**Problem 3: Poisson Distribution in Insurance Claims**Assume that a policyholder is sixteen times more likely to file exactly two claims than to file exactly four claims in one year.Let $ X $ be the number of claims in a year, where $ X \sim \text{Poisson}(\lambda) $.Tasks:- (a) Find $\lambda$.- (b) Calculate $\Pr(X \geq 2)$.- (c) Determine the expected value $E(X!)$.

given data x follows poisson distribution P(x=X) = e-λ(λ)xx! P(x=2) = 16P(x=4)

Answered: a) 3. Assume that a policyholder is…

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Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where i varies over the portfolio according to a G(a, 3) mixing distribution. The respective densities of the two distributions are given by Sx (x) = (1/2) exp (-x/à), x > 0, 2 > 0; S(2) =- T(a) 2-l exp(-8i), 2 > 0. Given that the Pareto pdf given by fx (x) = (5 +x)ª+1 * > 0, a > 0, 8> 0. (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a, 5). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, 5), then E (X) = .

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a) 3. Assume that a policyholder is sixteen times more likely to file exactly two claims as to file exactly four claims in one year. Let X be the number of claims in a year and X~ Poisson (X). (a) Find X (b) Pr(X > 2) 16 (c) E(X!)