Say I have a uniform distribution $P \sim \text{Unif}[1,2]$ and a constant $m$.
If I divide $m$ by $P$, will I still have a uniform distribution such that$\frac{m}{P} \sim \text{Unif}[\frac{m}{2}, m]$?
I am asking this question as it seems deeply intuitive that since P takes on a value equally likely from 1 to 2, that taking a constant over P would equally give an equal distribution but now around the new support.
Thank you!
We let $P\sim $ Unif$[1,2]$. Then one can calculate that $Y:=1/P$ has the following density function:
$$f_Y(y)=\begin{cases}\dfrac1{y^2},&y\in[1/2,1]\\0,&\text{else.}\end{cases}$$
So $Y$, or $m/P$, is not uniformly distributed. The random variable $m/P$ does take values in between $m/2$ and $m$.