Degenerate Zero-Truncated Poisson Random Variables

Recently, the degenerate Poisson random variable with parameter $$\alpha > 0$$ , whose probability mass function is given by $$P_{\lambda}(i) = e_{\lambda}^{-1} (\alpha) \frac{\alpha^{i}}{i!} (1)_{i,\lambda}$$ $$(i 0,1,2,\dots)$$ was studied. In theory, zero-truncated distributions are certain discrete supports set of positive integers. These also known as conditional or distributions. this paper, we introduce variables functions a natural extension distributions, and investigate various properties those variables.