IJSRED-V2I5P56

International Journal of Scientific Research and Engineering Development-– Volume 2 Issue 5, Sep – Oct 2019
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 654
Moment-Generating Functions and Reproductive Properties of
Distributions
Ngwe Kyar Su Khaing*, Soe Soe Kyaw**,Nwae Oo Htike***
*(Department of Mathematics, Technological UniversityYamethin,Yamethin, Myanmar
Email: ngwekhaingjune@gmail.com)
** (Department of Mathematics, Technological UniversityYamethin, Yamethin, Myanmar
Email: dawsoesoekyaw@gmail.com)
***(Department of Mathematics, Technological UniversityYamethin, Yamethin, Myanmar
Email: warso2015feb@gmail.com)
----------------------------------------************************----------------------------------
Abstract:
In this paper, an important mathematical concept which has many applications to the probabilistic
models are presented. Some of the important applications of the moment- generating function to the theory
of probability are discussed. Each probability distribution has a unique moment-generating function,
which means they are especially useful for solving problems like finding the distribution for sums of
random variables. Reproductive properties of probability distributions with illustrated examples are also
described.
Keywords —Density function, distributions, moment-generating function, probability, random
variable.
----------------------------------------************************----------------------------------
I. INTRODUCTION
In probability theory, an experiment with an
outcome depending on chance which is called a
random experiment. It is assumed that all possible
distinct outcomes of a random experiment are
known and that they are elements of a fundamental
set known as the sample space. Each possible
outcome is called a sample point and an event is
generally referred to as a subset of the sample space
having one or more sample points as its elements
[5].
The behavior of a random variable is
characterized by its probability distribution, that is,
by the way probabilities are distributed over the
values it assumes. A probability distribution
function and a probability mass function are two
ways to characterize this distribution for a discrete
random variable. The corresponding functions for a
continuous random variable are the probability
distribution function (pdf) and the probability
density function [5].
Assume that X is a random variable; that is,
X is a function from the sample space to the real
numbers. In computing various characteristics of
the random variable X, such that ( ) ( )E X or V X , we
work directly with the probability distribution of X.
The moment-generating function ( )XM t is the
value which the function XM is the value which the
function XM assumes for the real variable t. The
notation, indicating the dependence on X, is used
because we consider two random variables, X and
Y, and then investigate the moment -generating
function of each, X YM and M . The moment-
generating function is written as an infinite series or
improper integral, depending on whether the
random variable is discrete or continuous [4].
RESEARCH ARTICLE OPEN ACCESS

II. SOME DISTRIBUTION FUNCTIONS
In the nondeterministic or random mathematical
models, parameters may also be used to
characterize the probability distribution. With each
probability distribution we may associate certain
parameters which yields valuable information about
the distribution [5].
A. Definition
Let X be a discrete random variable with
possible value 1 nx , ,x , .K K Let
( ) ( )i ip x P X x , i 1,2,... .= = = Then the expected value
of X, denoted by ( )E X , is defined as
( ) ( )i i
i 1
E X x p x
∞
=
= ∑ (1)
if the series ( )i i
i 1
x p x
∞
=
∑ converges absolutely.
This number is also referred to as the mean value
of X.
B. Definition
Let X be a continuous random variable with
probability density function f. The expected value
of X is defined as
( ) ( )E X xf x dx,
+∞
−∞
= ∫ (2)
if the improper integral is a absolutely convergent,
that is,
( )x f x dx.
+∞
−∞
∫
C. Binomial Distribution
Consider an experiment ε and let A be some
event associated with .ε Suppose that ( )P A p= and
hence ( )P A 1 p.= − Consider n independent
repetitions of .ε Hence the sample space consists of
all possible sequences { }1 2 na ,a ,...,a , where each ia
is either A or A, depending on whether A or A
occurred on the ith
repetition of .ε Furthermore,
assume that ( )P A p= remains the same for all
repetitions [2].
Let the random variable X be defined as follows:
X = number of times the event A occurred. Then X
is called a binomial random variable with
parameters n and p. Its possible values are
obviously 0,1,2, ,n.… Equivalently X has a binomial
distribution. The individual repetitions of ε will be
called Bernoulli trails.
D.Uniform Distribution
Suppose that X is continuous random variable
assuming all values in the interval [ ]a,b , where both
a and b are finite. If the pdf of X is given by
( )
1
F x , a x b
b a
= ≤ ≤
−
(3)
0, elsewhere,=
Then X is uniformly distributed over the interval
[ ]a,b .
x a= x b=
Fig. 1 X has uniformly distribution
E. The Poisson Distribution
Let X be a discrete random variable assuming
the possible values: 0,1,2, ,n....… If
( )
k
e
P X k , k 0,1,2
k!
, ,n,... .
−α
α
= = = … (4)
then X has a Poisson distribution with parameter
0.α >
F. Geometric Distribution
Assume, as in the discussion of the binomial
distribution, that we perform ε repeatedly, that the
repetitions are independent and that an each
repetition ( )P A p= and ( )P A 1 p q= − = remain the
same. Suppose that we repeat the experiment until

