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Chapter 5: Problem 84

Let \(X_{1}\) and \(X_{2}\) be quantitative and verbal scores on one aptitude exam, and let \(Y_{1}\) and \(Y_{2}\) be corresponding scores on another exam. If \(\operatorname{Cov}\left(X_{1}, Y_{1}\right)=5, \operatorname{Cov}\left(X_{1}, Y_{2}\right)=1, \operatorname{Cov}\left(X_{2}, Y_{1}\right)=2\), and \(\operatorname{Cov}\left(X_{2}, Y_{2}\right)=8\), what is the covariance between the two total scores \(X_{1}+X_{2}\) and \(Y_{1}+Y_{2}\) ?

Short Answer

Expert verified
The covariance between the two total scores is 16.

Step by step solution

01

Understanding Covariance of Sums

The covariance between two sums, like \(X_1 + X_2\) and \(Y_1 + Y_2\), can be calculated using the formula:\[ \operatorname{Cov}(X_1 + X_2, Y_1 + Y_2) = \operatorname{Cov}(X_1, Y_1) + \operatorname{Cov}(X_1, Y_2) + \operatorname{Cov}(X_2, Y_1) + \operatorname{Cov}(X_2, Y_2). \] This formula is derived from the property of covariance being distributive over addition.
02

Substituting Known Values

Substitute the given covariance values into the formula:- \(\operatorname{Cov}(X_1, Y_1) = 5\)- \(\operatorname{Cov}(X_1, Y_2) = 1\)- \(\operatorname{Cov}(X_2, Y_1) = 2\)- \(\operatorname{Cov}(X_2, Y_2) = 8\)Plug these into the covariance formula:\[ \operatorname{Cov}(X_1 + X_2, Y_1 + Y_2) = 5 + 1 + 2 + 8. \]
03

Calculating the Total Covariance

Now, add the individual covariances together:\[ \operatorname{Cov}(X_1 + X_2, Y_1 + Y_2) = 5 + 1 + 2 + 8 = 16. \] This is the covariance between the total scores \(X_1 + X_2\) and \(Y_1 + Y_2\).

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Key Concepts

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

Quantitative Scores
Quantitative scores are numerical values that measure a person's ability to understand and manipulate numerical data. These scores are often derived from standardized tests. They evaluate mathematical skills including algebra, arithmetic, and data interpretation. In educational settings, quantitative scores help to estimate the mathematical proficiency of an individual.
  • They are commonly used to assess math-related skills.
  • Quantitative scores are crucial for fields that require analytical skills.
  • They help schools or employers understand your capacity to work with numbers.
Understanding your performance in quantitative scores can guide personal improvement and highlight areas where more practice is needed.
Verbal Scores
Verbal scores refer to the measure of an individual's ability to understand and use language effectively. Just like quantitative scores, they are assessed in exams that test literacy skills such as vocabulary, reading comprehension, and sentence structure. High verbal scores suggest strong communication and critical reading skills.
  • They are essential for gauging your aptitude in language-based tasks.
  • Verbal scores are a crucial indicator in fields that demand strong communication skills.
  • Scores often help identify strengths in understanding and processing written material.
In exams, verbal scores reveal an individual's command of the language and are vital for roles that require meticulous reading and writing abilities.
Covariance Formula
The covariance formula is a mathematical tool used to measure how much two random variables vary together. In simpler terms, it tells us if an increase in one variable tends to lead to an increase or decrease in another. For two variables, say \(X\) and \(Y\), the covariance is calculated as:\[ \operatorname{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)] \]where \(E\) is the expected value, and \(\mu_X\) and \(\mu_Y\) are the means of \(X\) and \(Y\) respectively.
  • Positive covariance indicates that variables tend to increase together.
  • Negative covariance indicates that as one variable increases, the other tends to decrease.
  • Zero covariance implies no linear relationship in the increase or decrease between variables.
The formula uses the sum of the products of the deviations of each pair of data points.
Mathematical Statistics
Mathematical statistics is a branch of math that deals with data collection, analysis, interpretation, and presentation. It provides methodologies for making inferences about population parameters based on sample data. Covariance is a fundamental concept within this field, as it helps to understand relationships between variables.
  • It involves techniques to model and understand random processes.
  • Statisticians use it for hypothesis testing, estimation, and data prediction.
  • Covariance plays a key role in statistical inference and multivariate analysis.
Studying mathematical statistics helps in making data-driven decisions and develop models that predict future trends based on past data.

