# Question: Let X1 X2 Xn be a

Let X1, X2, . . . , Xn be a random sample from a gamma distribution with known α and with θ = 1/τ. Say τ has a prior pdf that is gamma with parameters α0 and θ0, so that the prior mean is α0θ0.

(a) Find the posterior pdf of τ, given that X1 = x1, X2 = x2, . . . , Xn = xn.

(b) Find the mean of the posterior distribution found in part (a), and write it as a function of the sample mean and α0θ0.

(c) Explain how you would find a 95% interval estimate of τ if n = 10, α = 3, α0 = 10, and θ0 = 2.

(a) Find the posterior pdf of τ, given that X1 = x1, X2 = x2, . . . , Xn = xn.

(b) Find the mean of the posterior distribution found in part (a), and write it as a function of the sample mean and α0θ0.

(c) Explain how you would find a 95% interval estimate of τ if n = 10, α = 3, α0 = 10, and θ0 = 2.

## Answer to relevant Questions

Consider a random sample X1, X2, . . . , Xn from a distribution with pdf Let θ have a prior pdf that is gamma with α = 4 and the usual θ = 1/4. Find the conditional mean of θ, given that X1 = x1, X2 = x2, . . . , Xn = ...Let X1, X2 be a random sample from the Cauchy distribution with pdf Consider the non-informative prior h(θ1, θ2) ∝ 1 on that support. Obtain the posterior pdf (except for constants) of θ1, θ2 if x1 = 3 and x2 = 7. For ...To determine whether the bacteria count was lower in the west basin of Lake Macatawa than in the east basin, n = 37 samples of water were taken from the west basin and the number of bacteria colonies in 100 milliliters of ...Let X-bar, Y-bar, S2X, and S2Y be the respective sample means and unbiased estimates of the variances obtained from independent samples of sizes n and m from the normal distributions N(μX, σ2X) and N(μY, σ2Y), where μX, ...When placed in solutions of varying ionic strength, paramecia grow blisters in order to counteract the flow of water. The following 60 measurements in microns are blister lengths: (a) Construct an ordered stem-and-leaf ...Post your question