Question

The time (in minutes) until the next bus departs a major bus depot follows a distribution with f(x) = where x goes from 25 to 45 minutes.

a. What type of distribution does the time until the next bus departure follow?

b. Calculate the mean (μ) and standard deviation (σ) for the time distribution.

c. If X represents the time until the next bus departure, what is the probability P(X < 35)?

d. How would a change in the distribution parameters affect the waiting time?

Answer 1

a. The time until the next bus departure follows a uniform distribution. b. The mean is 35 minutes and the standard deviation is approximately 4.33 minutes. c. The probability of waiting less than 35 minutes is 0.5. d. Changing the distribution parameters would affect the waiting time.

Step-by-step explanation:

a. The time until the next bus departure follows a uniform distribution because the probability density function is constant within the range of 25 to 45 minutes.

b. The mean (μ) of the distribution can be calculated by taking the average of the lower and upper limits: (25 + 45) / 2 = 35 minutes. The standard deviation (σ) can be calculated using the formula: σ = (b - a) / sqrt(12), where a = 25 and b = 45. Therefore, σ = (45 - 25) / sqrt(12) ≈ 4.33 minutes.

c. To calculate the probability P(X < 35), we need to find the area under the probability density function curve from 25 to 35 minutes. Since the distribution is uniform, the probability is equal to the width of the interval divided by the total width of the distribution: P(X < 35) = (35 - 25) / (45 - 25) = 0.5.

d. A change in the distribution parameters, such as the mean or standard deviation, would affect the waiting time. For example, increasing the mean would result in longer waiting times on average, while increasing the standard deviation would result in greater variability in waiting times.