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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
{\displaystyle 0\leq \theta \leq 1}
P
(
Z
>
θ
E
[
Z
]
)
≥
(
1
−
θ
)
2
E
[
Z
]
2
E
[
Z
2
]
.
{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq (1-\theta )^{2}{\frac {\operatorname {E} [Z]^{2}}{\operatorname {E} [Z^{2}]}}.}
Proof : First,
E
[
Z
]
=
E
[
Z
1
{
Z
≤
θ
E
[
Z
]
}
]
+
E
[
Z
1
{
Z
>
θ
E
[
Z
]
}
]
.
{\displaystyle \operatorname {E} [Z]=\operatorname {E} [Z\,\mathbf {1} _{\{Z\leq \theta \operatorname {E} [Z]\}}]+\operatorname {E} [Z\,\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}].}
The first addend is at most
θ
E
[
Z
]
{\displaystyle \theta \operatorname {E} [Z]}
E
[
Z
2
]
1
/
2
P
(
Z
>
θ
E
[
Z
]
)
1
/
2
{\displaystyle \operatorname {E} [Z^{2}]^{1/2}\operatorname {P} (Z>\theta \operatorname {E} [Z])^{1/2}}
Cauchy–Schwarz inequality . The desired inequality then follows. ∎
The Paley–Zygmund inequality can be written as
P
(
Z
>
θ
E
[
Z
]
)
≥
(
1
−
θ
)
2
E
[
Z
]
2
Var
Z
+
E
[
Z
]
2
.
{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq {\frac {(1-\theta )^{2}\,\operatorname {E} [Z]^{2}}{\operatorname {Var} Z+\operatorname {E} [Z]^{2}}}.}
This can be improved. By the Cauchy–Schwarz inequality ,
E
[
Z
−
θ
E
[
Z
]
]
≤
E
[
(
Z
−
θ
E
[
Z
]
)
1
{
Z
>
θ
E
[
Z
]
}
]
≤
E
[
(
Z
−
θ
E
[
Z
]
)
2
]
1
/
2
P
(
Z
>
θ
E
[
Z
]
)
1
/
2
{\displaystyle \operatorname {E} [Z-\theta \operatorname {E} [Z]]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])^{2}]^{1/2}\operatorname {P} (Z>\theta \operatorname {E} [Z])^{1/2}}
which, after rearranging, implies that
P
(
Z
>
θ
E
[
Z
]
)
≥
(
1
−
θ
)
2
E
[
Z
]
2
E
[
(
Z
−
θ
E
[
Z
]
)
2
]
=
(
1
−
θ
)
2
E
[
Z
]
2
Var
Z
+
(
1
−
θ
)
2
E
[
Z
]
2
.
{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq {\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}}}.}
This inequality is sharp; equality is achieved if Z almost surely equals a positive constant.
References
R. E. A. C. Paley and A. Zygmund, "On some series of functions, (3)," Proc. Camb. Phil. Soc. 28 (1932), 190-205, (cf. Lemma 19 page 192).
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 
![{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq (1-\theta )^{2}{\frac {\operatorname {E} [Z]^{2}}{\operatorname {E} [Z^{2}]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d215506063bd4c6d27d04808bcb94387d7931d7)
![{\displaystyle \operatorname {E} [Z]=\operatorname {E} [Z\,\mathbf {1} _{\{Z\leq \theta \operatorname {E} [Z]\}}]+\operatorname {E} [Z\,\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1771609dffe911af2dcaa45558284564b75f5ef)
![{\displaystyle \theta \operatorname {E} [Z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5ac7cb0d70052cc97d4764e91cc2f0bc62bbd7)
![{\displaystyle \operatorname {E} [Z^{2}]^{1/2}\operatorname {P} (Z>\theta \operatorname {E} [Z])^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c221fb2f965cdbb850e7e5c2daa860ba95110865)
![{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq {\frac {(1-\theta )^{2}\,\operatorname {E} [Z]^{2}}{\operatorname {Var} Z+\operatorname {E} [Z]^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03eb36ae16ffe077072e03d9db8855632f5dd47e)
![{\displaystyle \operatorname {E} [Z-\theta \operatorname {E} [Z]]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])\mathbf {1} _{\{Z>\theta \operatorname {E} [Z]\}}]\leq \operatorname {E} [(Z-\theta \operatorname {E} [Z])^{2}]^{1/2}\operatorname {P} (Z>\theta \operatorname {E} [Z])^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9dea4705e5d7f39aaa9765aa7b89f47353559b)
![{\displaystyle \operatorname {P} (Z>\theta \operatorname {E} [Z])\geq {\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bce3bc5bd01a6479c20603a21fd1daa8162c826)