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 inequality—in many instances this bound is very crude.