EXAMPLE
: Suppose
X
1
,
X
2
, ...,
X
n
are independent
Poisson
random variables and define the sum
Y
X
1
X
2
...
X
n
. It follows
from previous properties of expected value and variance that
E
Y
n
and
Var
Y
n
. Because the variance equals the mean, it is tempting
to conclude
Y
has the
Y
Poisson
n
distributions. But we need to
show more than that the first two moments of
Y
match with the
Poisson
n
distribution.
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∙
To show
Y
Poisson
n
we use the MGF approach with
i
t
exp
exp
t
−
1
for each
i
:
Y
t
i
1
n
exp
exp
t
−
1
exp
∑
i
1
n
exp
t
−
1
exp
n
exp
t
−
1
and the final expression is the MGF of the
Poisson
n
distribution.
∙
We can extend this result to allow different means: if
X
i
has the
Poisson
i
distribution then the sum has the
Poisson
1
2
...
n
distribution.
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