Conditional expectation of Exp(theta)

  • Thread starter ghostyc
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  • #1
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Given X follows an exponential distribution [tex]\theta[/tex]

how could i show something like

[tex]\operatorname{E}(X|X \geq \tau)=\tau+\frac 1 \theta[/tex]

?

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

thanks.

casper
 

Answers and Replies

  • #2
525
5
i have get the idea of using Memorylessness property here,
but how can i combine the probabilty 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.
 
  • #3
26
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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.
[tex]\Pr(T > s + t\; |\; T > s) = \Pr(T > t) \;\; \hbox{for all}\ s, t \ge 0. [/tex]

i just cant convert from expectation to the probability...

damn
 
  • #4
26
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That's what I did so far.

But I just cant use the memoryless property to do it ....

http://img138.imageshack.us/img138/5945/tempz.jpg [Broken]
 

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  • #5
525
5
That's what I did so far.

But I just cant 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.
 

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