Probability theory: a regenerative process

[dirk_mec1]

Homework Statement

http://img291.imageshack.us/img291/4844/14820448wb9.png

The Attempt at a Solution

First of all I'm trying to find the expected time of a cycle. In a cycle two things can happen:

1) the car lives long enough to reach A with probability 1-F(A)
2) the car fails to reach age A with probability F(A)

Knowing this how can you compute the liftime of a cycle? I thought of something like:

E(T) =A*(1-F(A)) + F(A) * ? (I don't know what to fill in the question mark)

 

[EnumaElish]

E(T) is the expected value of T ~ F conditional on T < A. But you can directly solve for the expected resale price:

Let event 1 be {the car fails before A}. If event 1 occurs then resale value = 0 (do you see this?)
Let event 2 be {the car lives to A} (that is, it lives to see A, and possibly more). If event 2 occurs then resale value = R(A).

Suppose the probability of event 1 was G. Convince yourself that the prob. of event 2 must be 1 - G.

Then E(resale value) = G * 0 + (1-G) * R(A).

What you should think about is how to find G.

 

[dirk_mec1]

E(T) is the expected value of T ~ F conditional on T < A.

So the expected life of a cycle is: $$E[T|T<A] = \int_0^{A} x f(x)\ \mbox{d}x$$

But you can directly solve for the expected resale price:

Let event 1 be {the car fails before A}. If event 1 occurs then resale value = 0 (do you see this?).

Yes It's stated in the exercise.

Let event 2 be {the car lives to A} (that is, it lives to see A, and possibly more). If event 2 occurs then resale value = R(A).

Suppose the probability of event 1 was G. Convince yourself that the prob. of event 2 must be 1 - G.

That's clear.

Then E(resale value) = G * 0 + (1-G) * R(A).

What you should think about is how to find G.

I presume that G must be F(A). But the question asks for the costs so how do I incorporate that in the resale value given that there are two different outcomes?

 

[EnumaElish]

Once you figure the expected resale value, you can use it to figure the expected net cost, defined as cash expenses - expected resale value.