Suppose that the number of events that occur in a specified time is a Poisson random varia...

Suppose that the number of events that occur in a specified time is a Poisson random variable with parameter λ. If each event is counted with probability p, independently of every other event, show that the number of events that are counted is a Poisson random variable with parameter .λp. Also, give an intuitive argument as to why this should be so. As an application of the preceding result, suppose that the number of distinct uranium deposits in a given area is a Poisson random variable with parameter λ =10. If, in a fixed period of time, each deposit is discovered independently with probability  find the probability that (a) exactly 1, (b) at least 1, and (c) at most 1 deposit is discovered during that time.