Conditional expectation of exponential random variable

[new_for_ever]

For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value
from searching in internet I found that

E{X|X>a}=a+E{x}​

but I can not prove it
Help please

 

[mathman]

It can be done by a straightforward integration.
∫xe^-uxdx/∫e^-uxdx with limits a,∞ for both the numerator and the denominator.

 

[new_for_ever]

Thanks for reply, it is correct but why??

 

[mathman]

The integral in the numerator (properly normalized) from 0 to ∞ is the definition of E(X) for an exponentially distributed random variable. Changing the limits to (a,∞) means assuming X>a. The denominator supplies the normalization.