Let X be a Poisson variable with parameter lambda. Prove that the Moment Generating Function of Y = (X-lambda)/sqr(lambda) is equal to My(t) = exp{lambda*exp{t/lambda)}-sqr(lambda)*t-lambda}.
I substituted Y to get the fy(Y). I tried calculating the summation from y=0 to infinity of e^((X-lambda)/sqr(lambda)*t)[lambda^[(X-lambda)/sqr(lambda)]e^-lambda/lambda! but I aint getting nowhere... Thank you!
It's easiest to start from the MGF of $X$, which is$$M_X(t)=\exp(-\lambda)\sum_{k\ge0}\frac{(\lambda\exp t)^k}{k!}=\exp[\lambda(\exp t-1)].$$Then$$M_Y(t)=M_{\frac{X}{\sqrt{\lambda}}-\sqrt{\lambda}}(t)=M_X\left(\frac{t}{\sqrt{\lambda}}\right)\exp(-t\sqrt{\lambda})=\exp\left[\lambda\left(\exp\frac{t}{\sqrt{\lambda}}-1\right)-t\sqrt{\lambda}\right].$$In the context of the central limit theorem, we note the small-$t$ approximation$$M_Y(t)=\exp\left[\frac12t^2+o\left(t^2\right)\right].$$