Suppose that you arrive at a bus stop​ randomly, so all arrival times are equally likely. The bus arrives regularly every 90 minutes without delay. What is the expected value of your waiting​ time?

To find the expected waiting time for the bus, you can assume that your arrival time at the bus stop is uniformly distributed between two consecutive bus arrivals. Since the bus arrives every 90 minutes, the time interval we are looking at is from 0 to 90 minutes.

The expected waiting time is the average waiting time, which can be calculated by integrating the waiting time over the interval from 0 to 90 minutes and then taking the average by dividing by the length of the interval (90 minutes).

We can calculate the expected value (E) for a uniform distribution with the following formula:

\[ E = \frac{a + b}{2} \]

where 'a' is the minimum value and 'b' is the maximum value of the uniform distribution.

For this example: - a = 0 minutes (if you happen to arrive just as the bus does) - b = 90 minutes (the longest possible wait is if you just missed the bus)

Plugging these values into the formula gives:

\[ E = \frac{0 + 90}{2} \] \[ E = \frac{90}{2} \] \[ E = 45 \]

Therefore, the expected value of your waiting time is 45 minutes.

Extra: The concept of expected value is a fundamental idea in probability and statistics. It is essentially the long-run average value of repetitions of the experiment it represents. In this case, if you were to arrive at the bus stop an infinite number of times at random times, the average waiting time you would experience is 45 minutes.

The uniform distribution is often applied in scenarios where each outcome within a certain range has an equal probability of occurring. In our example, arriving at any moment between two bus arrivals is equally likely, so we can use the uniform distribution to model the situation. The uniform distribution is characterized by its constant probability density function over its range of possible values. That is why we can simply take the midpoint of the minimum and maximum waiting times to find the expected waiting time.

In more complex situations where the outcomes are not uniformly distributed, other methods of determining expected value would be necessary, involving integrating the probability density function over all possible values or summing the individual probabilities for discrete distributions.