Standard deviation of a negatively correlated sum

Question

Two random variables $X$ and $Y$ have variances $9$ and $16$ , and their correlation coefficient is $-\tfrac38$ . What is the standard deviation of $X+Y$ ?

Short answer

$4$ . $\operatorname{Var}(X+Y)=9+16+2\rho\sigma_X\sigma_Y=25+2\left(-\tfrac38\right)(3)(4)=25-9=16$ , so the standard deviation is $\sqrt{16}=4$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

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

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free