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Chebyshev's Inequality and the Weak Law of Large Numbers for iid Two-Vectors

sample sizes
20
epsilon
0.5
random seed
123
mean
standard deviation
Chebyshev's inequality states that if
X
1
,
X
2
,...,
X
n
are independent, identically distributed random variables (an iid sample) with common mean
μ
and common standard deviation
σ
and
X
is the average of these random variables, then
P(|X-μ|>ϵ)
2
σ
n
2
ϵ
.
An immediate consequence is the weak law of large numbers, which states that
P(|X-μ|>ϵ)0
as
n
. These results are usually stated for real-valued random variables but also hold for random vectors, provided you interpret all absolute values as Euclidean distances and the variance as
2
σ
=E
X
i
-μ
2
|
. The blue dots in the image are the means of 100 different iid samples from a bivariate normal distribution with mean and standard deviation specified by the locators on the left
2
σ
is the square of the magnitude of this standard deviation. The orange dot is the common mean,
μ
, and the circle shown is centered at
μ
with radius
ϵ
. The fraction of blue dots outside the circle will usually be smaller than the theoretical upper bound given in Chebyshev's inequalityin many instances this bound is very crude.
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