I have a component with exponential failure function, i.e. the CDF is ($T$ being the time of failure):
$$F(t):= P(T \leq t) = 1 – e^{-\lambda t}$$
I want to find the expected time of failure of the component, given it has survived until time $\tau$, i.e., $E(T | T \geq \tau)$.
I understand that finding the expected value of a random variable generally can be achieved by integrating $1-CDF$ thus:
$$E(X) = \int_{x=0}^{x=\infty} (1-CDF(x)) dx$$
So, the expected time of failure generally would be:
$$E(T) = \int_{t=0}^{t=\infty} (1-F(t)) dt$$
(In my case, this would simply be $1$, as it is an exponential distribution).
I have tried to approach the problem by integrating the CDF after the specified time ($\tau$) and then offseting by $\tau$, however I am unsure if this is the right way to approach it:
$$E(T|T\geq \tau) = \tau + \int_{t=\tau}^{t=\infty} (1-CDF(t)) dt$$
What is the correct way to compute $E(T|T\geq \tau)$?
Best Answer
Your definition of expected value is at least non-standard, and may be wrong. For a continuous random variable $X$ with probability density function $f$,
$$E(X)=\int_{-\infty}^\infty xf(x)\ dx$$
Given the CDF, the PDF can be easily found by
$$f(x) = \frac{d}{dx}F(x)$$
So in your specific example, we have
$$f(t) = \lambda e^{-\lambda t}, t>0$$
$$E(T) = \int_0^\infty t\lambda e^{-\lambda t}\ dt=\frac{1}{\lambda}$$
Now, we can use Bayes' rule to find a CDF for the random variable with conditional.
$$F_{T\geq\tau}(t)=P(T\leq t|T\geq \tau) = \frac{P(\tau \leq T\leq t)}{P(T\geq \tau)} = \frac{F(t)-F(\tau)}{1-F(\tau)},t\geq\tau$$
$$=\frac{e^{-\lambda \tau}-e^{-\lambda t}}{e^{-\lambda \tau}}$$
And we can differentiate to get a new PDF
$$f_{T\geq\tau}(t) = \frac{\lambda e^{-\lambda t}}{e^{-\lambda \tau}}=\lambda e^{-\lambda (t-\tau)},t>\tau$$
And finally the expected value is
$$E(T) = \int_\tau^\infty t\lambda e^{-\lambda (t-\tau)}\ dt=\int_0^\infty (u+\tau)\lambda e^{-\lambda (u)}\ du=\tau+\frac{1}{\lambda}$$
This gives a very nice result for the exponential distribution shown, owing to its memoryless property. However this process should work (albeit with more difficulty) for other distributions.