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# Moment generating functions II.pdf

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University of Alberta

Statistics

STAT312

Douglas Wiens

Fall

Description

135
27. Moment generating functions II
Example 1: If ( ) with 0
and 0 (as ) then
P( ) (Poisson, mean ). (For this reason the
Poisson is sometimes known as the distribution
of the number of rare events.)
Proof: First calculate
X ³ ´
( ) = (1 )
³ ´
= 1 + (how?).
n ³ ´o
We aim to show (why?) that thisexp 1
as . Take logs:
³ ´
log 1 +
³ ³ ´´
= log 1 + 1
³ ³ ´´
log 1 + 1 ³ ´
= · 1
( 1)
log (1 + ) ³ ´
lim · lim 1
³0 ´
= 1 136
Example 2: Suppose 1 are i.i.d., each
exponentially distributed with densi( ) = ,
0. The m.g.f. of each is
Z µ ¶ 1
( ) = = · · · = 1
0
for | | . This density is often used to model
the time to the occurrence of an event like a
random shock (such as might cause an electrical
component to fail), and so the time to the
such shock is = P . We will see later
=1
how to obtain the density of (by induction for
instance); it will turn out to be the Erlang den-
sity
1
( )
( ) = ( 1)! 0
Now we have an easy proof of this:

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