/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Three different roads feed into ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 12

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. \begin{tabular}{l|ccc} & Road 1 & Road 2 & Road 3 \\ \hline Expected value & 800 & 1000 & 600 \\ Standard deviation & 16 & 25 & 18 \end{tabular} a. What is the expected total number of cars entering the freeway at this point during the period? [Hint: Let \(X_{i}=\) the number from road \(\left.i .\right]\) 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 \(X_{i}\) 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.

Short Answer

Expert verified
a) 2400 cars; b) Var = 1205 (assuming independence); c) 2400 cars, SD 鈮� 39.05.

Step by step solution

01

Define the Variables

Let's define the variables: \(X_1\) is the number of cars from Road 1, \(X_2\) from Road 2, and \(X_3\) from Road 3. The total number \(T\) of cars entering the freeway during the period is given by \(T = X_1 + X_2 + X_3\).
02

Calculate the Expected Total Number of Cars (a)

The expected value of the total number of cars is the sum of individual expected values:\[E(T) = E(X_1) + E(X_2) + E(X_3) = 800 + 1000 + 600 = 2400.\]
03

Determine the Variance of Total Cars (b)

If we assume independence, the variance is the sum of the individual variances:\[\text{Var}(T) = \text{Var}(X_1) + \text{Var}(X_2) + \text{Var}(X_3) = 16^2 + 25^2 + 18^2 = 256 + 625 + 324 = 1205.\]This computation assumes that the number of cars entering from different roads are independent.
04

Calculate the Total Variance with Covariance (c)

Considering the given covariances, the total variance is:\[\text{Var}(T) = \text{Var}(X_1) + \text{Var}(X_2) + \text{Var}(X_3) + 2 \cdot \operatorname{Cov}(X_1, X_2) + 2 \cdot \operatorname{Cov}(X_1, X_3) + 2 \cdot \operatorname{Cov}(X_2, X_3).\]Insert the given values:\[256 + 625 + 324 + 2 \times 80 + 2 \times 90 + 2 \times 100 = 1525.\]
05

Compute the Standard Deviation of Total Cars (c)

The standard deviation of the total number of cars is the square root of its variance:\[\sigma_T = \sqrt{1525} \approx 39.05.\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Random Variables
In probability and statistics, a random variable is a variable that takes on numerical values, each associated with a probability. It is essentially a way to quantify uncertain outcomes. For example, in this exercise, the number of cars entering the freeway from each road is a random variable. These are denoted as \( X_1 \), \( X_2 \), and \( X_3 \) for roads 1, 2, and 3 respectively. Each random variable can take on a range of values, depending on the flow of traffic, and each possible number of cars has a specific likelihood of happening.

The concept of random variables is foundational in statistics as it allows us to model real-world phenomena in quantitative terms. By understanding random variables, we can predict and analyze situations where outcomes are not deterministic.
Expected Value
The expected value, or mean, of a random variable provides a measure of the central tendency of the possible outcomes it can take on. It is essentially the long-term average or the weighted average of all possible values the variable can assume. For the total number of cars entering the freeway, the expected value is calculated by summing up the expected number of cars from each road:

  • For Road 1, the expected value \( E(X_1) = 800 \).
  • For Road 2, \( E(X_2) = 1000 \).
  • For Road 3, \( E(X_3) = 600 \).

The total expected number of cars, denoted as \( E(T) \), is then the sum of these individual expected values: \( E(T) = 800 + 1000 + 600 = 2400 \).

This tells us that, on average, you can expect 2400 cars to enter the freeway during the period being considered.
Variance
Variance is a statistical measure that tells us how much the values of a random variable are spread out or how much they deviate from the expected value. It gives the "spread" of the possible outcomes. The higher the variance, the more spread out the outcomes are around the expected value.

In our exercise, if we assume the number of cars entering from different roads is independent (which means the flow on one road does not affect the others), the variance of the total number of cars, \( ext{Var}(T) \), is calculated as the sum of the variances of the individual roads:
  • \( ext{Var}(X_1) = 16^2 \)
  • \( ext{Var}(X_2) = 25^2 \)
  • \( ext{Var}(X_3) = 18^2 \)

Thus, \( ext{Var}(T) = 256 + 625 + 324 = 1205 \).

If the roads are not independent and covariances are provided, the computation must be adjusted to account for these covariances, resulting in a total variance of 1525.
Covariance
Covariance is a measure that shows how much two random variables change together. It indicates the direction of the linear relationship between variables. A positive covariance means that if one variable increases, the other tends to increase as well, while a negative covariance indicates the opposite.

