What is the probability mass function? A probability mass function (pmf) gives the probability that a discrete random variable is exactly equal to some value (Wikipedia 2006). A probability mass function differs from a probability density function (Wikipedia 2006). This is because the probability density function is defined only for continuous random variables and does not result in a probability (Wikipedia 2006). Rather, its integral over a set of possible values of the random variable is a probability (Wikipedia 2006). Suppose that X is a discrete random variable, taking values on some countable sample space S ⊆ R (Wikipedia 2006). Then the probability mass function fX(x) for X is given by

(1)(Wikipedia 2006) Note that this explicitly defines fX(x) for all real numbers, including all values in R that X could never take (Wikipedia 2006). It assigns such values a probability of zero. (Alternatively, think of Pr(X = x) as 0 when x ∈ R\S.) (Wikipedia 2006). The discontinuity of probability mass functions reflects the fact that the cumulative distribution functionof a discrete random variable is also discontinuous (Wikipedia 2006). 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 (Wikipedia 2006). Example: This example is extracted from (Wikipedia 2006). 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

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 (Wikipedia 2006) References: Wikipedia. "Probability mass function", Wikimedia Foundation Inc, http://en.wikipedia.org/wiki/Probability_mass_function, Last accessed 14 February 2006. Brandon Hodgson