Solve only 016.18 not Q16.10
16.10 Let Y1, Y2,..., Yn denote a random sample from an exponentially distributed population with density f(y) = e^-y, 0 < y. (Note: the mean of this population is 1/λ.) Use the conjugate gamma (α, β) prior for λ to do the following. Show that the joint density of Y1, Y2,..., Yn is
f(y1, y2,..., yn) = T
Show that the marginal density of Y1, Y2,..., Yn is
T(n + α) m(y1, y2,..., yn) = T
-
+ * g
Show that the posterior density for λ | (y1, y2, .., yn) is a gamma density with parameters α* = n+α and β* = Σyi+β. Show that the Bayes estimator for λ = 1/ is Ȳ = (n+α-1) / Σyi + β(n+α-1)
Solve only 016.18: not Q16.10
16.18
Applet Exercise In Exercise 16.10, we found the posterior density for λ based on a sample of size n from an exponentially distributed population with mean 1/λ. Specifically, using the gamma density with parameters α and β as the prior for λ, we found that the posterior density for y1, y2, yn is a gamma density with parameters α* = n+α and β* = Σyi+1. Assuming that a sample of size n = 15 produced a sample such that Σyi = 30.27 and that the parameters of the gamma prior are α = 2.3 and β = 0.4, use the applet Gamma Probabilities and Quantiles to find 80% credible intervals for λ and 1/λ, the mean of the exponential population.
