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Chapter 5: Problem 110

Suppose that \(Y_{1}\) and \(Y_{2}\) have correlation coefficient \(\rho=.2 .\) What is is the value of the correlation coefficient between a. \(1+2 Y_{1}\) and \(3+4 Y_{2} ?\) b. \(1+2 Y_{1}\) and \(3-4 Y_{2} ?\) c. \(1-2 Y_{1}\) and \(3-4 Y_{2} ?\)

Short Answer

Expert verified
The correlation coefficient in all cases is 0.2.

Step by step solution

01

Understanding the Impact of Linear Transformations

The correlation coefficient between two variables \(Y_1\) and \(Y_2\), denoted as \(\rho\), remains unchanged under linear transformations. This means if each variable is transformed by a linear equation, like \(aY_1 + b\) or \(cY_2 + d\), the correlation \(\rho\) between \(Y_1\) and \(Y_2\) will be the same as the correlation between \(aY_1 + b\) and \(cY_2 + d\).
02

Solving Part a: Correlation of \(1+2Y_1\) and \(3+4Y_2\)

For the variables \(1 + 2Y_1\) and \(3 + 4Y_2\), both transformations are linear. Therefore, the correlation coefficient remains \(\rho = 0.2\).
03

Solving Part b: Correlation of \(1+2Y_1\) and \(3-4Y_2\)

Applying the transformation \(-4Y_2\) is still linear. Hence, the correlation coefficient between \(1 + 2Y_1\) and \(3 - 4Y_2\) is still \(\rho = 0.2\). The linearity of the transformations ensures the coefficient remains unchanged.
04

Solving Part c: Correlation of \(1-2Y_1\) and \(3-4Y_2\)

Similarly, for \(1 - 2Y_1\) and \(3 - 4Y_2\), the transformations are linear. Therefore, the correlation coefficient remains \(\rho = 0.2\) between these two variables.

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Key Concepts

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

Linear Transformations
Linear transformations are crucial in understanding correlation coefficients. Imagine you have two variables, let's call them \(Y_1\) and \(Y_2\), and you're looking at how closely they relate to each other. This relationship is expressed through the correlation coefficient \(\rho\). Linear transformations involve altering your variables by scaling them and/or shifting them. This can be done by equations such as \(aY_1 + b\) or \(cY_2 + d\). With linear transformations, though the values of \(Y_1\) and \(Y_2\) might change, the correlation \(\rho\) between them stays exactly the same. This is because linear transformations preserve the direction and strength of relationships between variables.

For example, if \(Y_1\) is modified to become \(1 + 2Y_1\) and \(Y_2\) becomes \(3 + 4Y_2\), as shown in a math problem, the correlation remains unaffected. Whether you add or subtract constants or multiply by factors, as long as these operations are carried out uniformly, the correlation coefficient remains constant.
Statistical Independence
Statistical independence might seem like a distant concept from correlation, but they are two fundamental ideas in statistics. Independence concerns the idea that one event or variable does not have any effect on another. In mathematics, independence between two variables, say \(X\) and \(Y\), means that the occurrence or value of one does not predict or influence the other. When two random variables are statistically independent, the correlation coefficient between them is zero. This indicates no linear relationship at all.

In the given scenario, observing a correlation coefficient of 0.2 implies some level of dependence, as a truly independent pair would have a coefficient of zero. The essence here is that knowing about \(Y_1\) gives us some insight into \(Y_2\), but it's important to note that this does not equate to causation. Just because two variables are dependent doesn't mean one causes the other to happen; it's just that they move together to some degree.
Mathematical Statistics
Mathematical statistics exists as the backbone of solving problems involving correlation coefficients and linear relationships. It combines math theory with statistical principles, offering tools to analyze and make inferences about data. The correlation coefficient is one such powerful tool, providing a numerical summary of the relationship between two quantitative variables.

Within the field of mathematical statistics, various techniques allow you to interpret and transform data. Linear transformations, for instance, are used in mathematical statistics to modify variables without affecting their mutual relationship. This ensures that conclusions drawn from statistical analysis are robust, consistent, and applicable to real-world scenarios.

Using mathematical statistics, we can also understand that a correlation of 0.2 indicates a weak positive relationship. While mathematical statistics offers ways to explore these weak correlations, it often requires deeper probing into whether this observed relationship has any practical significance or if it merely results from random chance.

