Conditional expectation of Exp(theta)

[ghostyc]

Given X follows an exponential distribution $$\theta$$

how could i show something like

$$\operatorname{E}(X|X \geq \tau)=\tau+\frac 1 \theta$$

?

i have get the idea of using Memorylessness property here,
but how can i combine the probability with the expectation?

thanks.

casper

 

[bpet]

i have get the idea of using Memorylessness property here,
but how can i combine the probability with the expectation?

If you write down mathematically the Memorynessless property then it might become obvious. Otherwise it's fine to express the conditional expectation as a ratio of integrals and evaluate it directly.

 

[ghostyc]

If you write down mathematically the Memorynessless property then it might become obvious. Otherwise it's fine to express the conditional expectation as a ratio of integrals and evaluate it directly.

$$\Pr(T > s + t\; |\; T > s) = \Pr(T > t) \;\; \hbox{for all}\ s, t \ge 0.$$

i just can't convert from expectation to the probability...

damn

 

[ghostyc]

That's what I did so far.

But I just can't use the memoryless property to do it ...

http://img138.imageshack.us/img138/5945/tempz.jpg

 

[bpet]

That's what I did so far.

But I just can't use the memoryless property to do it ...

Hint: rewrite the memoryless property into a form suitable to use on line 1 of your proof. Conditional probabilities -> conditional cdf -> conditional pdf.