Page 206 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 189

6.32 Suppose that , the mean rate of arrival, in the Poisson distribution is time-
dependent and is given by

vt v 1
ˆ :
w
Determine pmf p (0, t), the probability of exactly k arrivals in the time interval
k
[0, t). [Note that differential equations such as Equations (6.39) and (6.42) now
have time-dependent coefficients.]
6.33 Derive the jpmf of two Poisson random variables X 1 and X 2 , where X 1 ˆ X (0, t 1 ),

and X ˆ X(0, t 2 ), t 2 > t 1 , with the same mean rate of arrival . Determine prob-
2
ability P(X 1 t 1 \ X 2 t 2 ). This is the probability that the numbers of arrivals
in intervals [0, t 1 ) and [0, t 2 ) are both equal to or less than the average arrivals in
their respective intervals.

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