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Most popular questions from this chapter

In the production of a certain type of copper, two types of copper powder (types A and B) are mixed together and sintered (heated) for a certain length of time. For a fixed volume of sintered copper, the producer measures the proportion \(Y_{1}\) of the volume due to solid copper (some pores will have to be filled with air) and the proportion \(Y_{2}\) of the solid mass due to type A crystals. Assume that appropriate probability densities for \(Y_{1}\) and \(Y_{2}\) are $$\begin{array}{l} f_{1}\left(y_{1}\right)=\left\\{\begin{array}{ll} 6 y_{1}\left(1-y_{1}\right), & 0 \leq y_{1} \leq 1 \\ 0, & \text { elsewhere } \end{array}\right. \\ f_{2}\left(y_{2}\right)=\left\\{\begin{array}{ll} 3 y_{2}^{2}, & 0 \leq y_{2} \leq 1 \\ 0, & \text { elsewhere } \end{array}\right. \end{array}$$ The proportion of the sample volume due to type A crystals is then \(Y_{1} Y_{2} .\) Assuming that \(Y_{1}\) and \(Y_{2}\) are independent, find \(P\left(Y_{1} Y_{2} \leq .5\right)\)

How big or small can \(\operatorname{Cov}\left(Y_{1}, Y_{2}\right)\) be? Use the fact that \(\rho^{2} \leq 1\) to show that $$-\sqrt{V\left(Y_{1}\right) \times V\left(Y_{2}\right)} \leq \operatorname{Cov}\left(Y_{1}, Y_{2}\right) \leq \sqrt{V\left(Y_{1}\right) \times V\left(Y_{2}\right)}$$

Let \(X_{1}, X_{2},\) and \(X_{3}\) be random variables, either continuous or discrete. The joint moment generating function of \(X_{1}, X_{2},\) and \(X_{3}\) is defined by $$m\left(t_{1}, t_{2}, t_{3}\right)=E\left(e^{t_{1} X_{1}+t_{2} X_{2}+t_{3} X_{3}}\right)$$ a. Show that \(m(t, t, t)\) gives the moment-generating function of \(X_{1}+X_{2}+X_{3}\) b. Show that \(m(t, t, 0)\) gives the moment-generating function of \(X_{1}+X_{2}\) c. Show that $$\left.\frac{\partial^{k_{1}+k_{2}+k_{3}} m\left(t_{1}, t_{2}, t_{3}\right)}{\partial t_{1}^{k_{1}} \partial t_{2}^{k_{2}} \partial t_{3}^{k_{3}}}\right]_{t_{1}=t_{2}=t_{3}=0}=E\left(X_{1}^{k_{1}} X_{2}^{k_{2}} X_{3}^{k_{3}}\right)$$

Let \(Y_{1}\) and \(Y_{2}\) be jointly distributed random variables with finite variances. a. Show that \(\left[E\left(Y_{1} Y_{2}\right)\right]^{2} \leq E\left(Y_{1}^{2}\right) E\left(Y_{2}^{2}\right) .\) [Hint: Observe that \(E\left[\left(t Y_{1}-Y_{2}\right)^{2}\right] \geq 0\) for any real number t or, equivalently, $$t^{2} E\left(Y_{1}^{2}\right)-2 t E\left(Y_{1} Y_{2}\right)+E\left(Y_{2}^{2}\right) \geq 0$$ This is a quadratic expression of the form \(A t^{2}+B t+C\); and because it is nonnegative, we must have \(B^{2}-4 A C \leq 0 .\) The preceding inequality follows directly.] b. Let \(\rho\) denote the correlation coefficient of \(Y_{1}\) and \(Y_{2} .\) Using the inequality in part (a), show that \(\rho^{2} \leq 1\)

A firm purchases two types of industrial chemicals. Type I chemical costs \(\$ 3\) per gallon, whereas type II costs \(\$ 5\) per gallon. The mean and variance for the number of gallons of type I chemical purchased, \(Y_{1},\) are 40 and \(4,\) respectively. The amount of type II chemical purchased, \(Y_{2}\), has \(E\left(Y_{2}\right)=65\) gallons and \(V\left(Y_{2}\right)=8 .\) Assume that \(Y_{1}\) and \(Y_{2}\) are independent and find the mean and variance of the total amount of money spent per week on the two chemicals.
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