Moment-Generating Function 動差生成函數/力矩產生函數

M_x(t) = E(e^tx) = Expected value of exponential function exp(tx).

M_x(t) = 1 + E[1 + tX + (tX)²/2! + ... + (tX)ⁿ/n!

Derivatives of the moment-generation function with t=0:

M_x⁽ⁿ⁾(0) = E[Xⁿ],         n = 1, 2, 3, ...

first moment - mean 平均數

second moment - variance 變異數

third moment - skewness 偏度/歪度/偏態

fourth moment - kurtosis 尖度/峰度

參考資料

動差生成函數 (Moment Generation Function) (陳鍾誠網站)

Mathematical statistics with applications in R, Ramachandran & Tsokos, 2nd edition (2015),  p96-100 (Section 2.6)