CDF of $Z=\min(X,c)$ when $c$ is a constant

Question

For independent continuous (non-negative) random variables $X$ and $Y$, we may have CDF for $Z=\min(X, Y)$:

\begin{align*} F_Z(t) &= \Pr(X \le t \;\text{or}\; Y \le t) = \Pr(X \le t) + \Pr(Y \le t) - \Pr(X \le t, Y \le t)\\ & = F_X(t) + F_Y(t) - F_X(t) F_Y(t). \end{align*}

However, how can I express the CDF of $Z=\min(X, c)$ when I have a non-negative constant $c$ instead of RV $Y$?

Answer 1

The distribution $\min\{X,c\}$ can be obtained as follows:

$$F_{\min\{X,c\}}(t)=\mathbb P (\min\{X,c\}\le t)=1-\mathbb P (\min\{X,c\}>t)=1-\mathbb P (X>g,c>t)= \begin{cases} F_X(t) & t<c \\ 1 & t\ge c. \end{cases}$$

Note that even if $F_X$ is a continuous distribution, $F_{\min\{X,c\}}$ is a mixed distribution for any $c$ with $0<F_X(c)<1$.

Using the same method for $\min\{X,Y\}$:

$$ F_{\min\{X,Y\}}(t)=1-\mathbb P (\min\{X,Y\}>t)= 1- (1- F_X(t))(1- F_Y(t)).$$

If both $F_X$ and $F_Y$ are continuous distributions, then $\min\{X,Y\}$ also has a continuous distribution.