Combined height of two independent people

Question

One person's height is X \sim N(	ext{mean } 70,\ 	ext{variance } 16) inches; an unrelated person's is Y \sim N(	ext{mean } 65,\ 	ext{variance } 9) inches, independent of X. Let S = X + Y be their combined height. What are the mean and variance of S, and what is its distribution?

Accepted Answer

A sum of independent normals is again normal. Means add and, for independent pieces, variances add: \mathbb{E}[S] = 70 + 65 = 135, \qquad \operatorname{Var}(S) = 16 + 9 = 25. So S \sim N(135,\ 25), standard deviation \sqrt{25} = 5. Notice the combined standard deviation (5) is not the sum of the individual ones (4 + 3 = 7); you must add variances first. Takeaway. Sum of independent normals is normal with means added (135) and variances added (25), giving sd 5, not 7.

Combined height of two independent people

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