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Bayesian Statistics Problems 2 1. It is believed that the number of accidents in a new factory will follow a Poisson distribution with mean θ per month. The prior distribution of θ is given by a gamma distribution with mean µ having density p(θ) = hµh θµh−1 exp(−hθ) Γ(µh) θ ≥ 0. If µ = 2, h = 2 and there are 18 accidents in the first six months, derive the posterior distribution of θ and find its mean and variance. 2. A coin is known to be biased. The probability that it lands heads when tossed is θ. The coin is tossed successively until the first tail is seen. Let x be the number of heads before the first tail. (a) Show that the resulting geometric distribution is an exponential family and identify the natural parameter. (b) Hence show that if this sampling method is repeated n times giving P observations x1 , x2 , . . . , xn then s = ni=1 xi is a sufficient statistic. (c) Show that a beta distribution provides a conjugate prior. 3. The number of offspring X in a certain population has probability function α x = 0, p(x|α, β) = (1 − α)β(1 − β)x−1 x = 1, 2, . . . where α and β are unknown parameters lying in the unit interval. Obtain the likelihood function when r zeroes and n − r non-zero values x1 , x2 , . . . , xn−r are obtained from n independent observations on X. Suppose α and β have independent prior beta densities with parameters a and b and p and q respectively. Show that α and β have independent posterior beta distributions and identify the posterior parameters. 4. Observations x1 , x2 , . . . , xn are a random sample from a uniform distribution on the interval (0, θ). The prior for θ is taken to be a Pareto distribution with density ( p(θ) = αθ0α θα+1 0 θ > θ0 , θ ≤ θ0 . Find the posterior distribution of θ. (Hint: Think carefully about the likelihood.) 1