An actuary wishes to estimate various quantities related to the number of insurance

Chapter 9, Problem 42

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An actuary wishes to estimate various quantities related to the number of insurance claims and the dollar amounts of those claims for someone named Fred. Suppose that Fred will make N claims next year, where N| Pois(). But is unknown, so the actuary, taking a Bayesian approach, gives a prior distribution based on past experience. Specifically, the prior is Expo(1). The dollar amount of a claim is Log-Normal with parameters and 2 (here and 2 are the mean and variance of the underlying Normal), with and 2 known. The dollar amounts of the claims are i.i.d. and independent of N. (a) Find E(N) and Var(N) using properties of conditional expectation (your answers should not depend on , since is unknown and being treated as an r.v.!). (b) Find the mean and variance of the total dollar amount of all the claims. (c) Find the distribution of N. If it is a named distribution we have studied, give its name and parameters. (d) Find the posterior distribution of , given that it is observed that Fred makes N = n claims next year. If it is a named distribution we have studied, give its name and parameters.