Problem 3 [15 points = 6 + 9] Assume that a bus left the station at t = O and passengers' arrivals form a Poisson process with rate A = 8 per hour. Next bus arrives at a random time (T) independent on a number of passengers at the station. Density function of T is 4 4 fT(t) = F -t3 -exp(4t) for t > 0, 1. Find expectation, E [X (1')] 2. Evaluate variance, Var [X (t)] Notice that T is also measured in hours! Solution Problem 4 [15 points = 9 + 6] Assume that arrivals X = {X(t); t 2 0} form a Poisson process with the rate ) = 6 per hour. 1. Determine the mixed moment, E [X(2) . X(4)]. 2. Derive the covariance, COV [X(3), X(5)].Problem 5 [20 points = 8 + 12] Passengers arrive at a bus stop so that the arrival times are described by a Poisson process with the intensity A = 15 per hour. A bus departed at t = 0 and all passengers were on board, with no one left behind. Next bus arrives at random time T g 1 that does not depend on the process X = {X (t) : t 2 0}. Assume that T has cumulative distribution function, G(t)=4-t33-t4 valid for (0 S w 5 1). Consider a random variable, Y = X (T), the number of passengers accumulated by T at the station. 1. Derive expectation of Y. 2. Determine variance of 1". Solution