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Duronto Express arrives at the Bombay Central station according to a Poisson process of rate 3 trains/hour. Local Line trains arrive according to a Poisson process of rate 4 trains/hour.

Conditionally on the event that 8 trains arrive from 9 am to 9:40 am, what is the probability that no trains arrived between 9:10 and 9:20?

So far I used the Superposition Lemma and so the arrival of all the trains at the station is a Poisson process of rate 7 trains/hr

Now I want to find $P(Z(20)-Z(10)=0 | Z(40) = 8)$, where $Z(t) = X(t)+Y(t)$, but I am stuck at this point and don't know how to solve it from here

bluelagoon
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1 Answers1

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Hint : $P[Z(t+h) - Z(t)=k] =\frac {e^{\lambda h}{(\lambda h)}^k}{k}$

Note : Make sure you convert $h$ to the same time units as in $\lambda$

Vishaal Sudarsan
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  • I know that property but I cannot use it since I don't two independent time interval, one is in the (10,20] and other is (0,40] – bluelagoon Nov 23 '20 at 10:21
  • Use the definition of conditional probability and use this property to find the probability of the intersectional event and the denominator... – Vishaal Sudarsan Nov 23 '20 at 10:27