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Exponential Random Variable Problem

crazycraig

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Mar 28, 2012
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I'm having some formatting issues here. I'll try reposting from Windows in a little bit. Sorry that this looks ugly! Problem: A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years. What is the average length of time the boat can remain at sea? Some basic facts: Density for an exponential random variable: f(x)= λe-λx for x>=0 E(T)=1/λ if T is an exponential random variable Maybe relevant: P(S lessthan T)=λS / (λS + λT) for exponential random variables S and T, and this is similar for many exponential random variables. Where I've gotten: The boat can remain at sea until 2 parts break. Let T be the time that the boat can remain at sea. Since the sample space can be divided into the order in which the parts fail, we have: E(T)= Sum where 1 lessthanor=i,j,k lessthanor= 3 of: E(T|Ti lessthan Tj lessthan Tk)P(Ti lessthan Tj lessthan Tk) where E(T|Ti lessthan Tj lessthan Tk)=E(Tj|Ti llessthan Tj lessthan Tk) since the boat can remain at sea until two parts fail. Now, this would be a similar method that we've used with discrete random variables in class. Unfortunately, I'm a little rusty with my continuous probability. Is there somewhere to go from here, or maybe an easier way to approach the problem? I very much appreciate any help you can give! Thanks!
 
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