DefineXas the sum of a Poisson random variableYwith parameter θ and a nonnegative integer valued variableZindependent ofY.If the distribution ofZis known,Xhas a one-parameter convoluted Poisson distribution. Such distributions are here characterized as solutions to a system of differential equations specifying the rate of change of the cumulative distribution function relative to a parametric change. The characterization is used in point and interval estimation of θ. It is shown that for a certain subclass of convoluted Poisson distributions, the maximum likelihood estimate of θ is the unique solution of the likelihood equation and is easily approximated numerically.