a given a rate poisson process nt and an independent exponential random variable x what is the d

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Question

(a) Given a rate λ Poisson process N(t) and an independent exponential (α) random variable X, what is the distribution of number of arrivals N(X) in the interval [0, X]? (b) Events occur according to a Poisson process with rate λ. Each time an event occurs, we must decide whether or not to stop, with our objective being to stop at the last event to occur prior to some specified time T, where T > 1 λ . That is, if an event occurs at time t, 0 ≤ t ≤ T, and we decide to stop, then we win if there are no additional events by time T, and we lose otherwise. If we do not stop when an event 1 occurs and no additional events occur by time T, then we lose. Also, if no events occur by time T, then we lose. Consider the strategy that stops at the first event to occur after some fixed time s, 0 ≤ s ≤ T. Compute the probability of winning when using the preceding strategy, for an optimally chosen s.

          (a) Given a rate λ Poisson process N(t) and an independent exponential (α)
random variable X, what is the distribution of number of arrivals N(X) in the interval
[0, X]?
(b) Events occur according to a Poisson process with rate λ. Each time an event
occurs, we must decide whether or not to stop, with our objective being to stop at
the last event to occur prior to some specified time T, where T > 1
λ
. That is, if an
event occurs at time t, 0 ≤ t ≤ T, and we decide to stop, then we win if there are no
additional events by time T, and we lose otherwise. If we do not stop when an event
1
occurs and no additional events occur by time T, then we lose. Also, if no events occur
by time T, then we lose. Consider the strategy that stops at the first event to occur
after some fixed time s, 0 ≤ s ≤ T. Compute the probability of winning when using
the preceding strategy, for an optimally chosen s.

Added by Bhushan M.

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