Suppose you observe the values 6 7 9 9 16 and you believe

Question

Let X₁, X₂, . . ., XN be a random sample from the Poisson distribution with pdf f (x) = θxe-θ/x!, x = 0, 1, 2,... .

(a) Write down the likelihood function f (θ).

(b) Suppose the prior for θ is the gamma distribution with parameters r, λ. Let π(θ) denote the pdf for this prior. Find the posterior density f (θ)π(θ).

(c) Recognize this posterior density as the pdf for what known distribution wisth what parameters?

(d) Suppose you observe the values 6, 7, 9, 9, 16 and you believe the prior is gamma with parameters r = 15, λ = 3. Find the posterior density.

(e) Find a 95% credible interval for the θ