The two rules of combining
A linear combination is an expression such as . Its mean follows ordinary algebra. Its variance follows a different rule because spread is squared.
Keep signs for means; square them for variances
One expression, two rules
aX + bY
independent X and Y
↓ ask for mean or variance
MEAN — keep signs
aE(X) + bE(Y)
VARIANCE — square coefficients
a²Var(X) + b²Var(Y)
A minus sign never makes an independent variance subtract.
Symbols
- = multiplier of X (none)
- = fixed constant (same as aX)
Symbols
- = squared multiplier (none)
- = does not change the variance (none)
For independent X and Y,
Scale one value, then add independent values
Worked example
Current-paper video-game cost
Playing time X has mean 15 minutes and variance 9. The cost is $0.40 per minute. Each day, 35 people play independently.
For the total cost T from 35 independent players,
(9709/61/M/J/25 Q4)
Examiner note
A multiple is not the same as repeated copies
- uses the same observation twice, so .
- uses two independent observations, so .
- Their means are equal, but their variances are not.
X has mean 10 and standard deviation 3. Independently, . Find the standard deviations of and .
Show worked answer
Common mistake