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Lecture

# Moment generating functions II.pdf

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School
University of Alberta
Department
Statistics
Course
STAT312
Professor
Douglas Wiens
Semester
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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