In this paper, we start with a distribution of a random sum of N independent exponential random variables when the random number N has a discrete uniform distribution with positive integer parameter k and show that the probability density function (pdf) and cumulative distribution function (cdf) of this new distribution remain pdf and cdf for any positive real number k. This distribution is proportional to the upper tail probability of the gamma distribution. The hazard rate function of the new model can be constant, decreasing, and increasing depending on its parameters. The statistical properties of the distribution are studied. Estimation of the unknown parameters are discussed by the moments and maximum likelihood methods. The usefulness of the distribution is illustrated by two real data sets.
