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October 26th, 2013, 03:30 PM  #1 
Newbie Joined: Oct 2013 Posts: 13 Thanks: 0  Exponential Random Variable
Hi, I have a quick question. Let R and S be two independent exponentially distributed random variables with rates ? and ?. How would I compute P{S < t < S + R}? I am a little bit confused because of the variables on either side of the inequalities. I have tried conditioning on both S and R but I am not sure if I'm doing it right here. I can compute something like P{S < R} but the t is throwing me off! Any help is appreciated! 
October 26th, 2013, 06:28 PM  #2 
Senior Member Joined: Feb 2013 Posts: 281 Thanks: 0  Re: Exponential Random Variable 
October 27th, 2013, 10:53 AM  #3 
Newbie Joined: Oct 2013 Posts: 13 Thanks: 0  Re: Exponential Random Variable
Thank you, I think I understand where that came from. What exactly are the function? Is it as simple as f(R) = L*exp(LR) (L=lambda) and f(S)* = M*exp(MS) (M=mu)? With the outer integral, should it be from 0 to infinity? As the exponential random variable isn't defined for below zero. Because doing this gives me M/(LM)[exp(Mt)exp(Lt)] which I'm not sure is correct because I want to show that M*P{S < t < S + R} = L*P{R < t < R + S}. 
October 28th, 2013, 02:47 PM  #4 
Newbie Joined: Oct 2013 Posts: 13 Thanks: 0  Re: Exponential Random Variable
Bump, I still can't progress.

October 28th, 2013, 05:29 PM  #5 
Senior Member Joined: Feb 2013 Posts: 281 Thanks: 0  Re: Exponential Random Variable
I edited my post a bit. f(S) was not correct, it should be f(S,M). I think your result is okey. You can show this: L*P{S < t < S + R} = M*P{R < t < R + S}. (compare it with yours) 
October 28th, 2013, 07:04 PM  #6 
Newbie Joined: Oct 2013 Posts: 13 Thanks: 0  Re: Exponential Random Variable
Okay, I guess it could be a typo in the question. Should I get P{S < t < S + R} = M/(LM)(exp(Mt)exp(Lt))? 
October 29th, 2013, 03:23 AM  #7 
Senior Member Joined: Feb 2013 Posts: 281 Thanks: 0  Re: Exponential Random Variable
yes

October 30th, 2013, 11:26 AM  #8 
Newbie Joined: Oct 2013 Posts: 13 Thanks: 0  Re: Exponential Random Variable
Thank you so much!


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exponential, random, variable 
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