Se dice que una variable aleatoria continua 
![\displaystyle{f(x)=\frac{1}{\sigma \sqrt{2\pi}} \exp \left [ - \frac{(x-\mu)^2}{2 \sigma^2} \right ].}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7Bf%28x%29%3D%5Cfrac%7B1%7D%7B%5Csigma+%5Csqrt%7B2%5Cpi%7D%7D+%5Cexp+%5Cleft+%5B+-+%5Cfrac%7B%28x-%5Cmu%29%5E2%7D%7B2+%5Csigma%5E2%7D+%5Cright+%5D.%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{f(x)=\frac{1}{\sigma \sqrt{2\pi}} \exp \left [ - \frac{(x-\mu)^2}{2 \sigma^2} \right ].}")
Observemos que 
resulta que
![\displaystyle{ \int_{-\infty}^\infty f(x)\,dx = \frac{1}{\sigma \sqrt{2\pi}} \int_{-\infty}^\infty \exp \left [ - \frac{(x-\mu)^2}{2 \sigma^2} \right ] dx = \frac{1}{\sqrt{\pi}} \int_{-\infty}^\infty\exp(-y^2)\,dy=1}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+f%28x%29%5C%2Cdx+%3D+%5Cfrac%7B1%7D%7B%5Csigma+%5Csqrt%7B2%5Cpi%7D%7D+%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty++%5Cexp+%5Cleft+%5B+-+%5Cfrac%7B%28x-%5Cmu%29%5E2%7D%7B2+%5Csigma%5E2%7D+%5Cright+%5D+dx+%3D+%5Cfrac%7B1%7D%7B%5Csqrt%7B%5Cpi%7D%7D+%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%5Cexp%28-y%5E2%29%5C%2Cdy%3D1%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \int_{-\infty}^\infty f(x)\,dx = \frac{1}{\sigma \sqrt{2\pi}} \int_{-\infty}^\infty \exp \left [ - \frac{(x-\mu)^2}{2 \sigma^2} \right ] dx = \frac{1}{\sqrt{\pi}} \int_{-\infty}^\infty\exp(-y^2)\,dy=1}")
porque según este post, la integral gaussiana viene dada por la expresión
Calculemos ahora la función generadora de momentos. Tenemos
![\displaystyle{ \psi(t)= \mathbb{E}(e^{tX})=\int_{-\infty}^\infty e^{tx}f(x)dx= \frac{1}{\sigma \sqrt{2\pi}} \int_{-\infty}^\infty \exp \left [ tx- \frac{(x-\mu)^2}{2 \sigma^2} \right ]\,dx}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cpsi%28t%29%3D+%5Cmathbb%7BE%7D%28e%5E%7BtX%7D%29%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+e%5E%7Btx%7Df%28x%29dx%3D+%5Cfrac%7B1%7D%7B%5Csigma+%5Csqrt%7B2%5Cpi%7D%7D+%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cexp+%5Cleft+%5B+tx-+%5Cfrac%7B%28x-%5Cmu%29%5E2%7D%7B2+%5Csigma%5E2%7D+%5Cright+%5D%5C%2Cdx%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \psi(t)= \mathbb{E}(e^{tX})=\int_{-\infty}^\infty e^{tx}f(x)dx= \frac{1}{\sigma \sqrt{2\pi}} \int_{-\infty}^\infty \exp \left [ tx- \frac{(x-\mu)^2}{2 \sigma^2} \right ]\,dx}")
Completando el cuadrado en la expresión entre corchetes resulta
![\displaystyle{tx - \frac{(x-\mu)^2}{2 \sigma^2} = \mu t + \frac{1}{2} \sigma^2 t^2 - \frac{[x-(\mu + \sigma^2t)]^2}{2 \sigma^2},}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7Btx+-++%5Cfrac%7B%28x-%5Cmu%29%5E2%7D%7B2+%5Csigma%5E2%7D+%3D+%5Cmu+t+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Csigma%5E2+t%5E2+-+%5Cfrac%7B%5Bx-%28%5Cmu+%2B+%5Csigma%5E2t%29%5D%5E2%7D%7B2+%5Csigma%5E2%7D%2C%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{tx - \frac{(x-\mu)^2}{2 \sigma^2} = \mu t + \frac{1}{2} \sigma^2 t^2 - \frac{[x-(\mu + \sigma^2t)]^2}{2 \sigma^2},}")
y de aquí se deduce que
donde
![\displaystyle{C=\frac{1}{\sigma \sqrt{2\pi}} \int_{-\infty}^\infty \exp \left \{ - \frac{[x-(\mu + \sigma^2t)]^2}{2 \sigma^2}\right \}dx=1.}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7BC%3D%5Cfrac%7B1%7D%7B%5Csigma+%5Csqrt%7B2%5Cpi%7D%7D+%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty+%5Cexp+%5Cleft+%5C%7B+-++%5Cfrac%7B%5Bx-%28%5Cmu+%2B+%5Csigma%5E2t%29%5D%5E2%7D%7B2+%5Csigma%5E2%7D%5Cright+%5C%7Ddx%3D1.%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{C=\frac{1}{\sigma \sqrt{2\pi}} \int_{-\infty}^\infty \exp \left \{ - \frac{[x-(\mu + \sigma^2t)]^2}{2 \sigma^2}\right \}dx=1.}")
Resumiendo, la función generadora de momentos de una distribución normal de parámetros 
Ahora calculamos las dos primeras derivadas de la función generadora de momentos.
![\displaystyle{\psi^{\prime \prime} (t)= [ \sigma^2 + (\mu + \sigma^2 t)^2 ] \exp \left ( \mu t + \frac{1}{2} \sigma^2t^2 \right).}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cpsi%5E%7B%5Cprime+%5Cprime%7D+%28t%29%3D+%5B+%5Csigma%5E2+%2B+%28%5Cmu+%2B+%5Csigma%5E2+t%29%5E2+%5D+%5Cexp+%5Cleft+%28+%5Cmu+t+%2B+%5Cfrac%7B1%7D%7B2%7D+%5Csigma%5E2t%5E2+%5Cright%29.%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{\psi^{\prime \prime} (t)= [ \sigma^2 + (\mu + \sigma^2 t)^2 ] \exp \left ( \mu t + \frac{1}{2} \sigma^2t^2 \right).}")
Finalmente, los momentos de 
