For general t 0 e6 the third call arrives before time

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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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