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Probability mass function

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Probability mass function

In probability theory and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

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A probability mass function differs from a probability density function (pdf) in that the latter is associated with continuous rather than discrete random variables; the values of the latter are not probabilities as such: a pdf must be integrated over an interval to yield a probability.

Formal definition

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

f X ( x ) = Pr ( X = x ) = Pr ( { s ∈ S : X ( s ) = x } ) .

Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes x:

∑ x ∈ A f X ( x ) = 1

When there is a natural order among the hypotheses x, it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable) and to consider also values not in the image of X. That is, fX may be defined for all real numbers and fX(x) = 0 for all x ∉ X(S) as shown in the figure.

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 is related to 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.

Measure theoretic formulation

A probability mass function of a discrete random variable X can be seen as a special case of two more general measure theoretic constructions: the distribution of X and the probability density function of X with respect to the counting measure. We make this more precise below.

Suppose that ( A , A , P ) is a probability space and that ( B , B ) is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of B. In this setting, a random variable X : A → B is discrete provided its image is countable. The pushforward measure X ∗ ( P ) ---called a distribution of X in this context---is a probability measure on B whose restriction to singleton sets induces a probability mass function f X : B → R since f X ( b ) = P ( X − 1 ( b ) ) = [ X ∗ ( P ) ] ( { b } ) for each b in B.

Now suppose that ( B , B , μ ) is a measure space equipped with the counting measure μ. The probability density function f of X with respect to the counting measure, if it exists, is the Radon-Nikodym derivative of the pushforward measure of X (with respect to the counting measure), so f = d X ∗ P / d μ and f is a function from B to the non-negative reals. As a consequence, for any b in B we have

P ( X = b ) = P ( X − 1 ( { b } ) ) := ∫ X − 1 ( { b } ) d P = ∫ { b } f d μ = f ( b ) ,

demonstrating that f is in fact a probability mass function.

Examples

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 ) = { 1 2 , x ∈ { 0 , 1 } , 0 , x ∉ { 0 , 1 } .

This is a special case of the binomial distribution, the Bernoulli distribution.

An example of a multivariate discrete distribution, and of its probability mass function, is provided by the multinomial distribution.

References

Probability mass function Wikipedia
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