Probability mass function

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The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.

In probability theory, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. A pmf differs from a probability density function (pdf) in that the values of a pdf, defined only for continuous random variables, are not probabilities as such. Instead, the integral of a pdf over a range of possible values (a, b] gives the probability of the random variable falling within that range. See notation for the meaning of (a, b].

[edit] Mathematical description

The probability mass function of a fair die. All the numbers on the die have an equal chance of appearing on top when the die is rolled.

Suppose that X: S → R is a discrete random variable defined on a sample space S. Then the probability mass function f_X: R → [0, 1] for X is defined as

Note that f_X is defined for all real numbers, including those not in the image of X; indeed, f_X(x) = 0 for all x ∉ X(S).

Since the image of X is countable, the probability mass function f_X(x) is zero for all but a countable number of values of x. 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, the derivative is zero, just as the probability mass function is zero at all such points.

[edit] Example

Suppose that S is the sample space of all outcomes of a single toss of a fair coin, and X is the random variable defined on S assigning 0 to "tails" and 1 to "heads". Since the coin is fair, the probability mass function is

[edit] See also

• Discrete probability distribution

[edit] References

• Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9 (p36)

v • d • e Theory of probability distributions probability mass function (pmf) · probability density function (pdf) · cumulative distribution function (cdf) · quantile function raw moment · central moment · mean · variance · standard deviation · skewness · kurtosis moment-generating function (mgf) · characteristic function · probability-generating function (pgf) · cumulant