Question

The caller times at a customer service center has an exponential distribution with an average of 10 seconds. Find the probability that a randomly selected call time will be less than 25 seconds? (Round to 4 decimal places.) Answer:

Answer 1

0.9179 = 91.79% probability that a randomly selected call time will be less than 25 seconds

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)`

In this question:

`m = 10, \mu = (1)/(10) = 0.1`

Find the probability that a randomly selected call time will be less than 25 seconds?

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

`P(X \leq 25) = 1 - e^(-0.1*25) = 0.9179`

0.9179 = 91.79% probability that a randomly selected call time will be less than 25 seconds