Combined P&L of two independent desks

Question

Desk A's daily profit (in thousands of dollars) is $A \sim N(\text{mean } 3,\ \text{variance } 4)$ . Desk B's is $B \sim N(\text{mean } 1,\ \text{variance } 9)$ , independent of A. Let $S = A + B$ be the combined daily P&L.

What are the mean and variance of $S$ , and what is its distribution?

Accepted Answer

Adding independent normals keeps things normal, and the two moments combine by simple rules: - Means always add: \mathbb{E}[S] = 3 + 1 = 4. - For independent variables, variances add (never the standard deviations): \operatorname{Var}(S) = 4 + 9 = 13. So S \sim N(4,\ 13), with standard deviation \sqrt{13} \approx 3.61. Note you do not add the standard deviations (2 + 3 = 5 would be wrong); you add the variances and then take the square root. Takeaway. A sum of independent normals is normal; means add to 4 and variances add to 13, not the standard deviations.

Combined P&L of two independent desks

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