Question: Let Nt,t=0 be a Poisson process with intensitya. Let X be a random variable, independent of the Poisson process Nt, meaning that X is independent of all the "gap times" Lkused to define Nt. You are given that X is exponentially distributed with parametera ~.
(a) Find the distribution of NX, that is, the Poisson process N that the random time t=X. That is, find P(NX=k), for each k? Z+. Hint: first compute P(NX=k), which can be expressed as a two-dimensional integral. Then write P(NX=k) =P(NX=k)-P(NX=k+ 1).
(b) Suppose that Y is another random variable independent of the Poisson process, this time with the uniform distribution on the time interval [0, b]. What is the probability that NY= 0? Hint: as a check on your answer, the probability P(NY= 0) should tend to 1 asb? 0, and it should tend to 0 asb?