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Problems 315
and
p(x2|y1) g(x2; y1 + 2, 2),
where g(x; α, λ) is the gamma density function (see Example (d) of Section
A.1).
c. Show that
and
p xt|y (t) g(xt; αt + t,t + 1)
p xt+1|y (t) g(xt+1; αt + t + 1,t + 1),
where αt y1 +···+yt.
d. Conclude from (c) that the minimum mean squared error estimates of Xt and
Xt+1 based on Y1,...,Yt are
and
respectively.
Xt|t t + Y1 +···+Yt
t + 1
ˆXt+1 t + 1 + Y1 +···+Yt
,
t + 1
8.28. Let Y and X be two random variables such that Y given X is exponential with
mean 1/X, and X has the gamma density function with
g(x; λ + 1,α) αλ+1x λ exp{−αx}
, x > 0,
Ɣ(λ + 1)
where λ>−1 and α>0.
a. Determine the posterior distribution of X given Y .
b. Show that Y has a Pareto distribution
p(y) (λ + 1)α λ+1 (y + α) −λ−2 , y > 0.
c. Find the mean of variance of Y . Under what conditions on α and λ does the
latter exist?
d. Verify the calculation of p yt+1|y (t) and E Yt+1|y (t) for the model in Example
8.8.8.
8.29. Consider an observation-driven model in which Yt given Xt is binomial with
parameters n and Xt, i.e.,
n
p(yt|xt) x yt
t (1 − xt) n−yt , yt0, 1,...,n.
yt