Introduction

In the realm of multivariate random variables, linear transformations play a pivotal role in understanding the relationships between variables. This chapter delves into the effects of applying linear transformations on the covariance and correlation between two random variables. By comprehending these effects, you’ll gain insights into how changes in scale and orientation impact the statistical interactions between variables.

Effects of Linear Transformations on Covariance

When two random variables are subjected to linear transformations, the covariance between them also undergoes changes. For two random variables X and Y, and linear transformations $aX + b$ and $cY + d$, the covariance of the transformed variables can be expressed as:

$$Cov(aX + b, cY + d) = ac \times Cov(X, Y)$$

This relationship underscores that the covariance is scaled by the product of the coefficients in the linear transformations. When the coefficients are positive, the covariance maintains its direction. Conversely, if the coefficients are negative, the covariance changes its sign, indicating an inverse relationship.

Effects of Linear Transformations on Correlation

The correlation coefficient between two random variables measures the strength and direction of their linear relationship. When applying linear transformations, the correlation coefficient remains unchanged. In mathematical terms:

$$Corr(aX + b, cY + d) = Corr(X, Y)$$

This property holds true because correlation is inherently invariant under linear transformations. It’s important to note that correlation captures only linear relationships; non-linear transformations can alter the underlying relationship and should be considered with caution.

Example: Consider two random variables X and Y with covariance $Cov(X, Y) = 0.5$. Applying the linear transformation $aX + b$, where $a = 2$ and $b = 3$, results in a covariance of $Cov(2X + 3, Y) = 1.0$. The correlation between $X$ and $Y$ remains unaffected, emphasizing the preservation of correlation under linear transformations.