Algebra of expectation and variance
Linearity always; variance only with independence.
Linearity of expectation (always, independent or not):
Variance of a linear transformation (single variable, always):
The shifts but does not stretch.
Variance of a sum — INDEPENDENCE REQUIRED:
Note the squared coefficients and that subtraction still adds variances: .
Worked example. has . has . independent. Find and .
.
. (constant contributes 0)
- Mean: linearity ALWAYS holds.
- Variance: squared coefficients AND independence.
- (PLUS!).