BY:

GROUP 4

NOOR AZIZAHRATNAH KURNIATIZULFAHMI SOFYAN

JALIL SETIAWAN

MATHEMATICS DEPARTEMENT

MATHEMATICS AND SCIENCE FACULTYMAKASSAR STATE

UNIVERSITY

MO

MEN

T

GEN

ER

ATIN

G

FU

NC

TIO

N A

ND

C

HEB

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’S

MATHEMATICAL STATISTICGROUP TASK

PREFACE

Author thanks deliver to the Almighty God, because the guidance blessed

and so I can finish this paper entitled " MOMENT GENERATING FUNCTION

AND CHEBYSHEV’S THEOREM".

In writing this paper, the author is not a little experience obstacles, but

eventually these obstacles can be overcome with the help of various parties.

Therefore, the author does not forget to thank the people who have given a great

contribution in resolving this paper.

The author is fully aware that in the preparation of this paper there are

many flaws and is far be pushed for perfection. It was, the writer was expecting

criticism and constructive suggestions in order for the perfection of this paper.

Finally, with humility, I hope this paper can benefit the reader and the writer

himself in particular.

Makassar, November 4, 2010

Writer

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CONTENTS

Page

TITLE PAGE........................................................................................................i

PREFACE.............................................................................................................ii

CONTENTS..........................................................................................................iii

CHAPTER I INTRODUCTION

A. Background...............................................................................................1

B. Problem Statement....................................................................................1

C. Objective of Paper.....................................................................................1

D. Significant of Paper...................................................................................1

CHAPTER II MOMENT GENERATING FUNCTION AND CHEBYSHEV’S

THEOREM

A. Moment Generating Function...................................................................2

B. Chebyshev’s Theorem...............................................................................5

CHAPTER III CLOSING

A. Conclusion ................................................................................................8

B. Suggestion.................................................................................................9

REFERENCES......................................................................................................10

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CHAPTER I

INTRODUCTION

A. Background

Mathematics expectation concept (expected hope in mathematics) in

statistic that give helpfulness. Beside used it to development in advanced and

application statistic, also can in other , also as basic concept to defined or make

measurements in statistic, like mean, variance, covariance , coefisien, and

correlation.

Example concepts expectation is mean of random variable, variance and

covariance, chebyshev, theorem, Bienaime’s theorem, and moment generating

function. From that some concepts, in this paper writer just restricted it to moment

generating function and Chebyshev’s theorem. That will explain in next chapter

or in explanation chapter.

B. Problem Statement

According to the background before, writer can conclude the problem from

this paper is:

1. What is moment generating function?

2. What is a Chebyshev’s Theorem?

C. Objective of Paper

According to the problem statement before, writer can conclude the purpose

from this paper is:

1. To know the moment generating function.

2. To know a Chebyshev’s Theorem.

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CHAPTER II

MOMENT GENERATING FUNCTION AND CHEBYSHEV’S

THEOREM

A. Moment Generating Function

The moment generating function of X is defined by

Mx(t) = E (etx)

That is

Mx(t) = ∑i=1

n

etxi f (xi) = ∑ etx f(x) (discrete variable)

Mx(t) = ∫−x

x

etx f (x) dx (continuous variable)

We can show that the Taylor series expansion is [problem 3.15(a)]

Mx(t) = 1 + µt + µ’ t 2

2! + ………. + µ

’ tr

r ! + ……..

Since the coefficients in this expansion enable us to find the moments, the

reason for the name moment generating function is apparent. From te expansion

we can show that [Problem 3.15(b)]

µ’r =

dr

d t r Mx (t) t= 0

i.e. µ’r is the rth derivative of Mx(t) evaluated at t = 0. Where n confusion can

result we often write M(t) instead of Mx(t).

