Spread of a total built from three independent noise sources

Question

A recorded signal picks up three independent sources of noise with variances 2, 3, and 7. The total noise is N = N1 + N2 + N3. What is the standard deviation of the total noise?

Accepted Answer

The three noise sources are independent, so their variances add, no matter how different their individual sizes are: \mathrm{Var}(N) = \mathrm{Var}(N1) + \mathrm{Var}(N2) + \mathrm{Var}(N3) = 2 + 3 + 7 = 12. In words: each independent source contributes its own variance to the pile, and the total variance is just the sum. Convert to the standard deviation only at the end: \mathrm{SD}(N) = \sqrt{12} \approx 3.46. It is worth noticing that the largest source (variance 7) dominates: it alone would give \sqrt{7} \approx 2.65, and the two smaller sources push the total only up to about 3.46. Becau…

Spread of a total built from three independent noise sources

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