Mean and variance of the number of renewals of a censored Poisson process

Expressions are derived for computing the mean and variance of the number of renewals in observation times of any length and for any values of the scaling parameter of the censored Poisson process. These results depend on certain identities involving the complex roots of unity. Computations show the oscillatory behavior to be expected when the average interval between renewals of the censored Poisson process is of the order of the observation time. Both ordinary and equilibrium renewal processes are considered.