9709 · 6.2
Linear combinations of random variables
This topic gives us the rules for combining the means and variances of different random variables. It's like creating a new recipe by mixing ingredients, and we need to predict the properties of the final dish.
Need to know
What you need to know
- **Independence is essential:** This variance formula is only valid if X and Y are independent.
- **Constants are squared:** Remember to square the coefficients 'a' and 'b'. This is because variance is a measure of spread in squared units.
- **Variances always add:** For a difference, $\text{Var}(X - Y) = \text{Var}(X) + (-1)^2\text{Var}(Y) = \text{Var}(X) + \text{Var}(Y)$. The uncertainty in a difference is the sum of the individual uncertainties. This is a very common exam trap.
Explanation
Mixing & Matching Variables
- E(aX + bY) = aE(X) + bE(Y) always (X, Y independent or not).
- Var(aX + bY) = a²Var(X) + b²Var(Y) if X, Y independent.
- Sum of independent normals is normal.
- Standardise: Z = (X − μ)/σ for normal problems.