“moment generating function”的概念、定义、翻译、参考文献-科学参考

　

单词	moment generating function
释义	moment generating function Mathematics M(t) = E(e^tX) is the moment generating function (mgf) for a random variable X. When M(t) exists then, since $e^{t X} = \sum_{r = 0}^{\infty} \frac{t^{r} X^{r}}{r!}$ and E is an additive function, it follows that the coefficient of t^r is r!E(X^r), allowing the moments of X to be recovered from knowledge of the mgf. Economics The moment generating function of a random variable X is defined as $M_{X} (t) = E [e^{t X}], t ∈ R$ , when this expectation exists. When it exists in an open interval around t=0, the mgf can be used to calculate the moments of distribution of X as $m_{n} = E [X^{n}] = \frac{d^{n} M_{X} (t)}{d t^{n}} \| t = 0$ .
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