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?