Cut off the exponential random variable

Question

Let $X$ be an exponential random variable with parameter (mean) $\mu = 2$, its density is given by $$f(x)=\frac{1}{2}e^{-\frac{1}{2}x}, \quad \text{for } x > 0$$ For $K > 0$, define a function $\mathrm{sat}_{K}$ that cuts off the values at level $K$ by $$\mathrm{sat}_{K}(x) = \begin{cases} x, & \text{if}\ x \leq K, \\ K, & \text{if}\ x > K, \end{cases}$$ and let $Y_K = \mathrm{sat}_{k}(X)$. Explicitly compute the distribution function $F_Y$ of $Y_K$ and sketch it. What type of random variable is $Y_K$ ?

Answer 1

If $y\ge K$ you have $$ F_{Y_K}(y)=P\{Y_K\le y\}=P\{X\in\mathbb{R}\}=1 $$ since $Y_K\le K\le y$ for any value of $X$. Now, if $y\le K$, $$ F_{Y_K}(y)=P\{Y_K\le y\}=P\{X\le y\}=F_X(y)=1-e^{-y/2} $$ so basically you are creating a jump in your distribution function at $y=K$.