1. A random variable Y follows a zero-truncated Poisson distribution with rate > > 0 if it has mass function Aye - d P( Y = y) = (1 - e-d)y!' for y = 1, 2, ... (a) Show that the zero-truncated Poisson distribution belongs to an exponential family. What are the canonical parameter 0, the cumulant b(0), and the dispersion o? (b) Using the cumulant, find the mean / of Y. Show that it coincides with E[Z | Z > 0] where Z ~ Poisson(1). (c) Obtain the inverse canonical link for this distribution