Linear Transformations of Random Variables (AI HL)
If you already know the mean and variance of a random variable \(X\), you don't need to rebuild its whole distribution to find the mean and variance of \(Y = aX + b\) - two short rules do the job directly. This page covers both rules, how they extend to a sum or difference of two independent variables, and the sign errors that catch people out - with worked examples and the mistakes that lose the most marks. It's part of the broader Probability & Distributions topic.
22 questions on this sub-topic.
The two rules
This builds on IB syllabus reference SL4.7 - the concept of discrete random variables, their probability distributions, and expected value \(E(X)\). Neither rule below is in the formula booklet, but both follow directly from the definitions of \(E(X)\) and \(\text{Var}(X) = \Sigma xP(X=x)\).
Expected value of \(aX+b\)
\(E(aX+b) = aE(X) + b\)
Scale and shift the mean exactly as you scale and shift \(X\) itself.
Variance of \(aX+b\)
\(\text{Var}(aX+b) = a^2\text{Var}(X)\)
Adding a constant \(b\) doesn't spread the data out any further, so it drops out entirely; \(a\) gets squared.
Need the full syllabus wording and formula-booklet reference table? See Probability & Distributions. For calculator routines, see the parent topic's GDC guidance.
Worked examples
A discrete random variable \(X\) has \(E(X) = 5\) and \(\text{Var}(X) = 4\). Let \(Y = 3X - 2\).
(a) Find \(E(Y)\).
(b) Find \(\text{Var}(Y)\).
Worked solution
(a) \(E(Y) = E(3X-2) = 3E(X) - 2\) M1
\(= 3(5) - 2 = 13\) A1
(b) \(\text{Var}(Y) = \text{Var}(3X-2) = 3^2\,\text{Var}(X)\) M1
\(= 9 \times 4 = 36\) A1
A stall sells items. Revenue \(R\) has \(E(R) = 80\) and \(\text{Var}(R) = 25\). Costs \(C\) (independent) have \(E(C) = 30\) and \(\text{Var}(C) = 9.\) Profit \(P = R - C.\)
(a) Find \(E(P).\)
(b)(i) Find \(\text{Var}(P).\)
(b)(ii) Find the standard deviation of profit.
Worked solution
(a) Subtract the means: \(E(P) = E(R) - E(C) = 80 - 30.\) M1
\(=50.\) A1
(b)(i) Variances add: \(\text{Var}(P) = \text{Var}(R) + \text{Var}(C) = 25 + 9.\) M1
\(=34.\) A1
(b)(ii) \(\text{SD}(P) = \sqrt{34} \approx 5.83.\) A1
Common mistakes
- Forgetting to square \(a\) in the variance rule. \(\text{Var}(aX+b) = a^2\text{Var}(X)\), not \(a\,\text{Var}(X)\) - a negative \(a\) still gives a positive multiplier once it's squared.
- Carrying the constant \(b\) into the variance. A constant shift changes where the data is centred but not how spread out it is, so \(b\) simply disappears from \(\text{Var}(aX+b)\) - writing \(\text{Var}(Y) = a^2\text{Var}(X) + b\) is a very common slip.
- Subtracting variances when combining independent variables. Even for \(P = R - C\), the variances of two independent variables always add: \(\text{Var}(R-C) = \text{Var}(R) + \text{Var}(C)\), never \(\text{Var}(R) - \text{Var}(C)\).
Ready to practise properly?
22 linear-transformation questions, marked instantly like the real exam.
Quick answers
What is E(aX + b) in terms of E(X)?
\(E(aX+b) = aE(X) + b\) - scale and shift the mean the same way you scale and shift \(X\).
What is Var(aX + b) in terms of Var(X)?
\(\text{Var}(aX+b) = a^2\text{Var}(X)\) - the constant \(b\) has no effect on variance, and \(a\) is squared.