I struggle to understand the transformation of a random variable with uniform distribution. For example:
Let $X\sim \text{Uniform}(0,1)$ and $T=-2\ln(X)$ and I want to find the CDF of $T$, then I know that I can compute $$P(T\leq t)=P(-2\ln(X)\leq t)=P\left(X\leq e^\frac{-t}{2}\right)$$ $$=\int\limits_{-\infty}^{e^\frac{-t}{2}}\mathbb{1}_{\{0,1\}}\mathbb{d}t$$
But how do I compute this? How can I get a nice integral without the indicator function and the $\displaystyle e^\frac{-t}{2}$ as the upper bound?
Thank you for your answer