Expected lifetime in an Exponential Distribution

Date: 10/28/1999 at 07:11:16
From: Kevin Williams
Subject: Exponential distribution

The lifetime X of an electronic component has an exponential 
distribution such that P(X <= 1000) = 0.75. What is the expected 
lifetime of the component?

Date: 10/28/1999 at 09:35:28
From: Doctor Mitteldorf
Subject: Re: Exponential distribution

Dear Kevin,

They're telling you that the probability has an exponential form. What 
does that mean? P(x) proportional to exp(-ax) seems to be the logical 
interpretation. Then we could figure out  a  from the specification 
that P(x) = 0.75. But even this is a bit ambiguous. How do weinterpret 
P(x)? Does it mean, for example, that the probability that the part 
lasts 1000 hrs or less is proportional to exp(-ax)?

Well, it can't mean that. Obviously, the probability that it lasts 
(1000 hours or less) is GREATER THAN the probability that it lasts 
(500 hours or less), because the former subsumes the latter. The 
sensible way to interpret the equation is that the probability of the 
part failing at any given moment time = x is proportional to exp(-ax). 
The reason that the probability goes DOWN with increasing time is that 
there are fewer and fewer parts left operational that have an 
opportunity to fail as the time grows longer.

So we'll interpret the question to mean the only thing it can mean 
sensibly: that the probability of failure at any moment time = x is 
proportional to exp(-ax).

How do we find a? Well, before we can even think about that, we need 
to find the constant of proportionality. We can do that by saying that 
the probability of failure for all times 0 to infinity must add up to 
1. Therefore, if p(x) = C*exp(-ax), we can evaluate C by demanding 
that the integral of p(x) from 0 to infinity should equal 1. Do this 
first. Solve for C.

The next step is to find a. You can do this, now that you know the 
constant of proportionality, by taking another integral, and setting 
that integral = 0.75. What integral is that?

Now that you know a, you can take the final step, and calculate the 
mean failure time. That is the same as the expected lifetime. To 
calculate the expected lifetime, we take the integral x*p(x), again 
from x = 0 to x = infinity. This is like saying: take a weighted 
average of all the times that it could fail (x), using the 
probabilities associated with those times (P(x)) as the weights.

That's a lot of steps. Let me know if you get stuck anywhere along the 
way; and if not, will you write back and let me know what you got 
for an answer?

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/