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−
1
−
1
−
1
−
1
x
,
x
0,1,2,
....
∙
When
1 we get the geometric density [because
Γ
1
1 and
Γ
1
x
x
!]. Can show that, as
→
0, the density converges to the
Poisson
density.
∙
It can be shown that
E
X
Var
X
1
2
33

This** preview**
has intentionally

Thus, we can write
Var
X
E
X
1
and so
is a measure of overdispersion. For a given
, as
increases,
Var
X
/
E
X
increases.
∙
Later, when we turn to statistics, we will see that if we fix
(for
example,
1 in the geometric) – that is, we assume we know it –
then estimating
from a sample of data is easy. The problem is
considerably harder when
and
both need to be estimated.
∙
We will write
X
~
NegBin
,
.
34