Consider x1, . . . , xn observations from an uniform distribution on the interval (0, θ). The prior distribution for θ is Pareto(α, θo) with α known, and the corresponding posterior distribution for θ is a Pareto with scale t = max(θo, maxi=1,...,n xi) and shape α + n. (a) Set a prior such that the prior mean is 3/2 and the prior variance is 3/4. (b) Test the following Hypotheses using a Bayesian approach H0 : θ ≤ 3 H1 : θ > 3 when the following data is available x1 = 1.2 x2 = 1.8 x3 = 1.5 x4 = 2. and the prior computed in part (a) is assumed.

 

*The density function of a Pareto(α, θo) distribution with scale θo and shape α is Ï€(θ) =  α(θo)^α/(θ^(α+1)) when θ ≥ θo and Ï€(θ)=0 when θ < 0, where θo > 0 and α > 0. When α > 2, the mean is α θo/ (α−1) and the variance is α(θo)^2/((α−1)^(2) (α−2)) .