A occurs for the first time. Define the random
variable X as the number the repetitions required up
to and including the first occurrence of A. Thus X
assumes the possible values 1,2, .… Since X k= if
and only if the first ( )k 1− repetitions of ε result in
A while the kth
repetition results in A,
( ) k 1
P X k q p, k 1,2,... .−
= = = (5)
A random variable with probability distribution (5)
is said to have a geometric distribution[1].
G. The Normal Distribution
The random variable X, assuming all real values
x ,−∞ < < ∞ has a normal (or Gaussian) distribution
if its pdf is of the form
( )
2
1 1 x
f x exp , x .
22
 −µ 
 = − − ∞ < < ∞  σπσ   
(6)
The parameters andµ σ must satisfy the
conditions , 0.−∞ < µ < ∞ σ > X has distribution
( )2
N ,µ σ if and only if probability distribution is
given by (6).
x = µ
Fig. 2 X has normal distribution
H. The Exponential Distribution
A continuous random variable X assuming all
nonnegative values is said to have an exponential
distribution with parameter 0α > if its pdf given by
( ) , 0
0,
x
f x e xα
α −
= >
=
(7)
Fig. 3 X has exponential distribution
I.The Gamma Distribution
The Gamma Distribution, denoted by ,τ is
defined as follows:
( ) p 1 x
p x e dx,
0
∞
− −
τ = ∫ defined for p 0.> (8)
Let X be a continuous random variable assuming
only non-negative values. Then X has a Gamma
probability distribution if its pdf is given by
( )
( )r 1 x
f (x) x e , x 0
r
− −αα
= α >
τ
0= , elsewhere (9)
This distribution depends on two parameters, r and
,α of with r 0> and 0.α >
x
f (x)
r 2=
r 4=
r 1=
Fig. 4 X has Gamma distribution
III.THE MOMENT-GENERATING FUNCTIONS
A.Definitions
Let X be a discrete random variable with
probability distribution ( ) ( )i ip x P X x ,i 1,2,... .= = =
The Function XM , called the moment-generating
function of X, is defined by
( ) ( )jtx
X j
j 1
M t e p x .
∞
=
= ∑ (10)

If X is a continuous random variable with pdf f,
the moment-generating function is defined by
( ) ( )tx
XM t e f x dx.
+∞
−∞
= ∫ (11)
In either the discrete or the continuous case,
( )XM t is simply the expected value of tX
e .
( ) ( )tx
XM t E e .= (12)
( )XM t is the value which the function XM
assumes for the real variable t.
B. Example
Suppose that X is uniformly distributed over the
interval [ ]a,b . Therefore the moment-generating
function is given by
( )
b tx
X
a
e
M t dx
b a
=
−∫
bt at1
e e , t 0.
b a
 = − ≠
 −
(13)
C. Example
Suppose that X is binomially distributed with
parameters n and p. Then
( ) ( )
( ) ( )
n
n ktk k
X
k 0
n k n kt
k 0
n
M t e p 1 p
k
n
pe 1 p
k
−
=
−
=
 
= − 
 
 
= − 
 
∑
∑
( )
t n
e
p 1 p .
 
