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=\frac1{b-a}`

So the density function is


`f(x)=\begin{cases}\frac1{b-a}&\text{for }a\le x\le b\\\\0&\text{otherwise}\end{cases}`