How to find the moment generating function of a poisson translation.

Question

I want to find the mgf of $Y$, where $Y = ((X − λ)/√λ)$ and $X$ ~ Poisson$(λ)$. What steps should I take in order to arrive to the mgf of $Y$?

Answer 1

The mgf of $X \sim$ Poisson($\lambda$) is:

$$ E[e^{tX}] = m_X(t) = \exp (\lambda (e^t -1)) $$

For the transformation, $Y = \frac{X-\lambda}{\sqrt{\lambda}}$, we have:

$$ E[e^{tY}] = E[e^{t\frac{X-\lambda}{\sqrt{\lambda}} }] = e^{-t\sqrt{\lambda}} E[e^{\frac{t}{\sqrt{\lambda}} X }] = e^{-t\sqrt{\lambda}} m_X(t/\sqrt{\lambda}) = e^{-t\sqrt{\lambda}} \exp (\lambda (e^\frac{t}{\sqrt{\lambda}} -1)) $$