Question 3.EX.13: Let X be exponential with mean 1/λ; that is, fX(x) = λe^−λx,......
Let X be exponential with mean 1/λ; that is,
f_{X}(x) = λe^{−λx} , 0<x <∞
Find E[X|X>1].
The conditional density of X given that X > 1 is
\begin{aligned}f_{X \mid X>1}(X) & =\frac{f(x)}{P\{X>1\}}=\frac{\lambda \exp ^{-\lambda x}}{e^{-\lambda}} \quad \text { when } x>1 \\E[X \mid X>1] & =e^\lambda \int_1^{\infty} x \lambda e^{-\lambda x} d x=1+1 / \lambda\end{aligned}
by integration by parts. This latter result also follows immediately by the lack of memory property of the exponential.
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