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}\frac15&\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\\\\\frac 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)=\frac k5=0.25\implies k=1.25`