Some Theorems on Moment Generating Functions

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Theorem 3 – 8 : if Mx(t) is the moment generating function of the random

variable X and a and b (b ≠ 0 ) are constants, then the moment generating function

of ( X + a)/b is

M (x+a) / b (t) = eat/b Mx ( tb )

Theorem 3 – 9 : if X and Y are independent random variables having

moment generating function Mx(t) and My(t) respectively, then

M x + y (t) = Mx(t) My(t)

Generalizations of theorem 3 – 9 to more than two independent random

variables are easily made. In words, the moment generating function of a sum of

independent random variables is equal to the product of their moment generating

functions.

Theorem 3 -10 (Uniqueness theorem) : Suppose that X and Y are random

variables having moment generating function Mx(t) and My(t) respectively. Hen X

and Y have the same probability distribution if and only if Mx(t) = My(t)

identically.

Characteristic Function

If we let t=iw, where i is the imaginary unit, in the moment generating

functionwe obtain an important function called the characteristic function. We

denote this by

Φx (ω)=M X ( iω )=E (e iωX )

It follows that

ΦX (ω)=∑j=1

n

e iωx f ( x j )=∑ eiωx f (x) (discrete variable)

ΦX (ω)=∫−∞

∞

eiωx f ( x ) dx (continuous variable)

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The corresponding result become

ΦX (ω)=1+ iμω−μ'2

ω2

2 !+…+ir μ'

rωr

r !+…

Where

μ 'r= (−1 )r ir dr

dωr Φ X (ω )|ω=0

When no confusion can result we often write Φ (ω) instead of ΦX (ω).

Theorems for characteristic function corresponding to Theorem 3-8, 3-9, and

3-10 are as follows

Theorem 3:

If ΦX (ω) is the characteristic function of random variable X and a and

b (b ≠ 0) are constants, then the characteristic function of X+a

b is

Φ( X+a)/b(ω)=eaiω /bΦ X( ωb )

Theorem 2:

If X and Y are independent random variables having characteristic functions

ΦX (ω)and ΦY (ω) respectively, then

ΦX+Y (ω )=ΦX (ω)ΦY (ω )

More generally, the characteristic function of a sum of independent random

variables is equal to the product of their characteristic functions.

Theorem 3-13 (uniqueness theorem)

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Suppose that X and Y are random variables having characteristic functions

ΦX (ω) and ΦY (ω) respectively. Then X and Y have the same probability

distribution if and only if ΦX (ω)¿ΦY (ω) identically.

An important reason for introducing the characteristic function is that (37)

represents the Fourier transform of the density function f (x). From the theory of

Fourier transforms we can easily determine the density function from the

characteristic function. In fact,

f ( x )= 12π

∫−∞

∞

e−iωxΦ X(ω)dω

Which is often called an inversion formula or inverse Fourier transform. In

a similar manner we can show in the discrete case that the probability function

f (x) can be obtained from (36) use the Fouerier series, which is the analog of the

Fourier integral for the discrete case. See problem 3.39

Another reason for using the characteristic function is that it always exists

whereas the moment generating function may not exist.

B. Chebyshev’s Theorem

An important in theorem in probability and statistics which reveals a general

property of discrete or continuous random variables having finite mean and

variance is known under the name of Chebyshev’s Theorem or Chebyshev’s

Inequality. The Russian mathematician P.L. Chebyshev (1821 – 1894) discovered

that the fraction of the area between any two values symmetric about the mean is

related to the standard deviation. Since the area under a probability distribution

curve or in a probability histogram adds to 1, the area between any two numbers is

the probability of the random variable assuming a value between these numbers.

The following theorem, due to Chebyshev, gives a conservative estimate of

the probability that a random variable assumes a value within k standard

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deviations of its mean for any real number k. We shall provide the proof only for

the continuous case, leaving the discrete case as an exercise.

