The methods of generate a probability function from a probability density function has long been used in recent years. In general, the discretization process produces probability functions that can be rivals to traditional distributions used in the analysis of count data as the geometric, the Poisson and negative binomial distributions. In this paper, by the method based on an infinite series, we studied an alternative discrete Lindley distribution to those study in Gomez (2011) and Bakouch (2014). For both distributions, a simulation study is carried out to examine the bias and mean squared error of the maximum likelihood estimators of the parameters as well as the coverage probability and the width of the confidence intervals. For the discrete Lindley distribution obtained by infinite series method we present the analytical expression for bias reduction of the maximum likelihood estimator. Some examples using real data from the literature show the potential of these distributions.