Question

Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ (lambda) = 0.5.

What's the probability that a repair takes less than 5 hours? AND what's the conditional probability that a repair takes at least 11 hours, given that it takes more than 8 hours?

Answer 1

The probability distribution you are talking about is
f (x; lambda) = lambda exp^-(lambda x), x > 0

It is discussed at http://en.wikipedia.org/wiki/Exponential_distribution

The probability that a repair takes less than x is
F (x; lambda) = 1 - exp(-lambda x)
For x = 1 - exp(-2.5) = 0.918

The fraction taking 8 hours or more to repair is
exp(-4) = 0.01832
The fraction taking 11 hours to repair is
exp(-5.5) = 0.00409
That means that 409/1832 = 22% of the repairs taking 8 hours or more require more than 11 hours.