Question

A continuous random variable has the uniform distribution on the interval [a, b] if its probability density function f(x __________.

Answer 1

A uniform distribution has a constant density over its support:

$f(x)=\begin{cases}c&\text{for }a\le x\le b\\0&\text{otherwise}\end{cases}$

Any probability distribution sums to 1 over its support, so in this case $f(x)$ satisfies

$\displaystyle\int_{-\infty}^\infty f(x)\,\mathrm dx=1$

Replace $f(x)$ with its definition as given above, so you have

$1=\displaystyle\int_a^bc\,\mathrm dx=cx\bigg|_{x=a}^{x=b}=c(b-a)$
$\implies c=\dfrac1{b-a}$

So the density function is

$f(x)=\begin{cases}\dfrac1{b-a}&\text{for }a\le x\le b\\\\0&\text{otherwise}\end{cases}$