If $X\sim \exp(\lambda)$, then $E[X\mid X>20]=20+E[X]$
Can anyone give an intuitive proof of this property? In the context that $X$ is exponential and thereby memoryless. I am having trouble in visualizing continuous memoryless distributions. Any intuitive example for the same is also appreciated.
Best Answer
An exponential distributed random variable is the measure of waiting time until the arrival of some event, the likes of which occur independently and at some constant average rate (ie: a Poisson Process).
This means that at every point in time the chance of such an event happening is the same as every other point in time, regardless of whether another such event happened recently or not.
So the expected waiting time until the next event is not determined by when you start waiting.
So if you wait for twenty units with no event occurring, you are facing the exact same situation you were twenty units ago; only now with the added knowledge that 20 units have passed without an event. That 20 units doesn't affect the expected wait time until the next event there after; but if you are given knowledge of it ahead of time (or in retrospect), then...
Or by calculus.
$$\begin{align} \mathsf E(X\mid X>20) & = \dfrac{\int_{20}^\infty \lambda t e^{-\lambda t}\operatorname d t}{\int_{20}^\infty \lambda e^{-\lambda t} \operatorname d t} \\[1ex] & = \dfrac{e^{-20 \lambda}(20\lambda +1 )}{\lambda}\dfrac{1}{e^{-20\lambda}} \\[1ex] & = 20+\frac 1 \lambda \\[1ex] & = 20+\mathsf E(X)\end{align}$$