The "Law of the Unconscious Statistician" states that, for a random variable $X$ with density $f_X(x)$ and a function of it $h(X)$ we have that

$$E[h(X)] = \int_{-\infty}^{\infty} f_X(x)h(x) dx$$

In words, we do not need to obtain the distribution and density of $Z=h(X)$ in order to calculate its expected value.

How this strange name for this law came about? (it does sound a little dismissive).

The wikipedia page does not discuss the issue. The Quora forum has a related thread, but no real historical answer regarding the name is given.


1 Answer 1


This was discussed on Math Forum. According to Elliot, Halmos called it Fundamental Theorem of the Unconscious Statistician as early as 1946, and according to Bernier, Introduction to the Techniques of Operations Research by Hillier (1965) calls it the "Law of the Unconscious Statistician".

But apparently the nickname only took off after Ross's "Introduction to Probability Models" (1980). Proposition 4.1 of Chapter 2 has a footnote that reads:"This law got its name from "unconscious" statisticians who ave used it as if it were the definition of $E[g(X)]$". It did not go over too well, for example Berger and Casella write in Statistical Inference:"Ross 1988 refers to this as the 'law of the unconscious statistician'. We do not find this amusing". Apparently, they were not alone, Ross dropped the nickname from subsequent editions.


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