When a question asks you to state an assumption for combining two normal measurements, identify the separate random quantities. Normal marginal distributions alone do not justify adding their variances.
What independence allows
For independent \(X\sim N(\mu_X,\sigma_X^2)\) and \(Y\sim N(\mu_Y,\sigma_Y^2)\), \[aX+bY\sim N(a\mu_X+b\mu_Y,\;a^2\sigma_X^2+b^2\sigma_Y^2).\] A negative coefficient changes the mean calculation but its square makes a positive contribution to variance.
| Quantity | Mean | Variance |
|---|---|---|
| \(X+Y\) | \(\mu_X+\mu_Y\) | \(\sigma_X^2+\sigma_Y^2\) |
| \(X-Y\) | \(\mu_X-\mu_Y\) | \(\sigma_X^2+\sigma_Y^2\) |
| \(2X-3Y\) | \(2\mu_X-3\mu_Y\) | \(4\sigma_X^2+9\sigma_Y^2\) |
Bridge the idea
Two components are manufactured together and exposed to the same temperature. Is independence automatic? No: a shared influence may make their measurements dependent. If independence is supplied, use it; if the question requests a required assumption, state it for those particular measurements.
Exam wording: “The measurements of the two components are assumed to be independent.” Explain that this permits the stated variance calculation. Without this condition, the independent-sum rule is not justified.
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