We shall introduce a general probability mass function which includes several discrete probability mass functions. Especially, when the random variable X is Poisson, binomial, and negative binomial random variables as some special cases of the introduced distribution, the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) of the probability P(X ${leq}$ t) are considered. And the efficiencies of the MLE and the UMVUE of the reliability ar compared each other.