This is a compound probability distribution based on the theoretical concept of continuous mixtures of Poisson distribution. It has a single parameter and two irrational numbers. It is named as ‘Polynomial-Exponential Mixture of Poisson Distribution (PEMPD) which will play a very important role in the field of statistical modeling of over-dispersed count data. Statistical characteristics have been defined and derived according to the need of the proposed paper. By applying goodness of fit to some over-dispersed count data of secondary nature, it has been observed that it is a better alternative of Poisson-Lindley distribution (PLD) of Sankaran, Poisson-Mishra distribution (PMD) of Sah and Poisson-Modified Mishra (MMD) distribution of Sah and Sahani.