How to designate the density function uniform distribution (continuous) on the set $[-1,0] \cup [3,5]$

Question

How to designate the density function uniform distribution (continuous) on the set $[-1,0] \cup [3,5]$?

I know how to do it when we only have for example [3,5] but for this set $[-1,0] \cup [3,5]$ I don't know. Please help me.

Answer 1

The total interval length for your uniform distribution is $3$, since the first interval has length $1$ and the second interval has length $2$. If it helps, you can think of the uniform density in your problem as isomorphic to a uniform density on $[0,3]$. $$ f(x) = \begin{cases} 1/3 & x \in [-1,0] \cup [3,5] \\ 0 & \text{otherwise} \end{cases} $$

Answer 2

The density function of the uniform distribution on $K$ is given by $f(x)=\frac{1}{m(K)}1\!\!1_{K}(x)$. Where $m(K)$ is the measure of $K$.

The measure of the union of the two intervalls is $2+1=3$. So the density function is given by : $$f(x)=\frac{1}{3}1\!\!1_{[-1,0] \cup [3,5]}(x)$$