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| In [[mathematics]], the '''Paley–Zygmund inequality''' bounds the
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| probability that a positive random variable is small, in terms of
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| its [[expected value|mean]] and [[variance]] (i.e., its first two [[moment (mathematics)|moments]]). The inequality was
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| proved by [[Raymond Paley]] and [[Antoni Zygmund]].
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| '''Theorem''': If ''Z'' ≥ 0 is a [[random variable]] with
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| finite variance, and if 0 < ''θ'' < 1, then
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| :<math> | |
| \operatorname{P}( Z \ge \theta\operatorname{E}[Z] )
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| \ge (1-\theta)^2 \frac{\operatorname{E}[Z]^2}{\operatorname{E}[Z^2]}.
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| </math>
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| '''Proof''': First,
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| :<math>
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| \operatorname{E}[Z] = \operatorname{E}[ Z \, \mathbf{1}_{\{ Z < \theta \operatorname{E}[Z] \}}] + \operatorname{E}[ Z \, \mathbf{1}_{\{ Z \ge \theta \operatorname{E}[Z] \}} ].
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| </math>
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| The first addend is at most <math>\theta \operatorname{E}[Z]</math>, while the second is at most <math> \operatorname{E}[Z^2]^{1/2} \operatorname{P}( Z \ge \theta\operatorname{E}[Z])^{1/2} </math> by the [[Cauchy–Schwarz inequality]]. The desired inequality then follows. ∎
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| == Related inequalities ==
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| The Paley–Zygmund inequality can be written as
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| :<math>
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| \operatorname{P}( Z \ge \theta \operatorname{E}[Z] )
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| \ge \frac{(1-\theta)^2 \, \operatorname{E}[Z]^2}{\operatorname{var} Z + \operatorname{E}[Z]^2}.
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| </math>
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| This can be improved. By the [[Cauchy–Schwarz inequality]],
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| :<math>
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| \operatorname{E}[Z - \theta \operatorname{E}[Z]]
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| \le \operatorname{E}[ (Z - \theta \operatorname{E}[Z]) \mathbf{1}_{\{ Z \ge \theta \operatorname{E}[Z] \}} ]
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| \le \operatorname{E}[ (Z - \theta \operatorname{E}[Z])^2 ]^{1/2} \operatorname{P}( Z \ge \theta \operatorname{E}[Z] )^{1/2}
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| </math>
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| which, after rearranging, implies that
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| :<math>
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| \operatorname{P}(Z \ge \theta \operatorname{E}[Z])
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| \ge \frac{(1-\theta)^2 \operatorname{E}[Z]^2}{\operatorname{E}[( Z - \theta \operatorname{E}[Z] )^2]}
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| = \frac{(1-\theta)^2 \operatorname{E}[Z]^2}{\operatorname{var} Z + (1-\theta)^2 \operatorname{E}[Z]^2}.
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| </math> | |
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| This inequality is sharp; equality is achieved if Z almost surely equals a positive constant, for example.
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| == References==
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| * 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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| {{DEFAULTSORT:Paley-Zygmund inequality}}
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| [[Category:Probabilistic inequalities]]
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