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 $E(T|T\geq \tau)$?