Question

A bus comes by every 13 minutes. The times from when a person arives at the busstop until the bus arrives follows a Uniform distribution from 0 to 13 minutes. A person arrives at the bus stop at a randomly selected time. Round to 4 decimal places where possible.

a. The mean of this distribution is?

b. The standard deviation is?

c. The probability that the person will wait more than 8 minutes is?

d. Suppose that the person has already been waiting for 3 minutes. Find the probability that the person's total waiting time will be between 6 and 9.4 minutes.

e. 78% of all customers wait at least how long for the train?

Answer 1

The mean waiting time for the bus is 6.5 minutes, and the standard deviation is approximately 3.75 minutes. The probability that a person waits more than 8 minutes is 38.46%, while the conditional probability of waiting between 6 and 9.4 minutes after already waiting for 3 minutes is 26.15%. To be among the 78% of people who wait the longest, a person would have to wait at least 2.86 minutes.

Step-by-step explanation:

For a uniform distribution with a minimum value of 0 and a maximum value of 13, the mean (expected value) can be calculated as:

Mean = (0 + 13) / 2 = 6.5 minutes

The formula for the standard deviation (σ) of a uniform distribution is:

σ = (max - min) / sqrt(12)

In this case:

σ = (13 - 0) / sqrt(12) = 13 / sqrt(12) ≈ 3.75 minutes rounded to four decimal places.

The probability that the person waits more than 8 minutes is found by calculating the fraction of the distribution that lies beyond 8 minutes since in a uniform distribution the probability is evenly spread:

Probability = (13 - 8) / 13 ≈ 0.3846 or 38.46%

For part d, since the person has waited for 3 minutes already, we have a conditional probability situation where we are considering the remaining time frame of 10 minutes:

Probability (6 to 9.4 minutes) = (9.4 - 6) / 13 ≈ 0.2615 or 26.15%

For part e, to find the time at which 78% of customers wait at least that long, we consider the lower 22% (as 100% - 78% = 22%):

22% of 13 minutes = 0.22 * 13 ≈ 2.86 minutes