16-cell honeycomb: Difference between revisions

Revision as of 06:08, 22 January 2014

In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its mean and variance (i.e., its first two moments). The inequality was proved by Raymond Paley and Antoni Zygmund.

Theorem: If Z ≥ 0 is a random variable with finite variance, and if 0 < θ < 1, then

P (Z \geq θ E [Z]) \geq (1 - θ)^{2} \frac{E [Z]^{2}}{E [Z^{2}]} .

Proof: First,

E [Z] = E [Z 1_{{Z < θ E [Z]}}] + E [Z 1_{{Z \geq θ E [Z]}}] .

The first addend is at most $θ E [Z]$ , while the second is at most $E [Z^{2}]^{1 / 2} P (Z \geq θ E [Z])^{1 / 2}$ by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎

Related inequalities

The Paley–Zygmund inequality can be written as

P (Z \geq θ E [Z]) \geq \frac{(1 - θ)^{2} E [Z]^{2}}{var Z + E [Z]^{2}} .

This can be improved. By the Cauchy–Schwarz inequality,

E [Z - θ E [Z]] \leq E [(Z - θ E [Z]) 1_{{Z \geq θ E [Z]}}] \leq E [(Z - θ E [Z])^{2}]^{1 / 2} P (Z \geq θ E [Z])^{1 / 2}

which, after rearranging, implies that

P (Z \geq θ E [Z]) \geq \frac{(1 - θ)^{2} E [Z]^{2}}{E [(Z - θ E [Z])^{2}]} = \frac{(1 - θ)^{2} E [Z]^{2}}{var Z + (1 - θ)^{2} E [Z]^{2}} .

This inequality is sharp; equality is achieved if Z almost surely equals a positive constant, for example.

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

16-cell honeycomb: Difference between revisions

Revision as of 06:08, 22 January 2014

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