Linear rules for mean and variance
For constants a and b: E(aX + b) = a E(X) + b, and Var(aX + b) = a squared x Var(X). Adding b shifts the centre only. For independent X and Y, Var(X + Y) = Var(X) + Var(Y).
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Analogy
Converting temperatures from Celsius to Fahrenheit stretches and shifts the whole distribution in a predictable way.
Example
E(X) = 0.7, Var(X) = 0.61E(Y) = 3(0.7) + 2 = 4.1Var(Y) = 9 x 0.61 = 5.49The +2 does not change spread
In practice
Handy for changing units or finding the spread of a total of independent amounts.
Practice
Flashcards
- What is Var(aX + b)?a squared times Var(X). The b has no effect.
Questions people ask
Why does adding a constant not change the variance?
Adding the same number to every value moves the whole distribution along the line, but the gaps between values stay the same. Variance measures those gaps relative to the mean, which also moves. So the shift cancels out and spread is unchanged.
Do variances add when you add random variables?
Only when the variables are independent. Then Var(X + Y) equals Var(X) plus Var(Y). Note that standard deviations do not add, but variances do. If the variables are linked, a covariance term must be included as well, which can make the total variance larger or smaller than the plain sum.