Does shifting or rescaling change correlation

Question

Two random variables $X$ and $Y$ have correlation $\rho$ . What is the correlation between $X+5$ and $Y$ ? What is the correlation between $5X$ and $Y$ ?

Short answer

Both equal $\rho$ . Adding a constant changes neither the covariance nor the variance. Multiplying by $5$ scales the covariance by $5$ and the standard deviation by $5$ , which cancel; correlation is unchanged by any positive affine transformation (a negative multiplier would flip the sign).

Adapted from Crack, Heard on the Street, Q4.5, p.38; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Covariance and Correlation
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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