Expectation and Variance of a Sum of Two Random Variables, Including Covariance

In a hiring-manager interview for a data scientist role, the statistics question was: how do you compute the expectation and the variance of the sum of two random variables? Answer for two random variables $X$ and $Y$ in general, then say what changes when they are independent. ```hint Expectation first Write the expectation of the sum directly from the definition, and notice which assumption you never needed. ``` ```hint Expand the square Write the variance of the sum as the expectation of a squared deviation and expand it. One term will not disappear on its own. ``` ### Clarifying Questions - Are $X$ and $Y$ independent, correlated, or of unknown dependence? - Does the interviewer want a derivation, or the result and how it is used? ### What a Strong Answer Covers - The expectation result, and the fact that it needs no independence - The variance result with its covariance term, derived rather than recited - When the covariance term vanishes: independent or merely uncorrelated variables - Generalization to weighted sums and to differences - A data-science use, such as the variance of a difference between two group means ### Follow-up Questions - What is the variance of $X - Y$? - If $X$ and $Y$ each have standard deviation $\sigma$, what range of values can the standard deviation of $X + Y$ take? - Two independent fair dice are rolled. What are the mean and the variance of their total?