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October 26th, 2013, 03:30 PM   #1
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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!
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October 26th, 2013, 06:28 PM   #2
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Re: Exponential Random Variable

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October 27th, 2013, 10:53 AM   #3
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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/(L-M)[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}.
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October 28th, 2013, 02:47 PM   #4
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Re: Exponential Random Variable

Bump, I still can't progress.
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October 28th, 2013, 05:29 PM   #5
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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)
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October 28th, 2013, 07:04 PM   #6
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Re: Exponential Random Variable

Okay, I guess it could be a typo in the question. Should I get

P{S < t < S + R} = M/(L-M)(exp(-Mt)-exp(-Lt))?
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October 29th, 2013, 03:23 AM   #7
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Re: Exponential Random Variable

yes
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October 30th, 2013, 11:26 AM   #8
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Re: Exponential Random Variable

Thank you so much!
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