Correlation from the spread of a sum

Question

Random variables $X$ and $Y$ have standard deviations $1$ and $3$ , and $X+Y$ has standard deviation $4$ . What is the correlation between $X$ and $Y$ ?

Short answer

$1$ . Expand $\operatorname{Var}(X+Y)=\operatorname{Var}X+\operatorname{Var}Y+2\rho\,\sigma_X\sigma_Y$ : $16=1+9+6\rho$ , so $\rho=1$ . Equivalently, standard deviations add only under perfect positive correlation.

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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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