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The number of buses that arrive at the main station during 4:00 PM and 5:00 PM is a Poisson random variable with parameter 1. Each bus has 20 seats which might be empty or taken by a passenger. The probability that each seat is taken is p, independent of other seats. Moreover, passengers are not allowed to stand up. (a) Let M be the number passengers in bus number 1. Write the p.m.f. of My and determine its range. (b) Let p = 0.2, and find the probability that more than 11 passengers are riding on the bus. Compute this probability exactly and compare it against the Markov bound. (c) Let p = 0.8, and compute the probability that there are at most 10 passengers on the bus. Find this probability and compare it against what you get using Chebyshev inequality. (d) Consider a general p, and let R denote the total number of passengers arrive at the station during this time. Determine E[R] (e) Compute Var(R). The number of buses that arrive at the main station during 4:00 PM and 5:00 PM is a Poisson random variable with parameter 1. Each bus has 20 seats which might be empty or taken by a passenger. The probability that each seat is taken is p, independent of other seats. Moreover, passengers are not allowed to stand up. (a) Let M be the number passengers in bus number 1. Write the p.m.f. of My and determine its range. (b) Let p = 0.2, and find the probability that more than 11 passengers are riding on the bus. Compute this probability exactly and compare it against the Markov bound. (c) Let p = 0.8, and compute the probability that there are at most 10 passengers on the bus. Find this probability and compare it against what you get using Chebyshev inequality. (d) Consider a general p, and let R denote the total number of passengers arrive at the station during this time. Determine E[R] (e) Compute Var(R)