What are Moment Generating Functions 🌱

Last updated on August 10, 2020

The moment-generating function $m (t)$ for a random variable $Y$ is defined as the following:

m (t) = E (e^{t Y})

If $m (t)$ exists, then for any positive integer $k$ , the $k$ -th derivative of $m (t)$ is:

{\frac{d^{k} m (t)}{d t^{k}} |}_{t = 0} = m^{(k)} (0) = E (Y^{k})

This is true because of the following:

\begin{aligned} e^{t Y} & = 1 + \frac{t Y}{1!} + \frac{t^{2} Y^{2}}{2!} + \frac{t^{3} Y^{3}}{3!} + \dots + \frac{t^{n} Y^{n}}{n!} + \dots \\ m (t) = E [e^{t Y}] & = 1 + t E [Y] + \frac{t^{2}}{2!} E [Y^{2}] + \frac{t^{3}}{3!} E [Y^{3}] + \dots + \frac{t^{n}}{n!} E [Y^{n}] + \dots \end{aligned}

MGF Determines Distribution

Suppose $m_{X} (t), m_{Y} (t)$ are mgfs of the random variables $X$ and $Y$ respectively. If both mgfs exist and $m_{X} (t) = m_{Y} (t)$ for all values of $t$ , then $X$ and $Y$ have the same probability distribution

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