I have summarized the content of risk theory in to detail mainly based on CM2(Loss reserving and financial engineering) on the IFOA syllabus up to chapter 8. the summary contains moment generating functions, conditional expectation, statistical models, risk models and compound distribution . the su...
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
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