71. Let Sn 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

N(t)

i=1 g(Si)

has the same distribution as the compound Poisson random variable

N(t)

i=1 g(Ui), where U1,U2, . . . is a sequence of independent and identically distributed uniform

(0, t) random variables that is independent of N, a Poisson random variable with mean λt. Consequently, conclude that

Transcribed Image Text:

N(t) (N(t) (5) () = = g(x) dx Var g(Si) = * f g (x) dx 0 i=1 i=1