QUESTION 1
(a) Let X denote the number of claims lodged by a policy holder in a policy year. Given that
$X|p \sim (1, p)$; $p \sim Be(\alpha, \beta)$; $x = 0, 1$,
(i) what is the distribution of the number of claims in a policy year made on the insurance
company by a policy holder chosen at random? That is, find the marginal $f_x(x)$.
(ii) Confirm the mean of the mixture distribution in (a) using the formula for conditional ex-
pectations
$E(X) = E_p[E_{X|p}(X)]$.
(b) Given that the compound Rayleigh pdf is
$f_x(x) = 2\beta^\delta x (\beta + x^2)^{-(\delta + 1)}$, $x > 0$,
find
(i) $P(X > t)$, and
(ii) $E(X)$.
