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* Moment Generating Function                                             :fc:
:PROPERTIES:
:ID:       2ffc8b34-b2b5-4472-9295-714b5422679d
:FC_CREATED: 2021-11-30T01:46:02Z
:FC_TYPE:  normal
:FC_ALGO:  noop
:END:
:REVIEW_DATA:
| position | due                  |
|----------+----------------------|
|        0 | 2000-01-01T00:00:00Z |
|        1 | 2000-01-01T00:00:00Z |
|        2 | 2000-01-01T00:00:00Z |
:END:
Define the moment generating function
** Back

\[ M_X(t) := \mathbb{E}\!\left[e^{tX}\right], \quad t \in \mathbb{R}\]

Differentiating $M_X(t)$ $i$ times with respect to $t$ and setting $t=0$ we obtain the $i\textup{th}$ moment.