Question

The length of time for one individual to be served at a cafeteria is a random variable having an exponential distribution with a mean of 4 minutes. a. Find the value of λ. b. What is the probability that a person waits for less than 3 minutes?

Answer 1

a) 0.25

b) 52.76% probability that a person waits for less than 3 minutes

Explanation:

Exponential distribution:

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

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

In which
`\lambda = (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 = 4`

a. Find the value of λ.

`\lambda = (1)/(m) = (1)/(4) = 0.25`

b. What is the probability that a person waits for less than 3 minutes?

`P(X \leq 3) = 1 - e^(-0.25*3) = 0.5276`

52.76% probability that a person waits for less than 3 minutes