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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.
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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.
 Spring '08
 Zahrn
 The Land

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