# 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 *ƒ _{X}* (but not

*ƒ*), then the expected value of

_{g(X)}*g*(

*X*) is

where the sum is over all possible values *x* of *X*. If it is a continuous distribution and one knows its probability density function *ƒ _{X}* (but not

*ƒ*), then the expected value of

_{g(X)}*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}* (but not

*F*), then the expected value of

_{g(X)}*g*(

*X*) is given by a Riemann–Stieltjes integral

(again assuming *X* is real-valued).^{[1]}^{[2]}

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.^{[3]}

## 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.^{[4]}{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## References

- ↑ 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- ↑ Bengt Ringner (2009) "Law of the unconscious statistician", unpublished note, Centre for Mathematical Sciences, Lund University
- ↑ Virtual Laboratories in Probability and Statistics, Sect. 3.1 "Expected Value: Definition and Properties", item "Basic Results: Change of Variables Theorem".
- ↑ S.R.Srinivasa Varadhan (2002) Lecture notes on Limit Theorems, NYU (Section 1.4)