In the exercise, the covariances between numbers of cars from different roads adjust the calculation of the total variance. Provided covariances are:
  • \( ext{Cov}(X_1, X_2) = 80 \)
  • \( ext{Cov}(X_1, X_3) = 90 \)
  • \( ext{Cov}(X_2, X_3) = 100 \)

These values are included in the formula for total variance to capture the interaction between traffic flows on different roads. Thus, the variance of the total cars, \( ext{Var}(T) \), becomes:\[ ext{Var}(T) = 256 + 625 + 324 + 2 \times 80 + 2 \times 90 + 2 \times 100 = 1525 \]

Understanding covariance is important as it shows how traffic fluctuations from different roads can be related, impacting overall variance and thus, reliability of traffic predictions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In an area having sandy soil, 50 small trees of a certain type were planted, and another 50 trees were planted in an area having clay soil. Let \(X=\) the number of trees planted in sandy soil that survive 1 year and \(Y=\) the number of trees planted in clay soil that survive 1 year. If the probability that a tree planted in sandy soil will survive 1 year is \(.7\) and the probability of 1-year survival in clay soil is .6, compute an approximation to \(P(-5 \leq X-Y \leq 5\) ) (do not bother with the continuity correction).

Let \(X_{1}, X_{2}\), and \(X_{3}\) be the lifetimes of components 1,2 , and 3 in a three-component system. a. How would you define the conditional pdf of \(X_{3}\) given that \(X_{1}=x_{1}\) and \(X_{2}=x_{2}\) ? b. How would you define the conditional joint pdf of \(X_{2}\) and \(X_{3}\) given that \(X_{1}=x_{1}\) ?

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let \(X\) denote the number of hoses being used on the self-service island at a particular time, and let \(Y\) denote the number of hoses on the full-service island in use at that time. The joint pmf of \(X\) and \(Y\) appears in the accompanying tabulation. \begin{tabular}{ll|ccc} \(p(x, y)\) & & 0 & 1 & 2 \\ \hline \multirow{x}{*}{\(x\)} & 0 & \(.10\) & \(.04\) & \(.02\) \\ & 1 & \(.08\) & \(.20\) & \(.06\) \\ & 2 & \(.06\) & \(.14\) & \(.30\) \end{tabular} a. What is \(P(X=1\) and \(Y=1)\) ? b. Compute \(P(X \leq 1\) and \(Y \leq 1)\). c. Give a word description of the event \(\\{X \neq 0\) and \(Y \neq 0\\}\), and compute the probability of this event. d. Compute the marginal pmf of \(X\) and of \(Y\). Using \(p_{X}(x)\), what is \(P(X \leq 1)\) ? e. Are \(X\) and \(Y\) independent rv's? Explain.

Let \(X\) denote the number of Canon SLR cameras sold during a particular week by a certain store. The pmf of \(X\) is \begin{tabular}{l|ccccc} \(x\) & 0 & 1 & 2 & 3 & 4 \\ \hline\(p_{x}(x)\) & \(.1\) & \(.2\) & \(.3\) & \(.25\) & \(.15\) \end{tabular} Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let \(Y\) denote the number of purchasers during this week who buy an extended warranty. a. What is \(P(X=4, Y=2)\) ? [Hint: This probability equals \(P(Y=2 \mid X=4) \cdot P(X=4)\); now think of the four purchases as four trials of a binomial experiment, with success on a trial corresponding to buying an extended warranty.] b. Calculate \(P(X=Y)\). c. Determine the joint pmf of \(X\) and \(Y\) and then the marginal pmf of \(Y\).

Let \(X_{1}, \ldots, X_{n}\) be independent rv's with mean values \(\mu_{1}, \ldots, \mu_{n}\) and variances \(\sigma_{1}^{2}, \ldots, \sigma_{\omega}^{2}\). Consider a function \(h\left(x_{1}, \ldots, x_{n}\right)\), and use it to define a rv \(Y=h\left(X_{1}, \ldots, X_{n}\right)\). Under rather general conditions on the \(h\) function, if the \(\sigma_{i}^{\prime}\) 's are all small relative to the corresponding \(\mu_{i}\) 's, it can be shown that \(E(Y) \approx h\left(\mu_{1}, \ldots, \mu_{n}\right)\) and $$ V(Y) \propto\left(\frac{\partial h}{\partial x_{1}}\right)^{2} \cdot \sigma_{1}^{2}+\cdots+\left(\frac{\partial h}{\partial x_{n}}\right)^{2} \cdot \sigma_{n}^{2} $$ where each partial derivative is evaluated at \(\left(x_{1}, \ldots, x_{n}\right)=\) \(\left(\mu_{1}, \ldots, \mu_{n}\right)\). Suppose three resistors with resistances \(X_{1}, X_{2}\), \(X_{3}\) are connected in parallel across a battery with voltage \(X_{4}\). Then by Ohm's law, the current is $$ Y=X_{4}\left[\frac{1}{X_{1}}+\frac{1}{X_{2}}+\frac{1}{X_{3}}\right] $$ Let \(\mu_{1}=10\) ohms, \(\sigma_{1}=1.0 \mathrm{ohm}, \quad \mu_{2}=15\) ohms, \(\sigma_{2}=1.0 \mathrm{ohm}, \mu_{3}=20\) ohms, \(\sigma_{3}=1.5 \mathrm{ohms}, \mu_{4}=120 \mathrm{~V}\), \(\sigma_{4}=4.0 \mathrm{~V}\). Calculate the approximate expected value and standard deviation of the current (suggested by "Random Samplings," CHEMTECH, 1984: 696-697).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.