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

Answer:

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

Step-by-step explanation:

Exponential distribution:

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

$f(x) = \mu e^{-\mu x}$

In which $\mu = \frac{1}{m}$ 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:

$m = 10, \mu = \frac{1}{10} = 0.1$

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