Linear transformations of random variables (HL)

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

Formulas

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 $Y_5$ in group 5 mathematics. Your total subject grades across six subjects is not $6Y_5$, rather each subject grade may be different. Your total subject grades would instead be $Y_1 + Y_2 + Y_3 + Y_4 + Y_5 + Y_6$. This is linear combination, which has its own formulas for expected value and variance beyond the scope of this course.