Probability mass function

Probability mass function

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"] .

Mathematical description

Suppose that "X" is a discrete random variable, taking values on some countable sample space "S" ⊆ R. Then the probability mass function "f""X"("x") for "X" is given by:f_X(x) = egin{cases} Pr(X = x), &xin S,\0, &xin mathbb{R}ackslash S.end{cases}Note that this explicitly defines "f""X"("x") for all real numbers, including all values in R that "X" could never take; indeed, it assigns such values a probability of zero.

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 "x" ∈ R"S") the derivative is zero, just as the probability mass function is zero at all such points.


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 0.5 on the state space {0, 1} (this is a Bernoulli random variable), and hence the probability mass function is:f_X(x) = egin{cases}frac{1}{2}, &x in {0, 1},\0, &x in mathbb{R}ackslash{0, 1}.end{cases}

ee also

* Discrete probability distribution


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

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