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Combinations of Random Variables

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  Edexcel IAL Statistics 3 (WST03)

Combinations of Random Variables

The big idea: Means always add; variances add for independent variables — even when the variables are subtracted.

Summary — What This Chapter Covers
  • E(aX ± bY) = aE(X) ± bE(Y).
  • For independent X, Y: Var(aX ± bY) = a2Var(X) + b2Var(Y).
  • A linear combination of independent normal variables is normal.
  • X₁ + X₂ (two independent observations) has variance 2σ2; 2X has variance 4σ2.
1. Linear Combinations of Normal Variables
Combinations

Means add, variances add

Decide carefully whether the question means two separate observations (X₁ + X₂) or one observation doubled (2X).

Worked Example

X ~ N(50, 9) and Y ~ N(30, 16) are independent. Find P(X − Y > 25).

Memorise
Means

Add or subtract

Variances

Always add

Scaling
2X vs X₁ + X₂
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More Statistics 3 topics

Linear combinations of normal variables
Methods of sampling
Unbiased estimates and standard error
Goodness-of-fit tests
Spearman's rank correlation
Sampling
The central limit theorem
Contingency tables
Testing correlation coefficients
Estimation, Confidence Intervals and Tests