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 ƒX (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 ƒX (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 FX (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.
Read more about Law Of The Unconscious Statistician: From The Perspective of Measure
Famous quotes containing the words law of the, law of, law, unconscious and/or statistician:
“The law of humanity ought to be composed of the past, the present, and the future, that we bear within us; whoever possesses but one of these terms, has but a fragment of the law of the moral world.”
—Edgar Quinet (1803–1875)
“The basis of good manners is self-reliance. Necessity is the law of all who are not self-possessed.”
—Ralph Waldo Emerson (1803–1882)
“Lawyers are necessary in a community. Some of you ... take a different view; but as I am a member of that legal profession, or was at one time, and have only lost standing in it to become a politician, I still retain the pride of the profession. And I still insist that it is the law and the lawyer that make popular government under a written constitution and written statutes possible.”
—William Howard Taft (1857–1930)
“The unconscious is the ocean of the unsayable, of what has been expelled from the land of language, removed as a result of ancient prohibitions.”
—Italo Calvino (1923–1985)
“We are not concerned with the very poor. They are unthinkable, and only to be approached by the statistician or the poet.”
—E.M. (Edward Morgan)