Bus Arrival Probability Analysis

For a bus that arrives every hour but with an unknown departure time, answer the following questions: (1) What is the probability density function for this situation? (2) How do we draw a continuous distribution graph for this situation? (3) What is the likelihood of having to wait between 5 and 20 minutes for the bus? (4) What are the mean and standard deviation for this situation?

When analyzing a situation where a bus arrives every hour but with an unknown departure time, we are essentially delving into continuous random variables, particularly because the bus could depart at any moment within the hour. One common approach to model such scenarios is using a uniform distribution, which is simple yet powerful due to its equal probability allocation across the specified interval. The probability density function, in this instance, would be constant if we assume the departure time can be any minute within an hour uniformly. A key step in understanding this problem is visualizing the continuous distribution.

The graph of a uniform distribution would be a rectangle indicating constant probability across the duration of the interval, in this case, from 0 to 60 minutes. It allows for straightforward computation of probabilities over specific intervals which, in turn, aids in addressing questions about specific waiting times. The mean and standard deviation of this distribution provide further insights into the expected value and the variability of wait times, respectively.

The mean tells us the average or expected wait time while the standard deviation provides a measure of the spread of the wait times around this average. In uniform distributions, the mean is the midpoint of the interval, and the standard deviation can be easily calculated, highlighting the simplicity and elegance of handling uniform distributions in real-world scenarios like bus timetables.