= + −  
(14)
D. Example
Suppose that X has a Poisson distribution with
parameter .λ Thus
( )
( )
k
tk
X
k 0
kt
k 0
e
M t e
k!
e
e
k!
∞ −λ
=
∞
−λ
=
λ
=
λ
=
∑
∑
t t
e (e 1)
e e e−λ λ λ −
= = (15)
E. Example
Suppose that X has an exponential distribution
with parameter .α Therefore
( )
( )
tx x
X
0
x t
0
M t e e dx
e dx
∞
−α
∞
−α
= α
= α
∫
∫
, t .
t
α
= < α
α −
(16)
F. Example
Suppose that X has normal distribution ( )2
N , .µ σ
Hence ( )
2
tx
X
1 1 x
M t e exp dx.
22
∞
−∞
 −µ 
 = −   σπσ   
∫
Let
( )x
s; thus x s and dx ds.
−µ
= = σ + µ = σ
σ
Therefore
( ) ( )
( )
[ ]
2
2 2
s
2
X
t 2
2t 2 2
t
t 22
1
M t exp t s e ds
2
1 1
e exp s 2 ts ds
22
1 1
e exp s t t ds
22
1 1
e exp s t ds.
22
∞
−
−∞
∞
µ
−∞
∞
µ
−∞
σ ∞
µ+
−∞
= σ + µ  
π
  = − − σ  π  
  = − − σ − σ   π  
 
= − − σ 
π  
∫
∫
∫
∫
Let s t ;−σ = γ then ds d= γ and
( )
2 2 2
t
t
2 2
X
1
M t e e d
2
σ γ∞
µ+ −
−∞
= γ
π
∫
2 2
t
t
2
e .
 σ
µ+ 
 
 = (17)
G. Example
Let X have a Gamma distribution with
parameters and r.α Then

( )
( )
( )
( )
( )
r 1tx x
X
0
r
x tr 1
0
M t e x e dx
r
x e dx.
r
∞
− −α
∞
− α−−
α
= α
τ
α
=
τ
∫
∫
Let ( )x t u;α − = thus
( )
( )
du
dx ,
t
=
α −
and
( )
( ) ( )
( )
r 1
r
u
X
0
r
r 1 u
0
u
M t e du
t r t
1
u e du.
t r
−∞
−
∞
− −
α  
=  
α − τ α − 
α 
=  
α − τ 
∫
∫
Since the integral equals ( )r ,τ
( )
r
XM t .
t
α 
=  
α − 
(18)
If r 1,= the Gamma function becomes the
exponential distribution.
IV. PROPERTIES OF THE MOMENT-
GENERATING FUNCTIONS
The Maclaurin series expansion of the function
x
e ;
2 3 n
x x x x
e 1 x ... ... .
2! 3! n!
= + + + + + + Thus
( ) ( )2 n
tx tx tx
e 1 tx ... ... .
2! n!
= + + + + +
Now
( ) ( ) ( ) ( )
( )
( ) ( )
2 n
tX
X
2 2 n n
tX tX
M t E e E 1 tX ... ...
2! n!
t E X t E X
1 tE X ... ... .
2! n!
 
