Question

The time in hours required to repair a machine is an exponential distributed random variable with the average repair time of 2 hours, what is the probability that the repair time takes between 2 to 4 hours

Answer 1

0.4968 = 49.68% probability that the repair time takes between 2 to 4 hours

Explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

`f(x) = \mu e^(-\mu x)`

In which
`\mu = (1)/(m)` is the decay parameter.

The probability that x is lower or equal to a is given by:

`P(X \leq x) = \int\limits^a_0 {f(x)} \, dx`

Which has the following solution:

`P(X \leq x) = 1 - e^(-\mu x)`

The probability of finding a value higher than x is:

`P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^(-\mu x)) = e^(-\mu x)`

Average repair time of 2 hours

This means that
`m = 2, \mu = (1)/(2) = 0.5`

What is the probability that the repair time takes between 2 to 4 hours

More than 2:

`P(X > 2) = e^(-0.5*2) = 0.3679`

Less than 4:

`P(X \leq 4) = 1 - e^(0.05*4) = 0.8647`

Between two and four:

`P(2 \leq x \leq 4) = 0.8647 - 0.3679 = 0.4968`

0.4968 = 49.68% probability that the repair time takes between 2 to 4 hours