The density of M = Median($U_1, U_2, U_3$)

Question

$U_1, U_2, U_3$ are identically independent distributed from Unif(0,1). Let M = Median($U_1, U_2, U_3$). What is the density and cdf of M?

Answer 1

Using the theory of Order Statistics, we can easily find

$$f_{U_{(2)}}(u)=3!u(1-u)=6u(1-u)$$

$u \in (0;1)$

We recognize a known density: $U_{(2)}\sim Beta(2;2)$