Question

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

Answer 1

A uniformly distributed random variable on the interval
`[a,b]` will have a constant positive value for point within the given interval, and zero elsewhere. This constant satisfies


`\displaystyle\int_(-\infty)^\infty f_X(x)\,\mathrm dx=\int_a^b c\,\mathrm dx=1`

Integrating, you get


`\displaystyle\int_a^b c\,\mathrm dx=cx\bigg|_(x=a)^b=c(b-a)=1\implies c=\frac1{b-a}`

So the density function is


`f_X(x)=\begin{cases}\frac1{b-a}&\text{for }a\le x\le b\\[1ex]0&\text{otherwise}\end{cases}`