Stochastic

be the arrival times of a poisson process with rate

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Unformatted text preview: t there were n arrivals before time t is n!/tn 91 2.5. CHAPTER SUMMARY for all times 0 < t1 < . . . < tn < t, which is the joint distribution of (V1 , . . . , Vn ). The second fact follows easily from this, since there are n! sets {T1 , T2 , . . . Tn } or {U1 , U2 , . . . Un } for each ordered vector (T1 , T2 , . . . Tn ) or (V1 , V2 , . . . , Vn ). Theorem 2.14 implies that if we condition on having n arrivals at time t, then the locations of the arrivals are the same as the location of n points thrown uniformly on [0, t]. From the last observation we immediately get: Theorem 2.15. If s < t and 0 m n, then ✓ ◆⇣ ⌘ ⇣ n sm P (N (s) = m|N (t) = n) = 1 m t s ⌘n t m That is, the conditional distribution of N (s) given N (t) = n is binomial(n, s/t). Proof. The number of arrivals by time s is the same as the number of Ui < s. The events {Ui < s} these events are independent and have probability s/t, so the number of Ui < s will be binomial(n, s/t). 2.5 Chapter Summary A random variable T is...
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