Q5. (a) What is meant by a conjugate prior?
(b) It is believed that the number of accidents in a new factory will follow a Poisson
distribution with mean $\theta$ per month as follow:
$P(X = x|\theta) = \frac{e^{-\theta}\theta^x}{x!}$ $x = 0, 1, 2, ...$
The prior distribution of $\theta$ is given by a gamma distribution with density
$p(\theta) = \frac{\beta^\alpha}{\Gamma(\alpha)}\theta^{\alpha - 1}e^{-\beta\theta}$ $\theta > 0$
Derive the posterior probability distribution for $\theta|X$ including the normalising
constant.
(c) If $\alpha = 12$, $\beta = 4$ and there are 27 accidents in the first eight months, derive the
posterior distribution of $\theta$ and find its mean and variance.
(d) Show the posterior mean in part (c) can be written as a weighted average of
the prior mean and the sample mean.
(2 marks)
(8 marks)
(6 marks)
(4 marks)
