Probability mass function 

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 (abbreviated 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 (abbreviated 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.

Contents

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 fX: R → [0, 1] for X is defined as

f_X(x) = \Pr(X = x) = \Pr(\{s \in S: X(s) = x\})

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

Since the image of X is countable, the probability mass function fX(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.

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

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

See also

References

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

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Theory of probability distributions
probability mass function (pmf) · probability density function (pdf) · cumulative distribution function (cdf) · quantile function