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MSc Financial Mathematics
Statistical Methods and Data Analytics 2020
MATH0099
Problem Sheet 5
Problem 1. Let X1 , . . . , Xn be iid copies of a random variable X ~ N (θ, σ2) and suppose that the prior distribution of the mean θ is N (µ, τ2). The parameters σ2, µ and τ2 are known.
1. Find the joint pdf of X and θ .
2. Show that m(xjσ2, µ, τ2), the marginal distribution of X, is N (µ, σ2 /n + τ2).
3. Show that π(θjx, σ2, µ, τ2), the posterior distribution of θ, is normal with mean and variance given by
Problem 2. Let X1 ; . . . ; Xn be iid copies of a random variable X ~ Poisson(λ) and suppose that the prior distribution of λ is gamma(Q; β).
1. Find the posterior distribution of λ .
2. Calculate the posterior mean and variance.
Problem 3. Consider the so called LINEX loss function given by
where c is a positive constant.
1. For c = .2, .5, 1, plot L(θ, a) as a function of a - θ .
2. If X ~ Pθ(x), show that the Bayes estimator (or Bayes rule) of θ, using a prior α, is given by
3. Let X1 , . . . , Xn be iid N(θ, σ2), where σ2 is known, and suppose that θ has the noninformative prior Q(θ) = 1. Show that the Bayes estimator (or Bayes rule) is given by δB(X) = X - (cσ2/(2n)).
4. Calculate the posterior expected loss for δB(X) and X- using the LINEX loss function.
5. Calculate the posterior expected loss for δB(X) and X- using squared error loss.