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6.2 Linear combinations of random variables

Syllabus
9709–2028–2029
Topic
6.2
Level
A2

Learning objectives

6.2.1Normal distribution• use, when solving problems, the results that - E(aX + b) = aE(X) + b and Var(aX + b) = a2 Var(X) - E(aX + bY) = aE(X) + bE(Y) - Var(aX + bY) = a2 Var(X) + b2 Var(Y) for independent X and Y - if X has a normal distribution then so does aX + b - if X and Y have independent normal distributions then aX + bY has a normal distribution - if X and Y have independent Poisson distributions then X + Y has a Poisson distribution. Proofs of these results are not required.

Means follow signs; independent variances use squared coefficients

Quantity Result Condition
E(aX+b)E(aX+b) aE(X)+baE(X)+b always
Var⁡(aX+b)\operatorname{Var}(aX+b) a2Var⁡(X)a^2\operatorname{Var}(X) always
E(aX+bY)E(aX+bY) aE(X)+bE(Y)aE(X)+bE(Y) always
Var⁡(aX+bY)\operatorname{Var}(aX+bY) a2Var⁡(X)+b2Var⁡(Y)a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y) X,YX,Y independent
Starting distributions Resulting distribution
XX normal aX+baX+b is normal
independent normal X,YX,Y aX+bYaX+bY is normal
independent X∼Po(λ),Y∼Po(μ)X\sim Po(\lambda),Y\sim Po(\mu) X+Y∼Po(λ+μ)X+Y\sim Po(\lambda+\mu)

If independent X∼N(10,4)X\sim N(10,4) and Y∼N(6,9)Y\sim N(6,9), then X−Y∼N(10−6,  4+9)=N(4,13).X-Y\sim N(10-6,\;4+9)=N(4,13). The mean subtracts, but the variances add because the coefficient of YY is squared: (−1)2=1(-1)^2=1.

If independent counts A∼Po(2)A\sim Po(2) and B∼Po(3.5)B\sim Po(3.5) refer to compatible exposure, then A+B∼Po(5.5)A+B\sim Po(5.5). A difference of Poisson variables is not Poisson by this result.

Define the required total, difference or cost as a linear combination; calculate its mean; calculate its variance with squared coefficients and an independence check; identify the resulting distribution; only then standardise or evaluate its probability.

Constants shift expectation but contribute no variance. Never subtract variances for a difference of independent variables, and do not use the independent-variance or Poisson-sum result when dependence is present.