# Probability theory: a regenerative process

1. Oct 13, 2008

### dirk_mec1

1. The problem statement, all variables and given/known data

3. 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)

2. Oct 13, 2008

### 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.

3. Oct 13, 2008

### dirk_mec1

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

Yes It's stated in the exercise.

That's clear.

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?

4. Oct 13, 2008

### 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.