5) [20 points] An insurer's portfolio consists of two types of contracts: A and B. After collecting the premiums, he/she has a capital C. For type A, if there is a claim, then its size is x with probability PA(x), where PA1=0.4, PA2=0.4, and PA3=0.2. For type B, the claim probability is 0.02 and the number of policies is nB=500. If there is a claim, then its size is y with probability PB(y), where PB1=0.4, PB2=0.1, and PB3=0.5. Suppose that the contracts are independent. Denote SA as the total claim amount of the contracts of type A and SB as the total claim amount of the contracts of type B. S=SA+SB.
a) [5 points] Calculate E[SA] and VAR[SA].
b) [5 points] Calculate E[SB] and VAR[SB].
c) [5 points] Calculate E[S] and VAR[S].
d) [5 points] Apply the Central Limit Theorem (CLT) to determine the minimum capital C that covers all claims with probability 0.90.