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Homogeneous Poisson process N(t) counts events occurring in a time interval and is characterized by Ņ(0)-0...

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Homogeneous Poisson process N(t) counts events occurring in a time interval and is characterized by Ņ(0)-0 and (t + τ)-N(k) ~ Poisson(λτ), where τ is the length of the interval (a) Show that the interarrival times to next event are independent and exponentially distributed random variables (b) A random variable X is said to be memoryless if P(X 〉 s+ t | X 〉 t) = P(X 〉 s) y s,t〉0. that this property applies for the interarrival times if they are exponentially (c) Bonus question, extra marks are possible. Show that the property in (b) (d) Show that the conditional distribution of Ņ(s)-k given (t) n, where s 〈 t Show distributed. applies for the interarrival times only if they are exponentially distributed and k 〈 n, is binomial e) Suppose that patients arrive at an emergency room at the rate of λ-50 per day. i. What is the expected time (in hours) for 10 new patients to arrive. ii. What is the probability that an hour goes by without a new patient arriving?

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