Distribution function of a conditional random variable

Question

Let $(S,\mathcal{F},P)$ be a probability space, and $R$ and $X$ be two random variables. For some positive number $d$, we have that $$R=\begin{cases}X \text{ if }X<d,\\d \text{ if } X>d.\end{cases}$$

How to calculated the distibution function $F_R(t)$ based on $F_X(t)$?

Answer 1

Since $R = \min(X,d)$ you have:

$$F_R(t) = \begin{cases} F_X(t) & & \text{if } t < d, \\[6pt] 1 & & \text{if } t \geqslant d. \\[6pt] \end{cases} $$