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Definition

ProbabilityStochasticProcesses

For a rv $X$ , $\mathbb{E}[X^{2}]$ is the second moment of $X$ . If $\mathbb{E}[X^{2}]<\infty$ , then $X\in \mathscr{L}^{2}$ .

For rv $X$ , $k\in \mathbb{Z}_{+}$ , we say $\mathbb{E}[X^{k}]$ is the $k^{th}$ moment of $X$ (well-defined for $X\in \mathscr{L}^{k}$ ).

For a random variable $X$ , its moment-generating function (MGF) is $M_X(t)=E[e^{tX}], \mbox{ for } t\in\mathbb{R}$ if $M(t)<\infty$ for each $t\in\mathbb{R}$ .

1. Uniqueness: Let $X$ and $Y$ be RVs with MGFs $M_X(t),M_Y(t)$ . Then $M_X(t)=M_Y(t)\ \forall t\in (-\delta,\delta), \ \delta>0 \implies F_X(x)=F_Y(y)$ 2. Let $a,b\in\mathbb{R}, \ Y=a+bX$ . Then $M_Y(t)=E[e^{tY}]=E[e^{t(a+bX)}]=e^{at}M_X(tb)$ 3. Derivative: If MGF exists, its derivatives are: $M^{(k)}(t)=E\left[\frac{d}{dt^k}e^{tX}\right]=E\left[X^ke^{tX}\right]$ as a result: $E[X]=M'(0), \ \ E[X^2]=M''(0)$ and so on.

Linked from

Continuous-time Gaussian process motion planning via probabilistic inference

Wasserstein space

Summary of MATH 895