Linear transformations of random variables (HL)

Suppose your teacher gave you mark XX on an internal assessment, then using linear regression, IB moderates your mark to aX+baX + b. This is a linear transformation of a random variable.

Formulas

E(aX+b)=aE(X)+b\text{E}(aX + b) = a\text{E}(X) + b
Var(aX+b)=a2Var(X)\text{Var}(aX + b) = a^2\text{Var}(X)

The scaling and shifting effects on a random variable do appear on the expected value. Whereas standard deviations are scaled but variance is scaled twice. Shifts do not affect either variance or standard deviation, as they quantify how the values are dispersed around the expected value.

What linear transformation is not

Suppose you earn a grade Y5Y_5 in group 5 mathematics. Your total subject grades across six subjects is not 6Y56Y_5, rather each subject grade may be different. Your total subject grades would instead be Y1+Y2+Y3+Y4+Y5+Y6Y_1 + Y_2 + Y_3 + Y_4 + Y_5 + Y_6. This is linear combination, which has its own formulas for expected value and variance beyond the scope of this course.