Modulus of a random variable which follows a continuous uniform distribution, follows which distribution?

Question

Let $X \sim U(\theta,0),$ $\theta<0$ (continuous uniform distribution)

I want a transformation on $X$ so that it follows $U(0,1)$ distribution I did $|X|/\theta\sim U(0,1).$ Am I right?

Answer 1

Since $|X|>0$ and $\theta<0,$ $|X|/\theta$ will be negative.

But $X/\theta$ will serve.

For $x\in[0,1],$ $\quad\Pr(X/\theta\le x) = \Pr(X\ge \theta x) = \dfrac{0-\theta x}{\theta} = x.$