S3.1.1 - Linear combinations of Normal variables

Distribution of linear combinations If X ∼ N(μ, σ 2) and Y∼ N(μ, σ 2) independently, then x x y y of independent Normal random aX± bY ∼ N(aμ ± bμ, a2σ 2 + b2σ 2). variables. x y x y No proofs required.

Use linear combinations of normal variables to connect the rule to the data and decision in the question.

This matters because linear combinations of normal variables determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply linear combinations of normal variables to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Linear combinations of Normal variables is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.