Probability mass
In probability 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 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; the values of the probability density function are not probabilities as such: 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.
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
Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes x:
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