Question

If x has a continuous uniform distribution on the interval (0,5), what is the 25th percentile for x?

Answer 1

Given that $X\sim\mathrm{Unif}(0,5)$, it has PDF

$f_X(x)=\begin{cases}\dfrac15&\text{for }0\le x\l5\\\\0&\text{otherwise}\end{cases}$

and thus CDF

$F_X(x)=\mathbb P(X\le x)=\begin{cases}0&\text{for }x<0\\\\\dfrac x5&\text{for }0\le x<5\\\\1&\text{for }x\ge5\end{cases}$

The 25th percentile is the value $X=k$ such that $\mathbb P(X\le k)=0.25$. We have

$\mathbb P(X\le k)=\dfrac k5=0.25\implies k=1.25$