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Most popular questions from this chapter

Suppose that \(X\) is uniformly distributed between 0 and 1. Given \(X=x, Y\) is uniformly distributed between 0 and \(x^{2}\) a. Determine \(E(Y \mid X=x)\) and then \(V(Y \mid X=x)\). Is \(E(Y \mid X=x)\) a linear function of \(x\) ? b. Determine \(f(x, y)\) using \(f_{X}(x)\) and \(f_{Y \mid X}(y \mid x)\). c. Determine \(f_{y}(y)\).

A circular sampling region with radius \(X\) is chosen by a biologist, where \(X\) has an exponential distribution with mean value \(10 \mathrm{ft}\). Plants of a certain type occur in this region according to a (spatial) Poisson process with "rate" \(.5\) plant per square foot. Let \(Y\) denote the number of plants in the region. a. Find \(E(Y \mid X=x)\) and \(V(Y \mid X=x)\) b. Use part (a) to find \(E(Y)\). c. Use part (a) to find \(V(Y)\).

Show that if \(X, Y\), and \(Z\) are rv's and \(a\) and \(b\) are constants, then \(\operatorname{Cov}(a X+b Y, Z)=a \operatorname{Cov}(X, Z)+\) \(b \operatorname{Cov}(Y, Z)\)

. An ecologist selects a point inside a circular sampling region according to a uniform distribution. Let \(X=\) the \(x\) coordinate of the point selected and \(Y=\) the \(y\) coordinate of the point selected. If the circle is centered at \((0,0)\) and has radius \(R\), then the joint pdf of \(X\) and \(Y\) is $$ f(x, y)=\left\\{\begin{array}{cl} \frac{1}{\pi R^{2}} & x^{2}+y^{2} \leq R^{2} \\ 0 & \text { otherwise } \end{array}\right. $$ a. What is the probability that the selected point is within \(R / 2\) of the center of the circular region? [Hint: Draw a picture of the region of positive density \(D\). Because \(f(x, y)\) is constant on \(D\), computing a probability reduces to computing an area.] b. What is the probability that both \(X\) and \(Y\) differ from 0 by at most \(R / 2\) ? c. Answer part (b) for \(R / \sqrt{2}\) replacing \(R / 2\) d. What is the marginal pdf of \(X ?\) Of \(Y ?\) Are \(X\) and \(Y\) independent?

Let \(X_{1}\) and \(X_{2}\) be independent, each having a standard normal distribution. The pair \(\left(X_{1}, X_{2}\right)\) corresponds to a point in a two-dimensional coordinate system. Consider now changing to polar coordinates via the transformation, $$ \begin{gathered} Y_{1}=X_{1}^{2}+X_{2}^{2} \\ Y_{2}=\left\\{\begin{array}{cl} \arctan \left(\frac{X_{2}}{X_{1}}\right) & X_{1}>0, X_{2} \geq 0 \\ \arctan \left(\frac{X_{2}}{X_{1}}\right)+2 \pi & X_{1}>0, X_{2}<0 \\ \arctan \left(\frac{X_{2}}{X_{1}}\right)+\pi & X_{1}<0 \\ 0 & X_{1}=0 \end{array}\right. \end{gathered} $$ from which \(X_{1}=\sqrt{Y_{1}} \cos \left(Y_{2}\right), X_{2}=\sqrt{Y_{1}} \sin \left(Y_{2}\right)\). Obtain the joint pdf of the new variables and then the marginal distribution of each one. [Note: It would be nice if we could simply let \(Y_{2}=\arctan\) \(\left(X_{2} / X_{1}\right)\), but in order to insure invertibility of the arctan function, it is defined to take on values only between \(-\pi / 2\) and \(\pi / 2\). Our specification of \(Y_{2}\) allows it to assume any value between 0 and \(2 \pi\).]
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