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

Figure 1

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

Figure 2

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