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Unformatted text preview: ls per minute. Measure time in minutes and consider an interval of time beginning at time t = 0. Let N (t) denote the number of calls that arrive up until time t. For a fixed t > 0, the random variable N (t) is a Poisson random variable with parameter −2t i 2t, so it’s pmf is given by P {N (t) = i} = e i(2t) for nonnegative integers i. ! (a) Find the probability of each of the following six events: E1 =“No calls arrive in the first 3.5 minutes.” E2 =“The first call arrives after time t = 3.5.” E3 =“Two or fewer calls arrive in the first 3.5 minutes.” E4 =“The third call arrives after time t = 3.5.” E5 =“The third call arrives after time t.” (for general t > 0) E6 =“The third call arrives before time t.” (for general t > 0) (b) Derive the pdf of the arrival time of the third call. (c) Find the expected arrival time of the tenth call? Solution: Since λ = 2, N (3.5) has the Poisson distribution with mean 7. Therefore, P (E1 ) = −7 (7)0 P {N (3.5) = 0} = e 0! = e−7 = 0.00091. Even...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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