A local insurance company receives two type of claims: car insurance claims and home insurance claims. Each car insurance claim is from a good driver with probability 0.9, or from a bad driver with probability 0.1, independent of all other car insurance claims. Car insurance claims are received according to a Poisson process with rate 20 claims per day. Home insurance claims are received according to a Poisson process with rate 5 claims per day. The two processes are independent. a) Determine the probability that this company receives no claims in a month (30 days). b) Suppose that 8 claims are received on a given day. Determine the probability that exactly 3 of those claims are car insurance claims. c) Determine the probability density function of the time from time 0 until this company receives exactly 10 car insurance claims from bad drivers. d) Suppose that a car insurance claim from a bad driver costs this company $1000, and a car insurance claim from a good driver costs the company $200. Determine the probability that the company spends exactly $2000 on car insurance claims in one day.