Question

The commute time for people in a city has an exponential distribution with an average of 0.5 hours. What is the probability that a randomly selected person in this city will have a commute time between 0.4 and 1 hours? Answer: (round to 3 decimal places)

Answer 1

0.314 = 31.4% probability that a randomly selected person in this city will have a commute time between 0.4 and 1 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)`

In this question:

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

What is the probability that a randomly selected person in this city will have a commute time between 0.4 and 1 hours?

`P(0.4 \leq X \leq 1) = P(X \leq 1) - P(X \leq 0.4)`

In which

`P(X \leq 1) = 1 - e^(-2) = 0.8647`

`P(X \leq 0.4) = 1 - e^(-2*0.4) = 0.5507`

So

`P(0.4 \leq X \leq 1) = P(X \leq 1) - P(X \leq 0.4) = 0.8647 - 0.5507 = 0.314`

0.314 = 31.4% probability that a randomly selected person in this city will have a commute time between 0.4 and 1 hours