Moment-generating Function

$M\_X(t)=E\backslash left(e^\{tX\}\backslash right),\; \backslash quad\; t\; \backslash in\; \backslash mathbb\{R\},$

$E\backslash left(X^n\backslash right)=M\_X^\{(n)\}(0)=\backslash left.\backslash frac\{\backslash mathrm\{d\}^n\}\{\backslash mathrm\{d\}t^n\}\backslash right|\_\{t=0\}\; M\_X(t).$

$M\_X(t)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{tx\}\; f(x)\backslash ,\backslash mathrm\{d\}x$

$=\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash left(\; 1+\; tx\; +\; \backslash frac\{t^2x^2\}\{2!\}\; +\; \backslash cdots\backslash right)\; f(x)\backslash ,\backslash mathrm\{d\}x$

$=\; 1\; +\; tm\_1\; +\; \backslash frac\{t^2m\_2\}\{2!\}\; +\backslash cdots,$

$\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{tx\}\backslash ,dF(x)$

$S\_n\; =\; \backslash sum\_\{i=1\}^n\; a\_i\; X\_i,$

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