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Linear Combinations of Random Variables

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  Cambridge International AS & A Level Mathematics 9709 — Probability & Statistics 2

Linear Combinations of Random Variables

The big idea: Means always add (with their coefficients); variances add too — but coefficients are squared, and variances never subtract.

Summary — What This Chapter Covers
  • E(aX + b) = aE(X) + b; Var(aX + b) = a2Var(X).
  • E(aX + bY) = aE(X) + bE(Y).
  • Var(aX + bY) = a2Var(X) + b2Var(Y) for independent X and Y — even when b is negative.
  • Linear combinations of independent normal variables are normal.
  • The sum of independent Poisson variables is Poisson, with the parameters added.
1. Expectation and Variance of Combinations
Rules

Square the coefficients

Adding a constant shifts the mean but not the spread; multiplying by a stretches both, so the variance is multiplied by a2. For independent variables variances add, whether you add or subtract the variables: Var(X − Y) = Var(X) + Var(Y).

Be careful to distinguish 2X (one observation doubled, variance 4Var(X)) from X1 + X2 (two independent observations, variance 2Var(X)).

Results (no proof needed) E(aX + b) = aE(X) + b    Var(aX + b) = a2Var(X)    Var(aX + bY) = a2Var(X) + b2Var(Y) (independent)
Worked Example

X and Y are independent with E(X) = 10, Var(X) = 4, E(Y) = 6, Var(Y) = 1. Find E(2X − 3Y) and Var(2X − 3Y).

Practice Question 1

E(X) = 5 and Var(X) = 4. Find E(3X − 2), Var(3X − 2) and Var(X1 + X2 + X3) for three independent observations.

2. Combinations of Normal and Poisson Variables
Distributions

What shape the result has

If X is normal, so is aX + b. If X and Y are independent normal variables, aX + bY is normal — so find its mean and variance with the rules above and then standardise. A question such as "the probability that X exceeds Y" becomes P(X − Y > 0).

If X ~ Po(λ) and Y ~ Po(μ) independently, X + Y ~ Po(λ + μ). This is only for sums: 2X or X − Y is not Poisson.

Shapes X, Y independent normal ⇒ aX + bY normal    X ~ Po(λ), Y ~ Po(μ) independent ⇒ X + Y ~ Po(λ + μ)
Worked Example 1

The masses of apples are N(150, 102) grams. Find the probability that the total mass of 6 randomly chosen apples exceeds 950 g.

Worked Example 2
Practice Question 2
Memorise
Means

Combine exactly like the variables.

Variances
Poisson
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More Probability & Statistics 2 topics

The Poisson Distribution
Poisson probabilities, mean and variance
Expectation and variance of combinations
Probability density functions
Samples and the sampling distribution of the mean
Tests for binomial and Poisson parameters
Approximations using the Poisson and normal distributions
Combinations of normal and Poisson variables
Mean, variance, median and percentiles
Unbiased estimates