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Unformatted text preview: to result in a count, so the time of the ﬁrst count is 7h, as shown in
3.7(b) for h = 0.1. We deﬁne the following random variables to describe the time-scaled Bernoulli
process with time step h:
• Uj = hLj : the amount of time between the j − 1th count and the j th count
• Tj = hSj : the time the j th count occurs
• Nt = C t/h : the number of counts up to time t 82 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES The tilde’s on the random variables here are used to distinguish the variables from the similar
random variables for a Poisson process, deﬁned below.
Suppose λ is ﬁxed and that h is so small that p = λh is much smaller than one. Then the
random variables describing the scaled Bernoulli process have simple approximate distributions.
Each Lj is a geometrically distributed random variable with parameter p, so as explained in Section
3.4, the scaled version of Lj , namely Uj = hLj , approximately has the exponential distribution
with parameter λ = p/h. For t ﬁxed, Nt is the sum of t/h Bernoulli random var...
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