324 CHAPTER 6 Inferential Statistics
From what we’ve proved about the mean, we see already thatE(X) =
μ. In case the random variablesX 1 , X 2 , ...,have the same distribution,
theWeak Law of Large Numberssays a bit more:
Lemma.(The Weak Law of Large Numbers)Assume thatX 1 , X 2 ,
...,Xn,..., is an infinite sequence of identically distributed random
variables with meanμ (and having finite varianceσ^2 ). Then for each
> 0
nlim→∞P(∣∣
∣∣
∣X 1 +X 2 +···+Xn
n−μ∣∣
∣∣
∣> )
= 0.Proof. We setSn=X 1 +X 2 +···+Xn, and soAn=Sn/nhas mean
μand varianceσ^2 /n. By Chebyshev’s Inequality we have
P
Ç∣∣
∣∣
∣An−μ∣∣
∣∣
∣≥å
≤σ^2
n^2.
Since >0 is fixed, the result is now obvious.
Notice that an equivalent formulation of the Weak Law of Large
Numbers is the statement that for all >0 we have that
nlim→∞P(∣∣
∣∣
∣X 1 +X 2 +···+Xn
n−μ∣∣
∣∣
∣≤)
= 1.As you might expect, there is also a Strong Law of Large Numbers
which is naively obtained by interchanging the limit and probabilityP;
see the footnote.^6
Exercises
- Prove that ifXandY are discrete independent random variables,
thenE(XY) =E(X)E(Y).Is this result still true ifXandY are
notindependent?
(^6) That is to say, ifX 1 , X 2 , ...,Xn,...,is an infinite sequence of identically distributed random
variables with meanμ, then
P
ÅX 1 +X 2 +···+Xn
n →μ
ã
= 1.
There is no requirement of finiteness of the variances.