Expectation of Gaussian Distribution/Proof 1

Proof
From the definition of the Gaussian distribution, $X$ has probability density function:


 * $\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$

From the definition of the expected value of a continuous random variable:


 * $\ds \expect X = \int_{-\infty}^\infty x \map {f_X} x \rd x$

So: