# Let Sn denote the time of the nth event of the Poisson process 1 answer below »

Let S_{n} denote the time of the nth event of the Poisson process {N(t), t 0} having rate λ. Show, for an arbitrary function g, that the random variable g(S_{i} ) has the same distribution as the compound Poisson random variable g(U_{i} ), where U_{1},U_{2}, . . . is a sequence of independent and identically distributed uniform (0, t ) random variables that is independent of N, a Poisson random variable with mean λ_{t}. Consequently, conclude that

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Let Sn denote the time of the nth event of the Poisson process
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