- Paley–Zygmund inequality
In
mathematics , the Paley - Zygmund inequality bounds theprobability that a positive random variable is small, in terms ofits mean andvariance (i.e., its first two moments). The inequality wasproved byRaymond Paley andAntoni Zygmund .Theorem: If "Z" ≥ 0 is a
random variable withfinite variance, and if 0 < θ < 1, then:Pr lbrace Z geq heta, operatorname{E}(Z) brace geq (1- heta)^2, frac{(operatorname{E}(Z))^2}{operatorname{E}(Z^2)}.
Proof: First, :operatorname{E} Z = operatorname{E} lbrace Z , mathbf{1}_{Z < heta operatorname{E} Z} brace + operatorname{E} lbrace Z , mathbf{1}_{Z geq heta operatorname{E} Z} brace~.Obviously, the first addend is at most heta operatorname{E}(Z). The second one is at most:lbrace operatorname{E} Z^2 brace^{1/2} lbrace operatorname{E} mathbf{1}_{Z geq heta operatorname{E} Z} brace^{1/2} = Big( operatorname{E} Z^2 Big)^{1/2} Big(Pr lbrace Z geq heta, operatorname{E}(Z) braceBig)^{1/2} according to the
Cauchy-Schwarz inequality . ∎Related inequalities
The right-hand side of the Paley - Zygmund inequality can be written as:Pr lbrace Z geq heta, operatorname{E}(Z) brace geq frac{(1- heta)^2 , (operatorname{E}(Z))^2}{(operatorname{E}(Z))^2 + operatorname{Var} Z}.
The
one-sided Chebyshev inequality gives a slightly better bound::Pr lbrace Z geq heta, operatorname{E}(Z) brace geq frac{(1- heta)^2 , (operatorname{E}(Z))^2}{(1- heta)^2 , (operatorname{E}(Z))^2+ operatorname{Var} Z}.The latter is sharp.References
* R.E.A.C.Paley and A.Zygmund, "A note on analytic functions in the unit circle", Proc. Camb. Phil. Soc. 28, 1932, 266-272
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