User:Kimaaron/Sandbox

A place to try things first without worrying too much about other people changing it as happens on the main page sandbox and without having the material deleted 12 hours later.

Why the moment-generating function is defined this way

The reason for defining this function is that it can be used to find all the moments of the distribution. The series expansion of $e^{tX}$ is

${\begin{aligned}e^{tX}&=1+tX+{\frac {t^{2}X^{2}}{2!}}++{\frac {t^{3}X^{3}}{3!}}\cdots \end{aligned}}$

Hence

${\begin{aligned}M(t)&=E(e^{tX})&=1+tm_{1}+{\frac {t^{2}m_{2}}{2!}}+{\frac {t^{3}m_{3}}{3!}}+\cdots ,\end{aligned}}$

where m_i is the ith moment.

If we differentiate M(t) i times with respect to t and then set $t=0$ we shall therefore obtain the ith moment about the origin, m_i. This is summarized more compactly below in the section entitled Calculations of moments, but the more detailed explanation here gives a little more insight.