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AdviserProf. Dr. Kritsana Neammanee ,
Chulalongkorn UniversityMr. Suwat sriyotee , Mahidol
Wittayanusorn SchoolResearch FundNSTDAJSTPYSC
IntroductionIntroductionFrom Strong Law of Large
Numbers1 2 ... nX X X
n
almost surely convergence to E(Xi)
ProblemProblemWe can estimate
by E(Xi) which is equal to p when n converges to infinity.
Therefore; the problem is to know the error between those two values when n is known.
1 2 ... nX X Xn
ObjectiveObjective
-To implement a computer program to do the random experiment which different p parameters.
-To know the error bound on Strong Law of Large Numbers for Bernoulli random variables by analyzing the data from experiment.
MethodMethod
Picture showing the program implementation
Picture showing random experiment with p=1/2
MethodMethod
The data from the experiment is a maximum error of random variable values summation from expectation value
Therefore; we should divide the data by n to change them into the error bound on Strong Law of Large Numbers
X p
0
n
ii
X np
MethodMethod
Analyzing the changed data to obtain the equations and graphs
p=0.5y = 2.3093x-0.5034
R2 = 0.9962
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 2000 4000 6000 8000 10000 12000
n
Max
|[∑p(
xi)]/n
-E(x
i)|
p=0.25 y = 1.7936x-0.4977
R2 = 0.9917
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2000 4000 6000 8000 10000 12000
n
Max
|[∑p(
xi)]
/n-E
(xi)|
Graph Examples p=0.5 , p=0.25
MethodMethod
Let The trend line of data is tend to be the power graph, E=anb , so we use the properties of logarithm to simply it into linear equation as follows
we can draw the graph between log E and log n as a linear graph
AnalysisAnalysisE X p
by axlog( ) log( )by ax
log( ) log( ) log( )by a x
log( ) log( ) log( )y a b x
log( ) log( ) log( )y a b x
p=0.5
-1.8
-1.6-1.4
-1.2
-1-0.8
-0.6
-0.4-0.2
00 1 2 3 4 5
log n
log(
Max
|[∑p(
xi)]
/n-E
(xi)|
)
p=0.25
-2-1.8
-1.6-1.4-1.2
-1
-0.8-0.6-0.4
-0.20
0 1 2 3 4 5
log n
log(
Max
|[∑p(
xi)]
/n-E
(xi)|
)
Graph Examples p=0.5 , p=0.25
AnalysisAnalysis
Picture showing data analysis
AnalysisAnalysis
The error bound on strong law of large numbers of Bernoulli random variables do relate to the number of times doing the random experiment in form of
when a and b are the real numbers as the table.
ConclusionConclusion
bX p an
p b(notation) a(coefficient)
0.5 -0.5034 2.3093
0.25 -0.4977 1.7936
0.125 -0.5189 1.5959
0.0625 -0.568 1.828942
0.03125 -0.5232 1.0159
0.015625 -0.5752 1.046
0.007813 -0.5958 0.9052
0.003906 -0.6006 0.7173
0.001953 -0.6466 0.7509
0.000977 -0.6714 0.7037
0.1 -0.5102 1.4359
0.2 -0.4983 1.5455
0.3 -0.4984 1.9903
0.4 -0.4779 1.8628
0.6 -0.4811 1.8605
ConclusionConclusionp b(notation) a(coefficient)
0.7 -0.5274 2.2552
0.8 -0.5057 1.882
0.9 -0.5201 1.583434
0.15 -0.5141 1.712379
0.35 -0.4918 1.899328
0.45 -0.4959 1.997101
0.55 -0.4869 1.988841
0.65 -0.4832 1.812174
0.75 -0.5024 1.915579
0.85 -0.5084 1.631549
0.95 -0.5542 1.477406
0.96 -0.5417 1.223207
0.97 -0.5624 1.240224
0.98 -0.5582 0.994031
0.99 -0.5898 0.869161
Table showing notaion(b) and coefficient(a) in different p
K. Neammanee, “ทฤษฎีความน่าจะเป็นขึน้สงูและขอบเขตการประมาณค่า”,
พทัิกษ์การพมิพ,์ 2005.
R.G. Laha , V.K. Rohatfi, “Probability Theory”, Bowling Green State University,1979.
Feller,W , “An Introduction to Probability Theory and Its Application vol 1” , Newyork: Wiley,1968.
Feller,W , “An Introduction to Probability Theory and Its Application vol 2” , Newyork: Wiley,1971.
Abdi, H , “Encyclopedia for research methods for the social sciences” ,
Thousand Oaks(CA),2003
ReferenceReference