Chapter 4, Problem 4.26TE

Suppose that the number of events that occur in a specifiedime is a Poisson random variable with parameter $\lambda$. 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 $\lambda p$. 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 $\lambda =10$. If, in a fixed period of time, each deposit is discovered independently with probability $\frac{1}{50}$, find the probability that (a) exactly 1 (b) at least 1, and (C) at most I deposit is discovered during that time.