How about this: Suppose are iid and distributed according to . Then
.
A critical component on a submarine has an operating lifetime that is exponentially distributed with a mean of 6 months. As soon as a component fails, it is replaced bya new one having statistically identical properties. What is the minimum number ofspare components which the submarine ought to carry if it is leaving for a one yeartour of duty and it is desired that the probability of having an inoperable unit causedby failures exceeding the spare inventory be less than 0.02
Let V represent the critical component
=>
The probability of the first time the component fails is:
Kind of stuck here. Any hints/suggestions would be greately appreciated.
So
EDIT: Okay, I think I got it
I need to find the probability that
So
Applying the central limit theorem:
<0.02
Solving for n, I get n=0.9835
So they need to bring 2 components.
Have I made a mistake anywhere?
But the CLT isn't, typically, a good approximation for n = 2. The reasoning behind what you did only applies when the that you get is sufficiently large the the CLT to hold.
Your reasoning is essentially:
(1) The n I need is large, so apply the CLT
(2) Solve for n
(3) Get n = 2
You see the problem, yes?
Actually, it is the same as a chisquare with 2n degrees of freedom (not n).
You are supposed to choose n, or equivalently, the degrees of freedom of an appropriate chi-square. Here's one more hint:
where we are using the shape/scale parameterization of the Gamma (i.e. ). You can look up in a table an appropriate cutoff to get the n you want.