an insurance company has issued $1200$ non-life insurance policies. the number of claims for each policy has a Poisson distribution with an expectation of $0.016$. The distribution of the amount of claim is given in table:
$$\begin{array}{ll}\text {amount of claim : }&& 200& 400& 600& 800& 1000\\ \text {probability :}&& 0.1& 0.2& 0.3& 0.3& 0.1 \end{array}$$
a. what is the expectation and variance of the amount of claim.
b. if the insurer wants to be $95\%$ sure that there will be a gain in performance. find the required relative surcharge.
