2. Suppose that X|θ ~ U(0.0), the uniform distribution on the interval (09). Assuming squared error...
1. Suppose that X P(A), the Poisson distribution with mean λ Assuming squared error loss, derive that Bayes estimator of λ with respect to the prior distribution「(Q), first by explicitly deriving the marginal probability mass function of X, obtaining an expression for the posterior density of A and evaluating E(alx) and secondly by identifying g(Alx) by inspection and noting that it is a familiar distribution with a known mean.
Consider the normal distribution f(x|θ) = [1 / sqrt(2π)] exp(−1/2 (x − θ)^2 ) for all x. Let the prior distribution for θ be f(θ) = [1 / sqrt(2π)] exp[(−1/2) (θ^2)] for all θ. (a) Show that the posterior distribution is a normal distribution. With what parameters? (b) Find the Bayes’ estimator for θ.
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...