Probability theory

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Probability theory
The department of math of central south university
Probability and Statistics Course group
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Chapter 4 The law of large numbers (LLN) and the
central limit theorem(CLT)
4.1 Law of large numbers
Probability is the stable representation
of frequency.When the number of random
experiments increase infinitely, the frequency
vibrates around its probability,approximating to
a certain value,which is stated by the law of
large numbers theoretically.In the probability
theory,a very important tool is the normal
distribution,which can meet the need for many
applications.
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Just as stated in the CLT,the distribution of
the sums of some non-normal r.v.s approximately
submit to the normal distribution, under certain
conditions. These two theorems play a fundamental
and important role in probability and statistics
theory.
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The objective background of LLN
The stability of average results in a great deal
of random phenomena

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Theorem(Chebyshevs inequality)Let and
be the expectation and variance of r.v.
respectively.Then we have
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The definition of LLN
Definition Let be a sequence of r.v.s
with expectations (k1,2,...) existed. If for
anyegt0 such that
then the LLN holds for .
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Various kinds of LLN
Theorem(Chebyshevs LLN)Let be a sequence of
pair-wise uncorrelated r.v.s,with and
(k1,2,) as their expectations and variances
respectively.Then if there exists a constant C
satisfying C(k1,2,),we obtain
for any egt0.
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Proof
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The special case of Chebychevs LLN
Corollary Let be a sequence of pair-wise
uncorrelated r.v.s where and
(k1,2,).
Then
for any egt0.
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i.e. for sufficient large number n, is
almost non-random,and its value approximate to 1
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ExampleLet be a sequence of i.i.d. r.v.s,
which submit to Poisson distribution with
parameter ?,then the LLN holds for .
ProofSince , ,then sequence
satisfy all the conditions of Chebychevs LLN
therefore for any egt0,we have
i.e.the LLN holds for .
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Let

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Proof
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Theorem(Bernoullis LLN)Let n be the times of
repetitive experience,where p is the probability
of each event A and µn is the frequency,then for
any egt0,we have
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It follows from the Chebychevs theorem that
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It follows from the Bernoullis LLN that
the probability of having big difference between
µn and p is slim ,while repeating the
experiment to a sufficiently large number.
A method of determining the probability of an
event is provided by the Bernoullis LLN
(Theoretical insured)
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The solution of Buffons needle problem is based
on LLN
The length of needle L
The distance between the nearest tow lines a
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Solution
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Solution
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Example 5(Markovs LLN)Let be a sequence of
r.v.s,and it satisfy Markovs condition.
Then for any egt0,we obtain
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Solution
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Theorem(Khintchines LLN)Let be a sequence
of i.i.d. r.v.s. with (k1,2,). Then
for any egt0,we obtain
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RemarkIf we carry observations on a r.v.,and
treat the value of each observation as a
r.v.,then we can get a sequence of i.i.d. r.v.s.
It follows from the Khintchines LLN that the
arithmetic average of observations converges to
the r.v.s expectation in Pr,which provide a
feasible method of solving the expectation
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• Example 6Compute the definite integral using
• Markov Chain Monte Carlo

Solution Let
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• Concrete procedures as follow

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• Conclusions
• The objective background of LLN
• The definition of LLN
• Various LLN Chebyshevs LLN
• Bernoullis LLN Khintchines LLN
• Exercises
• P222 4.24, 4.25,4.26,4.29
  

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A short break to continue