Theorem: the probability that any random variable X having mean µ and variance

0 < σ2 < ∞. Then if P is any positive number,

P(|X−μ|≥ ϵ )≰1σ2

ϵ2

Or, with ϵ 2=kσ,

P(|X−μ|≥ kσ )≰1

k2

Or the value of k standard deviation of the mean is at least 1−1 /k2, that

is

P(μ−kσ< X<μ+kσ )≥1−1/k2

Proof:

By our previous definition of the various of X, we can write

σ 2=E [ ( X−μ )2 ]=∫−∞

∞

( X−μ )2 f ( x ) dx

¿ ∫−∞

μ−kσ

( X−μ )2f ( x ) dx+ ∫μ−kσ

μ+ kσ

( X−μ )2 f ( x ) dx+ ∫μ+ kσ

∞

( X−μ )2 f ( x ) dx

≥ ∫−∞

μ−kσ

( X−μ )2 f ( x ) dx+¿ ∫μ+ kσ

∞

( X−μ )2 f ( x ) dx¿

Since the second of the three integrals is nonnegative. Now, since |x−μ|≥ kσ

wherever x≥ μ+kσ or x≰ μ−kσ , we have ( x−μ )2 ≥ k2 σ2 in both remaining

integrals. It follows that

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σ 2≥ ∫−∞

μ−kσ

k2 σ2 f (x ) dx+¿ ∫μ+kσ

∞

k 2σ 2 f ( x ) dx¿

And that

∫−∞

μ−kσ

f ( x ) dx+¿ ∫μ+ kσ

∞

f ( x )dx ≰1k 2 ¿

Hence

P (μ−kσ <X<μ+kσ )= ∫μ−kσ

μ+kσ

f ( x ) dx≥ 1− 1k 2

And the theorem is established.

For k = 2 the theorem states that the random variable X has a probability of

at least 1−1

22=3/4 of falling with two standard deviations of the mean. That is,

three-fourths or more of the observations of any distribution lie in the interval

μ ± 2σ . Similarly, the theorem says that at least eight-ninths of the observations of

any distribution fall in the interval μ ±3 σ.

Example:

A random variable X has a mean μ=8, a variance σ 2=9, and unknown

probability distribution. Find

a. P (−4<X<20 )b. P (|X−8|≥ 6 )

Solution:

a. P (−4<X<20 )=P [8−( 4 ) (3 )< X<8+( 4 ) (3 ) ] ≥ 1516

b. P (|X−8|≥ 6 )=1−P (|X−8|<6 )=1−P (−6<X<−8<6 )

¿1−P [8−(2 ) (3 )< X<8+ (2 )(3)]≰ 14

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Chebyshev’s theorem holds for any distribution of observations and for this

reason, the results are usually weak. The value given by the theorem is a lower

bound only. That is, we know that the probability of a random variable falling

within two standard deviations of the mean can be no less than 3/4 , but we never

know how much more it might actually be. Only when the probability distribution

is known can we determine exact probabilities. For this reason we call the

theorem a distribution-free result. When specific distributions are assumed as in

future chapters, the result will be les conservative. The use of Chebyshev’s

Theorem is relegated where the form of the distribution is unknown.

CHAPTER III

CLOSING

A. Conclusion

Moment Generating Function

The moment generating function of X is defined by

Mx(t) = E (etx)

That is

Mx(t) = ∑i=1

n

etxi f (xi) = ∑ etx f(x) (discrete variable)

Mx(t) = ∫−x

x

etx f (x) dx (continuous variable)

Chebyshev’s Theorem

the probability that any random variable X having mean µ and variance

0 < σ2 < ∞. Then if P is any positive number,

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P(|X−μ|≥ ϵ )≰1σ2

ϵ2

Or, with ϵ 2=kσ,

P(|X−μ|≥ kσ )≰1

k2

Or the value of k standard deviation of the mean is at least 1−1 /k2, that

is

P(μ−kσ< X<μ+kσ )≥1−1/k2

B. Suggestion

In our group hope, it will be better if next time we can get the book that will

be literature is better than our group use in this time. Then, time to work this task

is longer than time that we use today. And every member of group give opinion in

the paper project.

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REFFERENCES

Spiegel, Murray R. Probability and Statistic. Schaum’s Outline Series.

Tiro, Muhammad Arif. 2008. Pengantar Teori Peluang. Makassar: Andira

Publisher.

Walpole, Ronald.E G. E. P. 2007. Probability and Statistic for Engineering &

Scientists. Pearson International Edition.

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