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动差生成函数 - Wikipedia

动差生成函数

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动差又被称为隨機變數 X動差生成函數定義為:

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

前提是这个期望值存在.

由動差生成函數可生成此機率分佈的各階動差。動差定義:

E[X^n]=\int_{-\infty}^\infty x^n\,dF(x) = \int_{-\infty}^\infty x^n\,f(x)dx

由動差生成函數微分,再將參數以零代入可得各階動差:

E[X^n]=M_X^{(n)}(0)=\left.\frac{\mathrm{d}^n}{\mathrm{d}t^n}\right|_{t=0} M_X(t).
  • E[X] = \int_{-\infty}^\infty x f(x)dx = M_X'(0)
  • E[X^2] = \int_{-\infty}^\infty x^2 f(x)dx = M_X''(0)

依此類推。


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