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M. Chen (IE@CUHK) ENGG2430C lecture 10 18 / 29 ATV poll problem: The LLN perspective Deﬁne Xi as a Bernoulli r.v. indicating the answer of i th family
pXi (1) = p , pXi (0) = 1 − p
Statement based on observation and intuition: Sample mean Yn
Yn = 1n
Xi
n i∑
=1 approaches E [Yn ] = p for as n goes to inﬁnity.
Rigorous Statement: Because Xi (1 ≤ i ≤ n) are i.i.d. r.v.s with mean
p , the sample mean Yn converges to p in probability as n goes to
inﬁnity.
That is, for any ε > 0,
lim P (Yn − p  ≥ ε ) = 0. n →∞
M. Chen (IE@CUHK) ENGG2430C lecture 10 19 / 29 ATV poll problem: The LLN perspective Deﬁne Xi as a Bernoulli r.v. indicating the answer of i th family
pXi (1) = p , pXi (0) = 1 − p , E [Xi ] = p , Var(Xi ) = p (1 − p ).
The sample mean is deﬁned as
Yn = 1n
Xi
n i∑
=1 1
Its mean and variance are: E [Yn ] = p , Var(Yn ) = n p (1 − p ) By the Weak Law of Large Number, we obtain precisely Yn = p as n
goes to inﬁnity. But in practice, n has...
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 Spring '14

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