Expectation and Variance of the Sum of Two Random Variables

Let X and Y be two random variables. How do you calculate $E[X + Y]$ and $\operatorname{Var}(X + Y)$? The report asks only for the two calculations. This version also asks you to derive each result and to state what each one assumes about X and Y. ```hint Start from the definition Write the variance of $Z = X + Y$ as the expected squared deviation of Z from its mean, and expand the square. ``` ```hint Where dependence enters As you derive each result, note which step, if any, needs X and Y to be independent. ``` ### Clarifying Questions - Are X and Y independent, or should the formulas hold for any joint distribution? - May we assume both variables have finite means and variances? - Is a derivation for discrete variables enough, or should it also cover continuous ones? ### What a Strong Answer Covers - Linearity of expectation, derived from the joint distribution, with the point that it needs no independence - The variance of a sum including the covariance term, derived by expanding the square - The definition of covariance, when the cross term vanishes, and the difference between uncorrelated and independent variables - The moment conditions each result needs, and a sanity check on extreme cases such as $Y = X$ or $Y = -X$ ### Follow-up Questions - What are $\operatorname{Var}(X - Y)$ and $\operatorname{Var}(aX + bY + c)$ for constants a, b and c? - Can $\operatorname{Var}(X + Y)$ be smaller than $\operatorname{Var}(X)$? Give an example. - In an A/B test, what is the variance of the difference between the treatment and control sample means, and why is there no covariance term? - Give two random variables that are uncorrelated but not independent.