202405232040
Status: #idea
Tags: Probability

Moment Generating Function

It is a special case of the Characteristic Function.
It is a function defined as:

M_{X} (t) = E [e^{t x}], if E (e^{t x}) < \infty for t \in (- h, h)

Which when we take the derivatives (and set $t$ to $0$ ), we obtain the moments of a given random variable. It uses the fact that $e^{t X}$ can be written as the following series $1 + t E [X] + \frac{t^{2}}{2!} E [X^{2}] + \dots$

This gives a compact and simple way to compute all the moments of a random variable from one equation instead of having to repeatedly compute them manually as follows for the $n_{t h}$ moment. For some functions with complicated pdfs the moment generating functions give us a straightforward way to compute arbitrarily many moments.

E [X^{n}] = \int_{- \infty}^{\infty} x^{n} p_{X} (x) d x

Note that as long as we take enough derivatives both are equivalent.

Convenient Trick

No one has time to derivate the $k$ times to get the $k^{t h}$ moment.
So when looking for the $k^{t h}$ moment simply look for the coefficient of $t^{k}$ and multiply that by $k!$ factorial. (In the Taylor's Expansion)

If there is no $t^{k}$ in the Taylor's expansion, you know that the moment is $0$ .

Pasted image 20240531093334.png

Relevant Links

A good video explaining it