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Let $N(t)$ be $PP(λ)$.Given that $N(t)=n$, compute the probability of

a) Last event before $t$ occurs before $3t/4$.

b) First event after $t$ occurs after $t+h$, $0<h$.

c) $S_1<2$, $S_3$>4 for $t=10, n=3$.

I know I should use order statistics to compute these probabilities. But cannot figure out how.

a) $P(S_{N(t)}<{3t}/4|N(t)=n)$

$={P(S_{N(t)}<{3t}/4, N(t)=n)}/P(N(t)=n)$

$=P(S_{N(t)}<{3t}/4)$

b)$P(S_{1}>h|N(t)=n)$

c)$P(S_{1}<2, S_3>4|N(10)=3)$

Also, any good resource with many solved examples on Poisson processes and Continuous time Markov Chains will be highly appreciated.

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    $\begingroup$ This might help: stats.stackexchange.com/a/412240/303650 $\endgroup$
    – fblundun
    Jan 16, 2021 at 21:19
  • $\begingroup$ What does $S_2$ mean? The time of the second event? $\endgroup$
    – Henry
    Jan 17, 2021 at 1:06
  • $\begingroup$ Yes, $S_i$ means occurence of the $i^{th}$ event. @Henry $\endgroup$
    – diabolik
    Jan 17, 2021 at 9:37
  • $\begingroup$ Any idea @Henry ? $\endgroup$
    – diabolik
    Jan 18, 2021 at 11:44

1 Answer 1

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Hints:

For (b) you can use the memoryless property of the Poisson process to say this is the same as the probability that the first event happens after $h>0$

Meanwhile, as the link from @fblundun indicates, $N(t)=n$ is the equivalent of the timing of $n$ events by time $t$ being independently and uniformly distributed in $[0,t]$. So the first and third questions become:

(a) all $n$ events happen in the first $\frac34$ of the time interval

(c) with three events, the earliest occurs in the first $\frac15$ and the latest in the last $\frac35$ of the time interval, the probability of which can be found by inclusion/exclusion

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