LetNt,t=0 be a Poisson process with intensitya. LetXbe a random variable, independent of the Poisson processNt, meaning thatXis independent of all the “gap times”Lkused to define Nt. You are given...

LetNt,t=0 be a Poisson process with intensitya. LetXbe a random variable, independent of the Poisson processNt, meaning thatXis independent of all the “gap times”Lkused to define Nt. You are given thatXis exponentially distributed with parametera ~.

(a) Find the distribution ofNX, that is, the Poisson processNtat the random time t=X. That is, findP(NX=k), for eachk?Z+. Hint: first computeP(NX=k), which can be expressed as a two-dimensional integral. Then writeP(NX=k) =P(NX=k)-P(NX=k+ 1).

(b) Suppose thatYis another random variable independent of the Poisson process, this time with the uniform distribution on the time interval [0, b]. What is the probability thatNY= 0? Hint: as a check on your answer, the probabilityP(NY= 0) should tend to 1 asb?0, and it should tend to 0 asb? 8.