3. Suppose an insurance company's number of claims up to time t follows a Poisson process...

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3. Suppose an insurance company's number of claims up to time t follows a Poisson process N(t) with rate A=8. Let W, be the i-th interarrival time and T = W, be the n-th claim arrival time. (a) Calculate P{N(3) = 2). (b) Calculate P{W 5}. (c) Calculate E[Ts). Assume that the company has two types of businesses,, namely health and car insurance. For any given claim, it comes from health insurance with a probability p = 0.6 and car insurance with a probability of q=0.4. Let N(t) and Ne(t) denote the number of claims from health insurance and car insurance up to time t respectively. (d) Calculate P{N(4) + Ne(4) = 8). (e) Calculate P{N(3) = 3}. (f) Calculate P{N(3) 5|N.(3) 2}. (g) Calculate P{N(3) = 2|N(3)=5}. Assume that the claim amount (X) for health insurance are independent and follows an exponentail distri- bution with parameter 0.002, and the one for car insurnance is (X) which are also independent and follows Gamma(10,5), and assume that X and X are independent. (h) Calculate expected total claim amount of the company in 2 years. (i) Calculate the expected total claim amount from the car insurance in 2 years. 3. Suppose an insurance company's number of claims up to time t follows a Poisson process N(t) with rate A=8. Let W, be the i-th interarrival time and T = W, be the n-th claim arrival time. (a) Calculate P{N(3) = 2). (b) Calculate P{W 5}. (c) Calculate E[Ts). Assume that the company has two types of businesses,, namely health and car insurance. For any given claim, it comes from health insurance with a probability p = 0.6 and car insurance with a probability of q=0.4. Let N(t) and Ne(t) denote the number of claims from health insurance and car insurance up to time t respectively. (d) Calculate P{N(4) + Ne(4) = 8). (e) Calculate P{N(3) = 3}. (f) Calculate P{N(3) 5|N.(3) 2}. (g) Calculate P{N(3) = 2|N(3)=5}. Assume that the claim amount (X) for health insurance are independent and follows an exponentail distri- bution with parameter 0.002, and the one for car insurnance is (X) which are also independent and follows Gamma(10,5), and assume that X and X are independent. (h) Calculate expected total claim amount of the company in 2 years. (i) Calculate the expected total claim amount from the car insurance in 2 years.