Question: Claims arrive at random times to an insurance company. The daily amount of claims is normally distributed with mean $1570 and standard deviation $450. Total claims on different days each have this distribution, and they are probabilistically independent of one another.
a. Find the probability that the amount of total claims over a period of 100 days is at least $150,000. (Use the fact that the sum of independent normally distributed random variables is normally distributed, with mean equal to the sum of the individual means and variance equal to the sum of the individual variances.)
b. If the company receives premiums totaling $165,000, find the probability that the company will net at least $10,000 for the 100-day period.