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A further generalized version of one parameter polynomial exponential distribution with binomial probability mass as a mixture called a Binomial Mixture One Parameter Polynomial Exponential Distribution (BMOPPE) is proposed in the article. The moments and stochastic orderings are studied. Maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) of the probability density function and the cumulative distribution function have been derived and compared in the mean squared error sense. Estimation issues (both MLE and UMVUE) of reliability functions- mission time and stress-strength have been considered, and asymptotic variances of MLEs and variances of UMVUEs have been derived. UMVUEs of the variance of UMVUE of reliability functions have also been derived. Simulation study results have been reported to validate the theoretical findings. Few data sets have been fitted and compared.