Alice enters a queue at a bank. The waiting time $X$ follows an exponential distribution function $f_X(x)=\frac{1}{10}e^{-x/10}$. She can wait for at most $15$ minutes. What’s the cumulative distribution function for her waiting time?
What I know: Her waiting time is $Y=\min(X,15)$, so the textbook formula to follow is $F_Y(y)=1-(1-F_X(x))(1-F_Z(z))$. But what’s $F_Z(z)$? It seems to be $$F_Z(z)=\begin{cases}0,&z<15;\\1,&z\ge15.\end{cases}$$ Is this correct?
Also the density function $f_Z(z)$ would have an area at $z=15$, otherwise how do you account for the sudden jump of $F_Z(z)$ to $1$ at $z=15$? Is it because it’s no longer a continuous distribution function?
If Alice's waiting time is less than 15 minutes the CDF is the same as $F_X(x)$ but if the waiting time is 15 minutes or more she leaves. Thus the resulting CDF is
$$F_Y(y)=[1-e^{-y/10}]\cdot\mathbb{1}_{(0;15)}(y)+\mathbb{1}_{[15;\infty)}(y)$$
as you can see, there is a "jump" in $y=15$ thus the rv is not absolutely continuous.