Cumulative Distribution Function

F_X: t \mapsto P(x \lt t) = \int_{-\infty } ^t f_X(u) du

For functions like min, you can first find the CDF using probability intervals and then differentiate it to get the PDF.

Two distinct probability distributions can’t have the same CDF, if two probability distributions have the same CDF, they’re the same. $\mapsto$
Inverse transform sampling

F_X(X) \sim \mathcal{U}(0,1)

Random Variable X follows the probability distribution, then input into

F_X

, results in a uniform

Let

t \lt t'

, then

F_X(t') - F_X(t) \ge 0

, since

t' - t > 0

and

f_X \ge 0

since it is a