A silly confusion using expected value? 
Consider a fuse with lifetime $T$, which is exponentially distributed. The fuse can be fabricated by two > processes:

*

*Process 1: Has an expected lifetime of 100 hours and costs 10 by fuse.


*Process 2: Has an expected lifetime of 150 hours and costs 20 by fuse.
If a fuse lasts less then 200 hours, the factory will have to pay a fine of 5$.
What process should be used?

I tried the following: I think I should compare the following integrals for expected value:
$$(10+5)\int_{0}^{200} x \cdot \lambda e^{-\lambda x}  \quad \quad \quad  \quad  \quad (20+5)\int_{0}^{200} x \cdot \lambda' e^{-\lambda' x} $$
For both processes. But it's not clear if $\lambda=100$ and $\lambda '=150$ although I believe it is. I guess this would enable us to measure how much money would be lost due to fines plus manufacturing costs. Does that make sense? Sorry if it is too nonsensical, I'm studying on my own.
 A: The expected fine for a fuse made using the
first process is 4.32.
5 * pexp(200, 1/100)
[1] 4.323324
5*(1 - exp(-200/100))
[1] 4.323324

The expected fine for a fuse made using the
second process is less, but of course, not
enough less to make up for the additional cost (10)
of making the more durable fuse.
5 * pexp(200, 1/150)
[1] 3.682014  
5*(1 - exp(-200/150))
[1] 3.682014


The cost and fine structure makes no sense:

*

*Even if a fuse made by the inferior process
always incurred a fine of 5, that would mean a
total cost
of 15.


*Thus, if the better fuses never incurred
a fine, they still cost more (20) than the others.
With manufacturing costs 10 and 20, a fine of 50 would be sufficient to encourage
manufacture of the better type of fuses.
With costs 10 and 11, a fine of 5, would suffice.
Notes: (a) If an exponential distribution has mean $\mu = 100,$ then its failure rate is $1/100,$ sometimes designated $\lambda.$ R uses the rate parameterization, hence 5 * pexp(200, 1/100) in my first bit if R code; followed by
a strictly numeric computation giving the same numerical answer.
(b) I have interpreted "cost by fuse" to mean the manufacturing cost per fuse. If you mean the sales prices per fuse then the net incomes per fuse are 5.68 and 16.32, but that does not make
it clear which type the manufacturer should make,
because we have no idea about costs of production.
10 - 5 * pexp(200, 1/100)
[1] 5.676676
20 - 5 * pexp(200, 1/150)
[1] 16.31799

