Gaussian Mean

The mean of a distribution $f(t)$ is defined as its first-order moment:

$$\mu \isdef \int_{-\infty}^\infty t f(t)dt$$ (D.42)

To show that the mean of the Gaussian distribution is $\mu$ , we may write, letting $g\isdef 1/\sqrt{2\pi\sigma^2}$ ,

$$\begin{eqnarray*} \int_{-\infty}^\infty t f(t) dt &\isdef & g \int_{-\infty}^\infty t e^{-\frac{(t-\mu)^2}{2\sigma^2}} dt\\ &=&g \int_{-\infty}^\infty (t+\mu) e^{-\frac{t^2}{2\sigma^2}} dt\\ &=&g \int_{-\infty}^\infty t e^{-\frac{t^2}{2\sigma^2}} dt + \mu\\ &=&\left.g(-\sigma^2) e^{-\frac{t^2}{2\sigma^2}} \right\vert _{-\infty}^{\infty} + \mu\\ &=& \mu \end{eqnarray*}$$

since $f(\pm\infty)=0$ .