B e s 4 n 1 2 solution e s 4 n 1 2 1 etime for 2 more

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(b)E[S4|N(1) = 2],
50.The number of hours between successive train arrivals at the station is uniformly dis-tributed on (0,1). Passengers arrive according to a Poisson process with rate 7 per hour.Suppose a train has just left the station. LetXdenote the number of people who get on thenext train. Find(a)E[X],
(b)Var(X).
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57.Events occur according to a Poisson process with rateλ= 2 per hour.(a)What is the probability that no event occurs between 8 P.M. and 9 P.M.?
(b)Starting at noon, what is the expected time at which the fourth event occurs?
(c)What is the probability that two or more events occur between 6 P.M. and 8 P.M.?
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59.There are two types of claims that are made to an insurance company.LetNi(t)denote the number of typeiclaims made by timet, and suppose that{N1(t), t0}and{N2(t), t0}are independent Poison processes with ratesλ1= 10 andλ2= 1. The amountsof successive type 1 claims are independent exponential random variables with mean $1000whereas the amounts from type 2 claims are exponential random variables with mean $5000.A claim for $4000 has just been received; what is the probability it is a type 1 claim?
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