April 2016
Intermediate to advanced
384 pages
11h 9m
English
book
Transportation Statistics and Microsimulation
by Clifford Spiegelman, Eun Sug Park, Laurence R. Rilett
Content preview from Transportation Statistics and Microsimulation
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272 ◾ Transportation Statistics and Microsimulation
that the gamma distribution with parameters α and β, Gamma(α,β), is
given as
p
e
λ
λ
αβ
λαβ
αλβ
α
(
)
=
(
)
≥>
−−1
00
Γ
,,,.
By entering the formula for the Poisson likelihood and the gamma prior
into Equation 14.3, we get
πθ θθxfxp
(
)
∝
(
)
(
)
∝
(
)
(
)
=
−−−
+−
−+
(
)
ee
e
x
x
λαλβ
α
λβ
λλ
λ
1
1
11
which is essentially the same formula as the gamma prior except that the
parameters are dierent (i.e., updated to x + α and (1 + 1/β)
−1
). e full
gamma posterior density is given as
πλ
λβ
α
α
λβ
α
x
e
x
x
x
(
)
=
+
(
)
+
(
)
+−
−+
(
)
+
1
11
11
Γ
.
(14.4)
Note that a gamma prior gives rise to a posterior distribution that again
belongs to a gamma family.
A prior ...