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Stat 333 Summer 2007
Probability generating function, Branching process
Practice Problems
READING: Ross’s textbook, Page 233 to Page 236.
The following are some problems regarding the generating function, the probability generating
function and the branching process.
1. Let
X
be a Poisson random variable with parameter
λ
.
Find the probability generating
function
G
X
(
s
) and indicate the interval of convergence. Use
G
X
(
s
) to show that
E
(
X
)=
Var
(
X
)=
λ
.
Solution
:S
in
c
e
X
is a Poisson random variable with parameter
λ
,then
P
(
X
=
n
)=
λ
n
e
−
λ
n
!
for
n
=0
,
1
,...
.
G
X
(
s
)=
E
[
s
X
]=
∞
±
n
=0
s
n
P
(
X
=
n
)=
∞
±
n
=0
(
sλ
)
n
e
−
λ
n
!
=
e
sλ
−
λ
and the series converges for

sλ

<
∞
i.e. the series converges for all values of
s
.
From the formulas in course notes, we have
E
(
X
)=
G
±
X
(1) =
λ
and
Var
(
X
)=
G
±±
X
(1) +
G
±
X
(1)
−
[
G
±
X
(1)]
2
=
λ
2
+
λ
−
λ
2
=
λ.
2. Suppose
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This note was uploaded on 01/19/2011 for the course STATISTICS STAT 333 taught by Professor Menzhongxian during the Fall '10 term at Waterloo.
 Fall '10
 MenZhongxian
 Probability

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