Probability mass function
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In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function in that the values of the latter, defined only for continuous random variables, are not probabilities; rather, its integral over a set of possible values of the random variable is a probability.
Mathematical description
Suppose that X is a discrete random variable, taking values on some countable sample space S ⊆ R. Then the probability mass function fX(x) for X is given by
- <math>f_X(x) = \begin{cases} \Pr(X = x), &x\in S,\\0, &x\in \mathbb{R}\backslash S.\end{cases}<math>
Note that this explicitly defines fX(x) for all real numbers, including all values in R that X could never take; indeed, it assigns such values a probability of zero. (Alternatively, think of Pr(X = x) as 0 when x ∈ R\S.)
The discontinuity of probability mass functions reflects the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable (i.e. where x ∈ R\S) the derivative is zero, just as the probability mass function is zero at all such points.
Examples
A simple example of a probability mass function is the following. Suppose that X is the outcome of a single coin toss, assigning 0 to tails and 1 to heads. The probability that X = x is just 0.5 on the state space {0, 1} (this is a Bernoulli random variable), and hence the probability mass function is
- <math>f_X(x) = \begin{cases}0.5, &x \in \{0, 1\},\\0, &x \in \mathbb{R}\backslash\{0, 1\}.\end{cases}<math>
Probability mass functions may also be defined for any discrete random variable, including constant, binomial (including Bernoulli), negative binomial, Poisson, geometric and hypergeometric random variables.