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Suppose your waiting time for a bus in the morning is uniformly distributed on \([0,8]\), whereas waiting time in the evening is uniformly distributed on \([0,10]\) independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time? [Hint: Define rv's \(X_{1}, \ldots, X_{10}\) and use a rule of expected value.] b. What is the variance of your total waiting time? c. What are the expected value and variance of the difference between morning and evening waiting times on a given day? d. What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week?

Step by step solution

01

Define the Random Variables

Let's define the waiting times: \(X_i\) as the morning waiting time and \(Y_i\) as the evening waiting time for day \(i\). Both are uniformly distributed, \(X_i \sim U(0,8)\) and \(Y_i \sim U(0,10)\), for \(i = 1, 2, ..., 5\). Over a week, there are 5 mornings and 5 evenings.

02

Find Expected Value of Daily Waiting Times

For a uniform distribution \(U(a,b)\), the expected value is \(E[X]=\frac{a+b}{2}\). Therefore, \(E[X_i]=\frac{0+8}{2}=4\) and \(E[Y_i]=\frac{0+10}{2}=5\).

03

Total Expected Waiting Time Over a Week

The total expected morning waiting time for a week is \(E\left[\sum_{i=1}^5 X_i\right] = 5 \cdot 4 = 20\). The total expected evening waiting time is \(E\left[\sum_{i=1}^5 Y_i\right] = 5 \cdot 5 = 25\). Therefore, the total expected waiting time is \(20 + 25 = 45\) minutes.

04

Find Variance of Daily Waiting Times

For a uniform distribution \(U(a,b)\), the variance is \(Var[X]=\frac{(b-a)^2}{12}\). Therefore, \(Var[X_i]=\frac{(8-0)^2}{12}=\frac{64}{12}=\frac{16}{3}\) and \(Var[Y_i]=\frac{(10-0)^2}{12}=\frac{100}{12}=\frac{25}{3}\).

05

Variance of Total Waiting Time Over a Week

Because the waiting times are independent, the variance of the sum is the sum of the variances. Thus, \(Var\left(\sum_{i=1}^5 X_i\right) = 5 \cdot \frac{16}{3} = \frac{80}{3}\) and \(Var\left(\sum_{i=1}^5 Y_i\right) = 5 \cdot \frac{25}{3} = \frac{125}{3}\). The total variance over a week is \(\frac{80}{3} + \frac{125}{3} = \frac{205}{3}\).

06

Expected Value and Variance of the Daily Difference

The difference for a given day is \(D_i = X_i - Y_i\). Thus, \(E[D_i] = E[X_i] - E[Y_i] = 4 - 5 = -1\). The variance is \(Var[D_i] = Var[X_i] + Var[Y_i] = \frac{16}{3} + \frac{25}{3} = \frac{41}{3}\).

07

Expected Value and Variance of the Weekly Difference

The total difference over a week is \(D = \sum_{i=1}^5 (X_i - Y_i)\). Thus, \(E[D] = 5 \cdot (-1) = -5\). The variance is \(Var[D] = 5 \cdot \frac{41}{3} = \frac{205}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value

When dealing with random variables, the expected value is like the mean or average value you might expect after many observations or trials. In simple terms, it's the center of the distribution of a random variable.

• The expected value for a uniform distribution, such as waiting times here, is calculated as the midpoint of the distribution's range.
• For the morning bus with a waiting time distribution of ([0,8]), the expected value is ( \[ E[X_i] = \frac{0 + 8}{2} = 4 \] minutes).
• Similarly, for the evening bus's waiting time distribution of ([0,10]), the expected value is ( \[ E[Y_i] = \frac{0 + 10}{2} = 5 \] minutes).

To find the total expected waiting time over a week, you combine the morning and evening expectations. Multiply each by the number of days (5), and sum up the results:

• Total morning expected time: ( \[ 5 \times 4 = 20 \] minutes),
• Total evening expected time: ( \[ 5 \times 5 = 25 \] minutes).
• Total expected time for both: ( \[ 20 + 25 = 45 \] minutes).

This total gives you an overall perspective of how much time to reasonably expect waiting for buses in a week.

Variance

Variance measures the spread or variability of a random variable around its expected value. It tells you how much the values fluctuate from the mean value, on average.

For a uniform distribution ( \( U(a,b) \) ), the variance is calculated by:( \[ Var[X] = \frac{(b-a)^2}{12} \] ).

• For the morning bus wait time ( \( U(0,8) \) ), the variance is ( \[ Var[X_i] = \frac{(8-0)^2}{12} = rac{64}{12} = rac{16}{3} \] ).
• For the evening wait time ( \( U(0,10) \) ), the variance is ( \[ Var[Y_i] = \frac{(10-0)^2}{12} = rac{100}{12} = rac{25}{3} \] ).

The weekly variance considers the sum of individual variances due to independent waiting times:

• Total variance for morning: ( \[ 5 \times \frac{16}{3} = \frac{80}{3} \] ).
• Total variance for evening: ( \[ 5 \times \frac{25}{3} = \frac{125}{3} \] ).
• Combined variance over a week: ( \[ \frac{80}{3} + \frac{125}{3} = \frac{205}{3} \] ).

Understanding variance is critical because it helps to grasp how consistent your waiting times will be through the week.

Random Variables

A random variable represents a numerical outcome of a random phenomenon. It transforms real-world phenomena, like waiting times, into a mathematical context that can be analyzed and used for predicting future events.

• In our scenario, we define two types of random variables per day: ( \( X_i \) ) for morning waiting times and ( \( Y_i \) ) for evening waiting times.
• Both ( \( X_i \) ) and ( \( Y_i \) ) are uniformly distributed within specified intervals, implying every time within the range is equally likely.

To make sense of their relationships and implications on overall wait times, we combine these variables:

• Individual day difference: ( \( D_i = X_i - Y_i \) ) tells us how much longer or shorter morning wait is compared to the evening.
• On a particular day, the expected difference is ( \( E[D_i] = E[X_i] - E[Y_i] = 4 - 5 = -1 \) minutes), indicating longer evening waits are expected.
• Weekly differences for mornings vs. evenings, total:( \( E[D] = 5 \times (-1) = -5 \) ),showing an overall trend of evening waits being consistently longer across a week.
• Variance of this difference, ( \( Var[D] = \frac{205}{3} \) ), helps quantify how much these differences can vary.

Random variables are essential tools in statistics, letting us model and predict real-world situations, such as daily delays and their weekly impacts.