Exponential RV conditional expectation E[X|X<t]

Question

For an exponential RV with mean $\mu$, we know that $E[X\mid X>t]$ is $t + \mu$. But how can you compute $E[X\mid X < t]$?

Answer 1

\begin{align} \mu = \operatorname{E}(X) & = \operatorname{E}(X\mid X<t)\Pr(X<t) + \operatorname{E}(X\mid X>t)\Pr(X>t) \\[10pt] & = \operatorname{E}(X\mid X<t)(1-e^{-t/\mu}) + (t+\mu) e^{-t\mu}. \end{align} Hence $$ \operatorname{E}(X\mid X<t) = \frac{\mu - (t+\mu)e^{-t/\mu}}{1 - e^{-t/\mu}}. $$

(We need not consider the event $X=t$ since its probability is $0$.)