Law of The Unconscious Statistician

In probability theory and statistics, the law of the unconscious statistician is a theorem used to calculate the expected value of a function g(X) of a random variable X when one knows the probability distribution of X but one does not explicitly know the distribution of g(X).

The form of the law can depend on the form in which one states the probability distribution of the random variable X. If it is a discrete distribution and one knows the probability mass function ƒ_X (not of g(X)), then the expected value of g(X) is

where the sum is over all possible values x of X. If it is a continuous distribution and one knows the probability density function ƒ_X (not of g(X)), then the expected value of g(X) is

(provided the values of X are real numbers as opposed to vectors, complex numbers, etc.).

Regardless of continuity-versus-discreteness and related issues, if one knows the cumulative probability distribution function F_X (not of g(X)), then the expected value of g(X) is given by a Riemann–Stieltjes integral

(again assuming X is real-valued).

However, the result is so well known that it is usually used without stating a name for it: the name is not extensively used. For justifications of the result for discrete and continuous random variables see expected value.