Suppose that Y1 and Y2 have correlation coefficient ρY1, Y2 and for constants a, b, c and d let

step 1 Suppose y1 and y2 have the correlation have the correlation coefficient raw y1 y2 and a b c and

Suppose that Y1 and Y2 have correlation coefficient ρY1, Y2...

Suppose that $Y_{1}$ and $Y_{2}$ have correlation coefficient $\rho_{Y_{1}, Y_{2}}$ and for constants $a, b, c$ and $d$ let $W_{1}=a+b Y_{1}$ and $W_{2}=c+d Y_{2}$ a. Show that the correlation coefficient between $W_{1}$ and $W_{2}, \rho_{W_{1}, W_{2}},$ is such that $$ \left|\rho_{Y_{1}, Y_{2}}\right|=\left|\rho_{W_{1}, W_{2}}\right| $$ b. Does this result explain the results that you obtained in Exercise $5.110 ?$

Suppose that $Y_{1}$ and $Y_{2}$ have correlation coefficient $\rho_{Y_{1}, Y_{2}}$ and for constants $a, b, c$ and $d$ let $W_{1}=a+b Y_{1}$ and $W_{2}=c+d Y_{2}$
a. Show that the correlation coefficient between $W_{1}$ and $W_{2}, \rho_{W_{1}, W_{2}},$ is such that
$$
\left|\rho_{Y_{1}, Y_{2}}\right|=\left|\rho_{W_{1}, W_{2}}\right|
$$
b. Does this result explain the results that you obtained in Exercise $5.110 ?$

Key Concepts

Linearity of Covariance

Linearity of covariance means that the covariance of linear combinations of random variables can be expressed as a combination of the covariances of the original variables. This property justifies why the scaling factors in an affine transformation affect both the numerator and the denominator of the correlation coefficient in the same way, leading to the preservation of the absolute correlation.

Correlation Coefficient

The correlation coefficient measures the linear relationship between two random variables and is computed as the covariance of the variables divided by the product of their standard deviations. This normalization makes it independent of the units of measurement and ensures that it lies between -1 and 1.

Affine Transformation

An affine transformation is a linear transformation followed by a translation, typically represented as a + bX for a random variable X. These transformations change the location and scale of the distribution but do not affect the underlying linear relationship between variables, as the scaling factors cancel out in the correlation computation.

Invariance under Location-Scale Transformations

This concept refers to the property that the correlation coefficient remains unchanged (up to sign) when the random variables are transformed by adding constants (location shifts) and multiplying by nonzero constants (scaling). This invariance is essential in understanding why the absolute value of the correlation between the transformed variables remains the same as the original.

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