 = = + + + + +
 
 
= + + + + +
Since XM is a function of the real variable t, the
derivative of ( )XM t with respect to t, that is, ( )M t .′
( ) ( ) ( )
( ) ( )
( )
2 3 n 1 n
2
X
t E X t E X
M t . E X tE X ... ... .
2! n 1 !
−
′ = + + + + +
−
Setting t 0,=
( ) ( )M 0 E X .′ =
Thus the first derivative of the moment-generating
function evaluated at t 0= yields the expected value
of the random variable [6]. The second derivative of
( )XM t is
( ) ( ) ( )
( )
( )
n 2 n
2 3
t E X
M t E X tE X ... ...,
n 2 !
−
′′ = + + + +
−
and setting t 0,=
( ) ( )2
M 0 E X .′′ =
The nth
derivative of MX(t) evaluated at t=0 is
( )(n) n
M (0) E X= .
The number ( )n
E X , n 1,2,...,= are called the nth
moments of the random variable X about zero.
The general Maclaurin series expansion of the
function XM is
( ) ( ) ( )
(n) n
X
X X X
2 n
2 n
1
M t
M t . M 0 M 0 t ... ... .
n!
t t
1 t ... ...
2! n!
′= + + + +
µ µ
= + µ + + + +
where ( )i
i E X , i 1,2,... .µ = = In particular,
( ) ( ) ( )( )
( ) ( )
22
2
V X E X E X
M 0 M 0 .
= −
′′ ′= −   
A. Theorem
Suppose that the random variable X has XM . Let
Y X .= α +β Then YM , the moment-generator
function of the random variable Y, is given by
( ) ( )t
Y XM t e M t .β
= α (20)
Proof:

( ) ( ) ( )
( )
( )
X tYt
Y
t tX
t
X
M t E e E e
e E e
e M t .
α +β
β α
β
 = =
  
=
= α
B. Theorem
Suppose that X and Y are independent random
variables. Let Z X Y.= + Let ( ) ( ) ( )X Y ZM t ,M t and M t
be the moment-generating functions of the random
variables X, Y and Z, respectively. Then
( ) ( ) ( )Z X YM t M t M t .= (21)
Proof:
( ) ( )
( )
( )
( ) ( ) ( ) ( )
2t
Z
X Y t
Xt Yt
Xt Yt
X Y
M t E e
E e
E e e
E e E e M t M t .
+
=
 =
  
=
= =
V. REPRODUCTIVE PROPERTIES OF
DISTRIBUTIONS
If two or more independent random variables
having a certain distribution are added, the resulting
random variable has distribution of the same type as
that of the summands. This result is the
reproductive property[6].
A.Example
Suppose that X and Y are independent random
variables with distributions ( ) ( )2 2
1 1 2 2N , and N , ,µ σ µ σ
respectively. Let Z X Y.= + Hence
( ) ( ) ( )
( ) ( )
2 2 2 2
1 2
Z X Y 1 2
2
2 2
1 2 1 2
t t
M t M t M t exp t, exp t,
2 2
t
exp t, .
2
   σ σ
   = = µ µ
   
   
 
= µ + µ σ + σ 
 
 
Thus Z has this normal distribution.
B. Example
The length of a rod is a normally distributed
random variable with mean 4 inches and
variance 0.01 inch2
. Two such rods are placed
end to end and fitted into a slot. The length of this
slot is 8 inches with a tolerance of 0.01± inch. The
probability that the two rods will fit can be
evaluated.
Letting 1L and 2L represent the lengths of rod 1
and rod 2, thus 1 2L L L= + is normally distributed
with ( )E L 8= and ( )V L 0.02.= Hence
[ ]
( ) ( )
7.9 8 L 8 8.1 8
P 7.9 L 8.1 P
0.14 0.14 0.14
0.714 0.714 0.526,
− − − 
≤ ≤ = ≤ ≤  
= Φ + − Φ − =
from the tables of the normal distribution [4].
C. Theorem
Let 1 nX ,...,X be independent random variables.
Suppose that iX has a Poisson distribution with
parameter i, i 1,2,...,n.α = Let 1 nZ X ... X .= + + Then Z
has a Poisson distribution with parameter
1 n... .α = α + + α
Proof:
For the case of n 2:=
( )
( ) ( )
( )t t
1 2
1 2
e 1 e 1
X XM t e , M t e
α − α −
= = .
Hence ( )
( )( )t
1 2 e 1
ZM t e .
α +α −
= This is the moment-
generating function of a random variable with
Poisson distribution having parameter 1 2.α + α By the
mathematical induction, the theorem is proved.
D.Example
Suppose that the number of calls coming into a
telephone exchange between 9 a.m. and 10 a.m., 1X ,
is a random variable with Poisson distribution with
parameter 3. Similarly, the number of calls arriving
between 10 a.m. and 11 a.m, 2X , also has a Poisson
distribution, with parameter 5. If 1 2X and X are

