User:SamuelTheGhost/Reversion towards the Mean

Universal definition

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Let X, Y be random variables with any joint distribution (discrete or continuous). Reversion towards the Mean is the property defined in the following theorem.[1] Assume means exists and that X and Y have identical marginal distributions. Then for all c in the range of the distribution, so that

we have that

with the reverse inequality holding for all

Proof

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First we look at some probabilities. By elementary laws:

and

But the marginal distributions are equal, which implies

So taking these three equalities together we get

Going on the conditional probabilities we infer that

Looking now at expected values we have

But of course

, so

Similarly we have

and again of course

, so

Putting these together we have

and, since the marginal distributions are equal, we also have

, so

which concludes the proof.

Reference

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  1. ^ Samuels (1991)
  • Myra L. Samuels (November 1991). "Statistical Reversion Toward the Mean: More Universal than Regression Toward the Mean". The American Statistician. 45 (4): 344–346. doi:10.2307/2684474. JSTOR 2684474.{{cite journal}}: CS1 maint: date and year (link)