.
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