Probability mass function |
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
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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
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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.
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