Impact of Linear Transformations on Correlation

This question focuses on how simple linear transformations affect the correlation between two random variables. The setup involves one variable being shifted by a constant and another being scaled, and asks how these operations change the original measure of linear association. It is a basic but foundational probability and statistics question, often seen in the early stages of quant interviews or technical screens, to ensure that candidates understand what correlation actually measures rather than just being able to quote the formula.

Answering it cleanly relies on understanding the definitions of covariance, variance, and standard deviation, and how they behave under shifts and rescaling. The key techniques include manipulating the algebraic form of correlation, tracking how constants propagate through covariance and variance, and recognizing which operations change only location and which change scale. Interviewers look for conceptual clarity over memorized rules: can the candidate quickly reason from first principles, avoid common misconceptions about "unit changes," and articulate why correlation behaves the way it does under these transformations?

What it tests

Correlation measures the strength and direction of a linear relationship between two random variables, and it is invariant under affine transformations of the form $aX + b$ (where $a \neq 0$) applied to either variable. This invariance arises because correlation standardizes both variables: it divides the covariance by the product of their standard deviations, so any scaling factor $a$ applied to $X$ affects both the covariance and the standard deviation of $X$ equally, and any constant shift $b$ does not affect either. Thus, correlation is unaffected by changes in location (shifts) and only cares about relative changes in scale (which cancel out). This property makes correlation a measure of pure linear association, independent of units or origins of the variables. The underlying reason is that both covariance and standard deviation are linear in scale and insensitive to shifts, so their ratio (the correlation) remains unchanged.