## Description

1. Suppose an engineer is 95% confident that the probability of rejecting a product is

going to be .5 ± .2. Use this information to construct a beta prior for θ. (Hint: Use

normal approximation for the beta distribution.

Note that if x ∼ Beta(α, β), then

E(x) = α

α+β

and var(x) = αβ

(α+β)

2(α+β+1) . )

2. Find the Jeffreys’ prior for the parameter α of the Maxwell distribution

p(y|α) = r

2

π

α

3/2

y

2

exp(−

1

2

αy2

)

and find a transformation of this parameter in which the corresponding prior is uniform.

3. Jeffreys’ prior for multiparameter models is given by

p(θ) ∝

p

det(I(θ)),

where I(θ) is the Fisher Information matrix whose ijth element is given by

−E(∂

2

log p(y|θ)/∂θi∂θj ). Suppose that for i = 1, · · · , n, yi ∼ pi(yi

|θi) and πi(θi) is

the Jeffreys’ prior for θi

. If the yi

’s are independent, show that the Jeffreys’ prior for

θ = (θ1, · · · , θn)

0

is Qn

i=1 πi(θi).

4. Suppose x ∼ Binomial(n, π) and y ∼ Binomial(n, ρ) are independent. Find the Bayes

rule for estimating π − ρ corresponding to the loss function L(π − ρ, a) = (π − ρ − a)

2

under the priors: π ∼ Beta(1, 3) and ρ ∼ Beta(3, 1).

5. Find the Bayes rule for estimating θ corresponding to the loss function L(θ, a) =

c1(a − θ) if a ≥ θ and L(θ, a) = c2(θ − a) if a ≤ θ.