The M-Wright function appears in the literature in many forms. Its symmetric extension to the entire real line is attractive to practitioners as it naturally generalizes the Gaussian distribution. In this paper, we provide an explanation of the different parametrizations of the M-Wright function in terms of a scaling property of a generalization of the standard Poisson process called fractional Poisson process (fPp). In particular, the M-Wright function is derived to be the limiting distribution of an appropriately scaled fPp. The -stable representation of the limiting distribution is showed, and special cases of our results are provided as well.