ABC is an insurance company that receives two type of claims: car insurance claims and home insurance claims. Each car insurance claim is from a good...

ABC is an insurance company that 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 ABC receives no claims in a week (7 days).

b) Suppose that 10 claims are received on a given day. Determine the probability that exactly 4 of those claims are car insurance claims.

c) Determine the probability density function of the time from time 0 until ABC receives exactly 10 car insurance claims from bad drivers.

d) Suppose that a car insurance claim from a bad driver costs ABC $1000, and a car insurance claim from a good driver costs ABC $200. Determine the probability that ABC spends exactly $2000 on car insurance claims in one