18.44 Chapter 5: MIT18_440S14_ProblemSet5.pdf
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18. 440 problem set five, due march 21: from textbook chapter four, problem 70: at time 0 a coin that comes up heads with probability p is ipped and falls to the ground. At times chosen according to a poisson process with rate , the coin is picked up and ipped. (between these times the coin remains on the ground. ) Show that p {x = i} increases monotonically and then decreases monotonically as i increases, reaching its maximum when i is the largest integer not exceeding . P {x = i}/p {x = i 1}: theoretical exercise 25: suppose that the number of events that occur in a speci ed time is a poisson random variable with parameter . If each event is counted with probability p, independently of every other event, show that the number of events that are counted is a poisson random variable with parameter p. Also, give an intuitive argument as to why this should be so.
