Correlation when standard deviations simply add

Question

Suppose $\sigma_X=4$ , $\sigma_Y=7$ and the standard deviation of $X+Y$ is $11$ . Find the correlation of $X$ and $Y$ .

Short answer

$1$ . From $121=16+49+2\operatorname{Cov}(X,Y)$ we get $\operatorname{Cov}=28=4\cdot7$ , so $\rho=1$ . Standard deviations add only under perfect positive correlation.

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Lessons behind this question

Covariance and Correlation
Populations, Samples, and Sampling Distributions

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