4. Given a Poisson process X(t), t > 0, of rate λ > 0, let us fix a time, say t-2, and let us con...

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4. Given a Poisson process X(t), t > 0, of rate λ > 0, let us fix a time, say t-2, and let us consider the first point of X t

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Answer 1

Answer #1

Note that W- 2 conditional on the time $\tau$ of the last arrival before 2, is simply the remaining time until the next arrival. Since the inter-arrival time starting at $\tau$ is exponential and thus memoryless, W-2 is independent of $\tau$ <= 2 and of all earlier arrivals

P(W - 2 > x | N(2) = 0}

we first condition on N(2) = 0 , we see that X1 > 2 and W-2 = X1-2

given N(t) = 0

= P(X1 > w -2 + 2 | N(2)= 0}

= P(X1 > w)

=e^(- $\lambda$ w)

hence it follows exponential distribution

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Answer 2

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