As a risk manager for an insurance company, you are responsible for determining the premium for a new policy that covers a homeowner against losses due to fire. You have collected data on the frequency and severity of fire losses in the area and have estimated the expected frequency and severity of losses. The expected frequency of losses is 0.05 per year, and the expected severity of losses is $50,000 per loss. The insurance company wants to have a 95% confidence level that it will not suffer losses greater than the premium collected in any given year. Assume that the number of losses and the amount of each loss are independent and identically distributed random variables that follow a Poisson distribution and a lognormal distribution, respectively.

(a) Calculate the premium that the insurance company should charge to meet its requirement of a 95% confidence level.
(b) Calculate the expected profit of the insurance company if it sets the premium at the level calculated in (a).

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