Math 304: Homework 5
1. Let Yhave a binomial distribution in which n= 20 and p=θ. The prior probabilities on θ
are P(θ= 0.3) = 2/3 and P(θ= 0.5) = 1/3. If y= 9, what are the posterior probabilities for
θ= 0.3 and θ= 0.5.
2. Obtain the Bayes estimates of the Bernoulli parameter pbased on ten independent trials,
assuming quadratic loss,
(a) if the prior is g(p)∝p(1 −p)4and there are six successes.
(b) if the prior is g(p)∝p5(1 −p) and there are six successes.
3. Given a random sample of size n= 5 from a Gam(1,1/θ). Assuming absolute error loss,
find the Bayes estimate of θif the prior pdf is g(θ) = θe−θ, θ > 0, and the sample sum is
Pn
i=1 = 10.
4. Let X1,...,Xnbe a random sample of size n= 10 from a gamma distribution with α= 3
and β= 1/θ. Suppose we believe that θhas a gamma distribution with α= 10 and β= 2. If
the observed ¯x= 18.2,
(a) What is the Bayes point estimate associated with squared error loss?
(b) What is the Bayes point estimate using the mode of the posterior distribution?
5. Let X1,...,Xn|θbe a random sample from a Bernoulli distribution, X|θ∼bin(1, θ), and
assume a uniform prior θ∼U(0,1). Derive a (1 −α)∗100% Bayesian interval estimate for θ.
6. Consider the Bayes model
Xi|θ, for i= 1,...,n ∼bin(1, θ ),0< θ < 1.
Take the prior to be g(θ)∝pI(θ), where I(θ) is Fisher information. This is known as a class
of priors called Jeffreys priors. Assuming quadratic loss, what is the Bayes estimator?
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