independent, the probability that more than 5 calls
come in between 9 a.m. and 11 a.m. can be
solved[3].
Let 1 2Z X X .= + From the above theorem, Z has a
Poisson distribution with parameter 3 5 8.+ = Hence
( ) ( )
( )k5 8
k 0
P Z 5 1 P Z 5
e 8
1
k!
1 0.1919 0.8088.
−
=
> = − ≤
= −
= − =
∑
E. Theorem
Suppose that 1 kX ,...,X are independent random
variables, each having distributions ( )N 0,1 .Then
2 2 2
1 2 kS X ,X ... X= + + has distribution 2
kX .
F. Example
Suppose that 1 nX ,...,X are independent random
variables, each with distribution ( )N 0,1 . Let
2 2
1 nT X ,...,X .= Since 2
T has distribution 2
nX .
( ) ( ) ( )2 2
H t P T t P T t= ≤ = ≤
( )
2
n zt
1
2 2
n
0 2
1
H t z e dz.
n
2
2
− −
=
 
τ 
 
∫
Hence
( ) ( )
( )
2
2
tn
12 22
n
2
t
n 1 2
n
2
h t H t
2t
t e
n
2
2
2t e
if t 0.
n
2
2
−−
−
−
′=
=
 
τ 
 
= ≥
 
τ 
 
V. CONCLUSIONS
The moment-generating function as defined above
is written as an infinite series or improper integral
depending on whether the random variable is
discrete or continuous. The method of moment-
generating functions to evaluate the expectation and
variance of a random variable with probability
distribution are used. And then we have discussed
a number of distribution for which a reproductive
property holds. We have seen that the moment-
generating function can be a powerful tool for
studying various aspects of probability
distributions. We found the use of the moment-
generating function very helpful in studying sums
of independent, identically distributed random
variables and obtaining various reproductive laws.
ACKNOWLEDGEMENT
I would like to thank Dr. Khin Mg Swe, Professor
(Retire),Department of Mathematics who shares
ideas and helpful suggestion. I also grateful to my
supervisor Dr. Daw Win Kyi, Professor
(Retire),Department of Mathematics in Yangon
University who motivates me to do this. I
appreciate to my parents and my teachers for their
patient, understanding and encouragement during
my work that has to successful finish.
REFERENCES
[1] B.A.Robert, Basic Probability Theory, Minela, New York:
Dover Publications, Inc, pp. 46-95, 1970.
[2] J.L. Devore, Probability and Statistics for Engineering and
the Science, Canada: Nelson Education, Ltd, pp. 130-183,
2009.
[3] M.R. Spiegel, Theory and Problems of Statistics,New York:
McGraw-Hill, pp. 139-146, 1961.
[4] P.L.Meyer, Introductory Probability and Statistical
Applications, Addison-Wesley Publishing Company, London
, pp. 209-222, 2004.
[5] T.T.Soong, Fundamentals of Probability and Statistics for
EngineeringNew York: John Wiley & Sons, Ltd, pp. 161-
219, 2004.
[6] W.Feller ,An Introduction to Probability Theory and Its
Applications, John Wiley & Sons, Inc,
USA, pp. 146-193,1967.

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