Page 521 - Probability and Statistical Inference

498 10. Bayesian Methods

–αθ
Assume the prior density h(θ) = ae I(θ > 0) where α (> 0) is known. Derive
the posterior pdf of v given that T = t where
10.2.2 (Example 10.2.1 Continued) Suppose that X , ..., X are iid with the
1
5
common pdf f(x; θ) = θe I(x > 0) given that = θ where (> 0) is the
-θx
unknown parameter. Assume the prior density h(θ) = 1/8e –θ/8 I(θ > 0). Draw
and compare the posterior pdf’s of given that with t = 15, 40,
50.
10.2.3 Suppose that X , ..., X are iid Poisson(θ) given that = θ where
1
n
v(> 0) is the unknown population mean. Assume the prior density h(θ) = e –
θ I(θ > 0). Derive the posterior pdf of given that T = t where T =

10.2.4 (Example 10.2.3 Continued) Suppose that X , ..., X are iid
1
10
Poisson(θ) given that = θ where (> 0) is the unknown population mean.
Assume the prior density h(θ) = ¼e I(θ > 0). Draw and compare the poste-
-θ/4
rior pdf’s of given that with t = 30, 40, 50.
10.3.1 Let X , X be independent, X be distributed as N(θ, 1) and X be
1
2
2
1
distributed as N(2θ, 3) given that = θ where (∈ ℜ) is the unknown param-
eter. Consider the minimal sufficient statistic T for θ given that = θ. Let us
suppose that the prior distribution of on the space Θ = ℜ is N(5, τ ) where
2
τ(> 0) is a known number. Derive the posterior distribution of v given that T
= t, t ∈ ℜ. {Hint: Given = θ, is the statistic T normally distributed?}
10.3.2 Suppose that X , ..., X are iid Exponential(θ) given that = θ
1 n
where the parameter v(> 0) is unknown. We say that has the inverted
gamma prior, denoted by IGamma(α, β), whenever the prior pdf is given by

where α, β are known positive numbers. Denote the sufficient statistic T =
given that = θ.

(i) Show that the prior distribution of is IGamma(α, β) if and only if
has the Gamma(α, β) distribution;
(ii) Show that the posterior distribution of v given T = t turns out to be
IGamma(n + α, {t + β } ).
-1 -1
2
10.3.3 Suppose that X , ..., X are iid N(0, θ ) given that = θ where the
n
1
parameter v(> 0) is unknown. Assume that 2 has the inverted gamma prior
IGamma(α, β), where α, β are known positive numbers. Denote the suffi-
cient statistic given that = θ. Show that the posterior distribu-
tion of given T = t is an appropriate inverted gamma distribution.