Psychology Wiki

Probability mass function

34,191pages on
this wiki

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory

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 descriptionEdit

Suppose that X is a discrete random variable, taking values on some countable sample space  SR. Then the probability mass function  fX(x)  for X is given by

f_X(x) = \begin{cases} \Pr(X = x), &x\in S,\\0, &x\in \mathbb{R}\backslash S.\end{cases}

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  xR\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 xR\S) the derivative is zero, just as the probability mass function is zero at all such points.


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

f_X(x) = \begin{cases}\frac{1}{2}, &x \in \{0, 1\},\\0, &x \in \mathbb{R}\backslash\{0, 1\}.\end{cases}

Probability mass functions may also be defined for any discrete random variable, including constant, binomial (including Bernoulli), negative binomial, Poisson, geometric and hypergeometric random

This page uses Creative Commons Licensed content from Wikipedia (view authors).

Around Wikia's network

Random Wiki