RISK. THEORY
LECTURE NOTES
SUMMARY.
1 c
,TOPIC 1
MOMENT GENERATING FUNCTIONS
The moment generating function associated with a random variable X is a
function of a real variable. It is used in:
(i) Calculating moments of the distribution
(ii) Identifying the distribution of sum ofindependent variables from the same
parametric family.
Definition:
The moment generating function of a random variable X , denoted M ( t) or
M x ( t) is a real valued function of the real variable t defined as:
M x ( t) = E ( etx )
= etx P r ( X = x ) : Discrete case
or
∞
= etx f x ( x ) dx : Continuous case
−∞
The domain of this function is the set of all values of t such that the sum
or the integral exists. There is the possibility that for some t the sum may be
divergent infinite series or the integral may be a divergent improper integral.
The moment generating function is useful when it is defined on an open interval
containing zero.
Example 1
1
If X is a random variable with P r ( X = i ) = 3 i = 1 , 2, 3 then M x ( t)
is given by:
M x ( t) = E ( etx )
= etx P r ( X = x )
1 t 1 2t 1 3t
= e + e + e
3 3 3
2
, for −∞ < t < ∞
Example 2
A random variable X has a Poisson distribution with parameter λ , find the
the moment generating function (M x ( t))
Solution
e− λ λ x
f (x) = x = 0 , 1, 2, ..., λ > 0
x!
M x ( t) = E ( etx )
= etx P r ( X = x )
e− λ λ x
= etx
x!
etx e− λ λ x
=
x!
Recall : Taylor series from elsewhere
∞
xk
ex =
k!
k =0
x x2 x3
=1+ + + + ...
1! 2! 3!
Applying the above result
∞
etx e− λ λ x
M x ( t) =
x =0
x!
∞
( λe t ) x
= e− λ
x =0
x!
3
LECTURE NOTES
SUMMARY.
1 c
,TOPIC 1
MOMENT GENERATING FUNCTIONS
The moment generating function associated with a random variable X is a
function of a real variable. It is used in:
(i) Calculating moments of the distribution
(ii) Identifying the distribution of sum ofindependent variables from the same
parametric family.
Definition:
The moment generating function of a random variable X , denoted M ( t) or
M x ( t) is a real valued function of the real variable t defined as:
M x ( t) = E ( etx )
= etx P r ( X = x ) : Discrete case
or
∞
= etx f x ( x ) dx : Continuous case
−∞
The domain of this function is the set of all values of t such that the sum
or the integral exists. There is the possibility that for some t the sum may be
divergent infinite series or the integral may be a divergent improper integral.
The moment generating function is useful when it is defined on an open interval
containing zero.
Example 1
1
If X is a random variable with P r ( X = i ) = 3 i = 1 , 2, 3 then M x ( t)
is given by:
M x ( t) = E ( etx )
= etx P r ( X = x )
1 t 1 2t 1 3t
= e + e + e
3 3 3
2
, for −∞ < t < ∞
Example 2
A random variable X has a Poisson distribution with parameter λ , find the
the moment generating function (M x ( t))
Solution
e− λ λ x
f (x) = x = 0 , 1, 2, ..., λ > 0
x!
M x ( t) = E ( etx )
= etx P r ( X = x )
e− λ λ x
= etx
x!
etx e− λ λ x
=
x!
Recall : Taylor series from elsewhere
∞
xk
ex =
k!
k =0
x x2 x3
=1+ + + + ...
1! 2! 3!
Applying the above result
∞
etx e− λ λ x
M x ( t) =
x =0
x!
∞
( λe t ) x
= e− λ
x =0
x!
3
