Sunday, 14 October 2018

probability - Independent Exponential Random Variables




I am currently trying to figure out a problem and it is using notation that I have never seen before so I am pretty stuck, any suggestions would be greatly appreciated!



Let X,Y,Z be independent exponential random variables with the same mean, σ. Find the value of σ so that \Pr[\max(X,Y,Z)>1]=0.05.



Any help with how to solve this problem or leading down the right path would be awesome as I am not to sure where to even start!



EDIT: So I have followed what André Nicolas has said and got a σ of 0.25. However I am still not sure where he got this formula from: The probability they are all \le a is (1-e^{-a/\sigma})^3.



I cant find anything like this is my notes or textbook, could any reference where he got this?



Answer



A start: The probability that \max(X,Y,Z)\gt 1 is 1 minus the probability they are all \le 1.



The probability they are all \le a is (1-e^{-a/\sigma})^3.


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