Connexions module: m13502 1
Chebyshev's Inequality
∗
Ewa Paszek
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
Abstract
This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the
Table of Contents. The notes were prepared by Ewa Paszek and Marek Kimmel. The development of
this course has been supported by NSF 0203396 grant.
1 Chebyshev's Inequality
In this paragraph the Chebyshev's inequality is used to show, in another sense, that the sample mean,
x
,
is a good statistic to use to estimate a population with mean
µ
; the relative frequency of successes in
n
Bernoulli trials,
y/n
, is a good statistic for estimating
p
; and the empirical distribution function,
Fn(x)
,
can be used to estimate the theoretical distribution function
F(x)
. The eect of the sample size
n
on these
estimates is discussed.
At the beginning, it is showed that the Chebyshev's inequality gives added signicance to the standard
deviation in terms of bounding certain probabilities. The inequality is valid for all distributions for which
the standard deviation exists. The proof is given for the discrete case, but it holds for the continuous case
with integrals replacing summations.
Theorem 1:
Chebyshev's Inequality
If the random variable
X
has a mean
µ
and variance
σ2
, then for every
k≥1
,
P(|X−µ| ≥ kσ)≤1
k2.
Proof:
Let
f(x)
denote p.d.f. of
X
. Then
σ2=Eh(X−µ)2i=X
x∈R
(x−µ)2f(x) = X
x∈A
(x−µ)2f(x) + X
x∈A
'
(x−µ)2f(x),
where
A= (x:|x−µ| ≥ kσ)
. The second term in the right-hand member of the equation is the
sum of nonnegative numbers and thus is greater than or equal to zero, Hence
σ2≥X
x∈A
(x−µ)2f(x).
∗
Version 1.3: Oct 8, 2007 3:19 pm GMT-5
†
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