Question
A device that continuously measures and records seismic activity is placed in a remote region. The time, $T$, to failure of this device is exponentially distributed with mean $3$ years. Since the device will not be monitored during its first two years of service, the time to discovery of its failure is $X = \max(T, 2)$.
I understand that the density function of $T$ is $f(t)=\frac{1}{3} e^{-\frac{t}{3}}$ and I'm pretty sure that $\max(T,2)= T$ if $T>2$ and $2$ elsewhere.
Here's where I'm stuck:
$$\text{E}(X)=\text{E}(\max(T,2))=\text{?}$$
I'm not sure how to figure this part out.
Best Answer
Comment. Here is a simulation in R statistical software with very nearly the correct numerical answer. A direct method based on $X$ is used. You can compare the result (to two or three places) with your analytic answer.
A little over half of the devices survive beyond 2.