91Ó°ÊÓ

Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the number of cars coming from each road onto the freeway is a random variable, with expected value and standard deviation as given in the table. a. What is the expected total number of cars entering the freeway at this point during the period? [Hint: Let \(X_{b}=\) the number from road i.] b. What is the variance of the total number of entering cars? Have you made any assumptions about the relationship between the numbers of cars on the different roads? c. With \(X_{l}\) denoting the number of cars entering from road \(i\) during the period, suppose that \(\operatorname{Cov}\left(X_{1}, X_{2}\right)=80\), \(\operatorname{Cov}\left(X_{1}, X_{3}\right)=90\), and \(\operatorname{Cov}\left(X_{2}, X_{3}\right)=100\) (so that the three streams of traffic are not independent). Compute the expected total number of entering cars and the standard deviation of the total.

Step by step solution

01

Identify Random Variables

Let's denote the number of cars entering the freeway from road 1 as \(X_1\), from road 2 as \(X_2\), and from road 3 as \(X_3\). The total number of cars entering the freeway is represented as \(T = X_1 + X_2 + X_3\).

02

Calculate Expected Value

To find the expected total number of cars, use the property of expectation: \(E(T) = E(X_1) + E(X_2) + E(X_3)\). Substitute the expected values from the table: \(E(X_1) = 100\), \(E(X_2) = 120\), and \(E(X_3) = 140\). Thus, \(E(T) = 100 + 120 + 140 = 360\).

03

Calculate Variance of Total Cars (Assuming Independence)

If numbers of cars from different roads are independent, the variance of total is the sum of variances: \(Var(T) = Var(X_1) + Var(X_2) + Var(X_3)\). With standard deviations given as 10, 15, and 20, we have \(Var(X_1) = 10^2 = 100\), \(Var(X_2) = 15^2 = 225\), and \(Var(X_3) = 20^2 = 400\). Therefore, \(Var(T) = 100 + 225 + 400 = 725\). To assume independence is made here.

04

Adjust Variance for Covariance

Given covariances are \(\operatorname{Cov}(X_1, X_2) = 80\), \(\operatorname{Cov}(X_1, X_3) = 90\), and \(\operatorname{Cov}(X_2, X_3) = 100\), adjust variance: \(Var(T) = Var(X_1) + Var(X_2) + Var(X_3) + 2\operatorname{Cov}(X_1, X_2) + 2\operatorname{Cov}(X_1, X_3) + 2\operatorname{Cov}(X_2, X_3)\). Therefore, \(Var(T) = 100 + 225 + 400 + 2(80) + 2(90) + 2(100) = 1175\).

05

Compute Standard Deviation of Total

The standard deviation of the total number of cars \(T\) is the square root of the variance: \(\text{SD}(T) = \sqrt{1175} \approx 34.3\).

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Key Concepts

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

Expected Value

The expected value, often referred to as the mean, gives us the average outcome of a random variable if an experiment is repeated many times. In our scenario with cars coming from three roads, it tells us how many cars we can expect to enter the freeway during the chosen period.
The expected value for the total number of cars, denoted as \(E(T)\), is calculated by summing the expected values of individual roads. Here, it's important to remember that expected value is additive. This means:

• \(E(T) = E(X_1) + E(X_2) + E(X_3)\)
• In our problem: \(E(T) = 100 + 120 + 140 = 360\)

Thus, on average, we can expect 360 cars to enter the freeway from all roads during the specified time period. The beauty of expected value lies in its simplicity and ability to provide a clear picture based on averages.

Variance

Variance measures the spread of random variables around their expected value. It tells us how much variability exists from the average. In the context of our exercise, variance gives us an understanding of how consistent the number of entering cars is over time.
When we calculate the variance for the total number of cars (\(Var(T)\)), we initially assume independence among roads. This means the variance is just a cumulative sum:

• \(Var(X_1) = 10^2 = 100\)
• \(Var(X_2) = 15^2 = 225\)
• \(Var(X_3) = 20^2 = 400\)
• Total \(Var(T) = 100 + 225 + 400 = 725\) under independence assumption

If the roads are not independent, additional calculations involving covariance terms are necessary to adjust and refine our understanding of variance. Variance is crucial as it provides insights into potential fluctuations in traffic numbers.

Covariance

Covariance is a measure of how two random variables change together. In our given problem, it indicates the relationship between the number of cars from different roads. Positive covariance tells us that as the number of cars from one road increases, the number will likely increase on the other road as well.
For our three roads, the covariance values given are:

• \(\text{Cov}(X_1, X_2) = 80\)
• \(\text{Cov}(X_1, X_3) = 90\)
• \(\text{Cov}(X_2, X_3) = 100\)

These values are essential when calculating the overall variance, especially since the roads are not independent. By incorporating covariance, the adjusted variance of total cars is calculated to be 1175 instead of 725. Understanding covariance helps us grasp the interconnected nature of our variables and refine our statistical calculations.

Standard Deviation

Standard deviation provides a more intuitive sense of dispersion by converting variance back into the original units of measurement (i.e., cars in our case). It represents the average distance of each number from the mean, or expected value.
In this exercise, the standard deviation of the total number of cars, denoted \(\text{SD}(T)\), is calculated by taking the square root of the variance:

• If the variance \(Var(T)\) is 1175, then \(\text{SD}(T) = \sqrt{1175}\)
• This calculates to approximately 34.3 cars

Thus, the standard deviation of about 34.3 indicates how much the total number of cars entering the freeway can deviate from the expected value, giving a clearer picture of variability and potential traffic congestion.