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November 15th, 2015, 11:35 AM   #1
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Joined: Feb 2015
From: london

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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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