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The expected value, often symbolized as \( E(X) \), is a fundamental concept in probability that gives us the long-term average of a random variable. For a uniform distribution, where the variable has an equal chance of taking any value in a specified interval, the calculation is straightforward. The expected value of a uniformly distributed random variable on the interval \([a, b]\) is the midpoint of that interval: \[ E(X) = \frac{a + b}{2} \]In our bus waiting time scenario, the morning waiting time \(X\), distributed uniformly from 0 to 8, has an expected value of 4 minutes. In the evening, the expected waiting time \(Y\) is 5 minutes for the interval \([0, 10]\). These calculations are crucial when we want to sum the expected values for multiple events. For instance, if you take the bus every morning and evening for seven days, your total expected waiting time is simply the sum of each day's expectation, multiplying them by 7: \[ 7 \times (4 + 5) = 63 \text{ minutes} \] This application shows how the expected value helps us understand long-term averages across independent events.

Variance is a measure of how spread out a set of values are. For a uniform distribution, the variance can be calculated using the formula:\[ \text{Var}(X) = \frac{(b-a)^2}{12} \]This tells us how much the observed values will deviate from their expected value. In the context of our example:- The variance of the morning bus waiting time, \(X\), on the interval \([0,8]\) is: \[ \text{Var}(X) = \frac{(8-0)^2}{12} = \frac{64}{12} = \frac{16}{3} \]- Similarly, the variance of the evening waiting time, \(Y\), on \([0,10]\) is: \[ \text{Var}(Y) = \frac{(10-0)^2}{12} = \frac{100}{12} = \frac{25}{3} \]For a week, we consider seven days of waiting time, so the variance of the total waiting time is scaled by 7:\[ 7 \times \left( \frac{16}{3} + \frac{25}{3} \right) = \frac{287}{3} \]This calculation captures the variability you would expect in the total waiting time over multiple days.

Random variables are numerical outcomes of random phenomena. They can represent different things, such as the time spent waiting for a bus. In this exercise, we define two random variables for each day: - \(X_i\) represents the waiting time in the morning.- \(Y_i\) represents the waiting time in the evening.Both are described by a uniform distribution because the waiting time is equally likely across defined intervals each day.By treating these times as random variables,- We can analyze the **expected waiting times** over multiple days, using their expected values.- We can also assess the **variability of the waiting times** across those days using their variances.This approach simplifies the process of calculating metrics over multiple scenarios, illustrating how random variables are powerful tools in probability theory.

Independent events are a cornerstone concept in probability, meaning the occurrence of one event does not alter the likelihood of another. In the context of the bus waiting times:- The morning waiting time \(X\) and the evening waiting time \(Y\) on any given day are independent.- This means the waiting time in the morning does not affect the waiting time in the evening.This independence simplifies calculations of combined metrics like expected value and variance. Since the waiting times are independent, when calculating the variance of the sum \(X+Y\), it becomes:\[ \text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) \]This principle is used to calculate the variance over multiple days as well, by summing each day's variances independently. Thus, understanding independence lets us more easily compute outcomes across repeated trials, such as waiting times over a week.