Day: September 24, 2020

For a random variable $latex X$, we can simply define a moment generating function as $latex MG(t) \triangleq E[e^{t X}]$. Then, $latex MG(t)^{(n)}|_{t=0} = E[X^{n}e^{tX}]|_{t=0}=E[X^n]$ is simply the $latex n$-th moment of $latex X$. Easy to verify that $latex \mathcal{N}(0,\sigma^2)$ has moment generating function of $latex e^{\frac{t^2\sigma^2}{2}}$ since $latex E[e^{tX}]=\frac{1}{\sqrt{2\pi\sigma^2}}\int e^{-\frac{x^2}{2\sigma^2}}e^{tx}dx=\frac{1}{\sqrt{2\pi\sigma^2}}\int e^{-\frac{1}{2\sigma^2}[(x-t\sigma^2)^2-t^2\sigma^4]}dx=e^{\frac{t^2\sigma^2}{2}}$ Central limit theorem…