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