6.32 Suppose that , the mean rate of arrival, in the Poisson distribution is time-
dependent and is given by
Determine pmf the probability of exactly k arrivals in the time interval
[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 X(0, t 1 ),
and with the same mean rate of arrival. Determine prob-
ability 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.
Some Important Discrete Distributions 189
vtv^1
w
:pk(0,t),
X X(0,t 2 ),t 2 >t 1 ,
P(X 1 t 1 \X 2 t 2 ).2