CDF of minimum of random variable and constant

Question

Let $X$ be a normally distributed random variable. I am trying to find the CDF of a minimum of $X$ and a constant $c$. In other words the CDF of the random variable $Y = \min (X,c)$.

With

$Y = \left\{\begin{matrix} X,& X < c\\ c,& X \geq c \end{matrix}\right.$

I started with considering that $P\{\min_i X_i > a\} = \prod_{i=1}^n P\{X_i > a\}$ for a group of random variables $X_i$ which led me to

$F_Y(x) = P(Y \leq x) = 1 - (1-F_X(x))(1-F_c(x))$

but am unsure if that is the correct approach and also how to determine $F_c(x)$.

Any further insights are much appreciated!

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Answer 1

$c$, being constant, does not have a CDF as such.   You merely need to determine if the value $x$ is less than $c$ or not.

$Y$ is distributed identically to $X$ for all values less than $c$, and the rest of the distribution lies exactly on $c$.

$$F_Y(x)= F_X(x)\mathbf 1_{x\lt c}+\mathbf 1_{x\geqslant c}$$

Answer 2

You know:

• for $x \lt c$: $F_Y(x)=\mathbb P(Y \le x) = \mathbb P(X \le x) =F_X(x) $
• for $x \ge c$: $F_Y(x)=\mathbb P(Y \le x) \le \mathbb P(Y \le c)= 1 $ , so $F_Y(x)=1$

The cumulative distribution functions for $X$ (red) and for $Y$ (blue) look like this: