My Math Forum Moment generating function

 November 15th, 2015, 11:35 AM #1 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 Moment generating function Some variable Z has a poisson distribution where a > 0 $\displaystyle Pz(i) = e^{-a}\frac{a^i}{i!}$ $\displaystyle Mz(t) = e^{-a(1-e^t)}$ <- Moment generating function of Z $\displaystyle Y = \frac{Z -a }{\sqrt{a}}$ I need to calculate the natural logarithm of the moment generating function of Y.Is my working below correct? $\displaystyle Y = \frac{1}{\sqrt{a}} Z - \sqrt{a}$ $\displaystyle My(t) = E[e^{t(\frac{1}{\sqrt{a}}Z - \sqrt{a})}]$ $\displaystyle My(t) = e^{-\sqrt{a}t} E[e^{\frac{1}{\sqrt{a}}tz} ]$ $\displaystyle My(t) = e^{-\sqrt{a}t} Mz(\frac{1}{\sqrt{a}}z)$ $\displaystyle My(t) = e^{-\sqrt{a}t} e^{-a(1-e^{\frac{1}{\sqrt{a}}z})}$ $\displaystyle In My(t) = -a (ta^{-0.5} -1 + e^{\frac{1}{\sqrt{a}}z})$

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