Linear transformations, combining variables, and the unbiased estimates that divide by n − 1.
Here is a distribution. a stretches it, b slides it. Watch the mean and the variance underneath: only one of the two controls touches the variance.
Slide b as far as you like. The variance never moves.
E(aX + b) = aE(X) + b Var(aX + b) = a2Var(X)
No b in the variance, and the a is squared. Adding a constant moves everything by the same amount, so no value gets further from any other and the spread is untouched. Multiplying by a stretches every gap by a, and variance is built from squared gaps.
| Expected value | Variance (independent) | |
|---|---|---|
| X + Y | E(X) + E(Y) | Var(X) + Var(Y) |
| X − Y | E(X) − E(Y) | Var(X) + Var(Y) |
| aX + bY | aE(X) + bE(Y) | a²Var(X) + b²Var(Y) |
Variances add even when you subtract. Taking a difference makes the result more variable, not less, because both sources of uncertainty are still there. Writing Var(X − Y) = Var(X) − Var(Y) can give a negative variance, which is impossible, and that is the quick way to spot the error.
A sample is used to estimate things about a population it came from.
Dividing by the smaller number gives the larger answer, which is the point: a sample sits closer to its own mean than to the population mean, so the raw version under-estimates and needs correcting upwards.
1. E(X) = 4 and Var(X) = 2. Find Var(3X + 5).
2. Var(X) = 2 and Var(Y) = 5, independent. Find Var(X − Y).
3. For the sample 4, 8, 6, 10, 12, give the unbiased estimate of the population variance.
Squaring the a and dropping the b. One line, and it is the single most tested thing in this sub-topic.
Adding variances for a difference. If an answer comes out negative, you have found your own mistake.
Choosing n − 1 when the question says estimate the population. Calculators give both; knowing which key you pressed is the mark.
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