TITLE: $X$ exponential random variable - probability
QUESTION [0 upvotes]: We know that $X$ is an exponential random variable with parameter $\gamma$.
I need to calculate $\mathbb{E}[X|X>1]$.
How I do it?

REPLY [0 votes]: The conditonal density is
$$f_{X|X>1}=e^{\theta}\theta e^{-\theta x}=\theta e^{\theta(1-x)}$$
To calculate the conditional expectation you get
$$E(X|X>1)=\int_1^{\infty}x\theta e^{\theta(1-x)}dx=1+\frac{1}{\theta}$$

But you can calculate your expectation quickly using memoryless property
$$E(X|X>1)=1+E(X)=1+\frac{1}{\theta}$$
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