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<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<=i,j,k<=3 of: E(T|Ti<Tj<Tk)P(Ti<Tj<Tk)
where E(T|Ti<Tj<Tk)=E(Tj|Ti<Tj<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!