Median of Gaussian Distribution

Theorem
Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution.

Then the median of $X$ is equal to $\mu$.

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


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

Note that $f_X$ is non-zero, sufficient to ensure a unique median.

By the definition of a median, to prove that $\mu$ is the median of $X$ we must verify:


 * $\displaystyle \map \Pr {X < \mu} = \int_{-\infty}^\mu \map {f_X} x \rd x = \frac 1 2$

We have: