Median of Gaussian Distribution

Theorem
Let $X \sim N \left({\mu, \sigma^2}\right)$ 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:


 * $f_X \left({x}\right) = \dfrac 1 {\sigma \sqrt{2 \pi} } \, \exp \left({-\dfrac { \left({x - \mu}\right)^2} {2 \sigma^2} }\right)$

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:


 * $\displaystyle \operatorname{Pr} \left({X < \mu}\right) = \int_{-\infty}^\mu f_X \left({x}\right) \rd x = \frac 1 2$

We have: