Median of Continuous Uniform Distribution

Theorem
Let $X$ be a continuous random variable which is uniformly distributed on a closed real interval $\closedint a b$.

Then the median $M$ of $X$ is given by:


 * $M = \dfrac {a + b} 2$

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


 * $\map {f_X} x = \dfrac 1 {b - a}$

Note that $f_X$ is non-zero, so the median is unique.

We have by the definition of a median:


 * $\displaystyle \map \Pr {X < M} = \frac 1 {b - a} \int_a^M \rd x = \frac 1 2$

We have, by Primitive of Constant:


 * $\dfrac {M - a} {b - a} = \dfrac 1 2$

So: