Law of the unconscious statistician
In probability theory and statistics, the law of the unconscious statistician (sometimes abbreviated LOTUS) 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 its probability mass function ƒX (but not ƒg(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 (but not Fg(X)), then the expected value of g(X) is given by a Riemann–Stieltjes integral
The above equation is sometimes known as the law of the unconscious statistician, as statisticians have been accused of using the identity without realizing that it must be treated as the result of a rigorously proved theorem, not merely a definition.
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.
In the continuous case, with f(x, y) being the joint probability density function,
From the perspective of measure
A technically complete derivation of the result is available using arguments in measure theory, in which the probability space of a transformed random variable g(X) is related to that of the original random variable X. The steps here involve defining a pushforward measure for the transformed space, and the result is then an example of a change of variables formula.
We say has a density if is absolutely continuous with respect to the Lebesgue measure . In that case
where is the density (see Radon-Nikodym derivative). So the above can be rewritten as the more familiar
This article needs additional citations for verification. (November 2011) (Learn how and when to remove this template message)
- Eric Key (1998) Lecture 6: Random variables , Lecture notes, University of Leeds
- Bengt Ringner (2009) "Law of the unconscious statistician", unpublished note, Centre for Mathematical Sciences, Lund University
- Ross, Sheldon M. (1992). Applied Probability Models with Optimization Applications. Dover Publications. p. 2. ISBN 0486673146.
- Virtual Laboratories in Probability and Statistics, Sect. 3.1 "Expected Value: Definition and Properties", item "Basic Results: Change of Variables Theorem".
- Ross, Sheldon M. (2010). Introduction to Probability Models (10th ed.). Elsevier, Inc. p. 46.
- S.R.Srinivasa Varadhan (2002) Lecture notes on Limit Theorems, NYU (Section 1.4)