Problem 11
Question
Recall the relation between degrees Celsius and degrees Fahrenheit degrees Fahrenheit \(=\frac{9}{5} \cdot\) degrees Celsius \(+32\). Let \(X\) and \(Y\) be the average daily temperatures in degrees Celsius in Amsterdam and Antwerp. Suppose that \(\operatorname{Cov}(X, Y)=3\) and \(\rho(X, Y)=0.8\). Let \(T\) and \(S\) be the same temperatures in degrees Fahrenheit. Compute \(\operatorname{Cov}(T, S)\) and \(\rho(T, S)\).
Step-by-Step Solution
Verified Answer
\( \operatorname{Cov}(T, S) = 9.72 \) and \( \rho(T, S) = 0.8 \).
1Step 1: Understanding the Conversion Formula
Since Fahrenheit (T) = \( \frac{9}{5} C + 32 \), for T and S, we have \( T = \frac{9}{5} X + 32 \) and \( S = \frac{9}{5} Y + 32 \). This conversion affects how we find the covariance and correlation between T and S.
2Step 2: Covariance with Linear Transformation
The covariance of linearly transformed variables includes scaling. Since \( T = \frac{9}{5} X + 32 \), and \( S = \frac{9}{5} Y + 32 \), the covariance is \( \operatorname{Cov}(T, S) = \left( \frac{9}{5} \right)^2 \times \operatorname{Cov}(X, Y) = \frac{81}{25} \times 3 = \frac{243}{25} = 9.72 \).
3Step 3: Understanding Correlation Preservation
Correlation, unlike covariance, is dimensionless and not affected by changes in units. Therefore, \( \rho(T, S) = \rho(X, Y) = 0.8 \).
4Step 4: Summary of Calculations
By scaling the covariance due to the linear transformations, and understanding the nature of correlation, we obtain: \( \operatorname{Cov}(T, S) = 9.72 \) and \( \rho(T, S) = 0.8 \).
Key Concepts
Linear TransformationTemperature ConversionStatistical Independence
Linear Transformation
Linear transformation is a powerful concept in mathematics and statistics that involves changing the underlying characteristics of a dataset through a linear formula. When dealing with covariance and correlation, it's crucial to understand how these transformations affect the variables.
A linear transformation takes the form: \( T = aX + b \), where \( a \) represents a scale factor and \( b \) is a translation factor. In the context of temperature conversion between Celsius and Fahrenheit, the formula \( T = \frac{9}{5}X + 32 \) shows how Celsius temperatures are linearly transformed into Fahrenheit.
One interesting aspect of linear transformations is their impact on statistical measures:
A linear transformation takes the form: \( T = aX + b \), where \( a \) represents a scale factor and \( b \) is a translation factor. In the context of temperature conversion between Celsius and Fahrenheit, the formula \( T = \frac{9}{5}X + 32 \) shows how Celsius temperatures are linearly transformed into Fahrenheit.
One interesting aspect of linear transformations is their impact on statistical measures:
- Scaling: When you apply a linear transformation using a scaling factor \( a \), the covariance of the transformed variables is scaled by \( a^2 \). For instance, if \( X \) and \( Y \) are temperatures in Celsius with a covariance of 3, transforming them to Fahrenheit results in a covariance \( \operatorname{Cov}(T, S) = \left(\frac{9}{5}\right)^2 \times 3 = 9.72 \).
- Translation: Adding a constant (like adding 32 when converting to Fahrenheit) does not affect the covariance, as covariance is only influenced by how much two variables change together.
Temperature Conversion
Temperature conversion is a practical application of a linear transformation between Celsius and Fahrenheit scales. The formula \( T = \frac{9}{5}C + 32 \) allows us to convert temperatures from Celsius to Fahrenheit, offering a perfect example of how linear transformations work in real life.
This transformation helps in various scientific and meteorological fields where temperatures are compared or analyzed across different units. The scaling factor \( \frac{9}{5} \) comes from the difference in the size of degrees between the two scales, while the added 32 aligns an offset due to different zero points (freezing point of water).
Understanding this conversion not only helps to seamlessly switch between units but also underscores how certain transformations affect statistical attributes like covariance and correlation. It demonstrates that while covariance changes due to scaling, correlation remains unaffected by the unit change, as it's a relative measure of association between variables.
This transformation helps in various scientific and meteorological fields where temperatures are compared or analyzed across different units. The scaling factor \( \frac{9}{5} \) comes from the difference in the size of degrees between the two scales, while the added 32 aligns an offset due to different zero points (freezing point of water).
Understanding this conversion not only helps to seamlessly switch between units but also underscores how certain transformations affect statistical attributes like covariance and correlation. It demonstrates that while covariance changes due to scaling, correlation remains unaffected by the unit change, as it's a relative measure of association between variables.
Statistical Independence
Statistical independence is a concept that describes the relationship between two variables. Two variables are independent if the occurrence of one does not affect the probability of the occurrence of the other.
In terms of covariance and correlation, if two variables are statistically independent, their covariance is zero. This is because independence implies that there is no linear relationship or mutual influence between the variables. However, zero covariance does not necessarily mean independence unless the relationship is strictly linear.
Understanding statistical independence is crucial in interpreting correlation, especially when transformed variables are involved. While linear transformations alter covariance by scaling, they do not create or eliminate statistical dependence or independence. Therefore, the knowledge of covariance and correlation is enhanced through recognizing if the variables imply independence or if transformations affect measurements but not the underlying relationships.
In terms of covariance and correlation, if two variables are statistically independent, their covariance is zero. This is because independence implies that there is no linear relationship or mutual influence between the variables. However, zero covariance does not necessarily mean independence unless the relationship is strictly linear.
Understanding statistical independence is crucial in interpreting correlation, especially when transformed variables are involved. While linear transformations alter covariance by scaling, they do not create or eliminate statistical dependence or independence. Therefore, the knowledge of covariance and correlation is enhanced through recognizing if the variables imply independence or if transformations affect measurements but not the underlying relationships.
Other exercises in this chapter
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