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A Level Maths 9709 · Paper 6 Probability & Statistics 2 · 6.2 · ~4 min read

Linear combinations of random variables

Expectation and variance of aX + b and of sums and differences of independent random variables, the difference between 3X and X₁ + X₂ + X₃, and the facts that sums of independent normal (or Poisson) variables are normal (or Poisson).

E(aX + b)Var(aX + b)E(X ± Y)Var(X ± Y)3X vs X₁ + X₂ + X₃Sums of normalsSums of PoissonsComparing two quantities
RSby Riyaz Sir
mentorriyaz.in
Paper tag = where it is examinedFORMULA LIST (MF19) given in the examM1A1B1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Expectations always add: E(aX + bY + c) = aE(X) + bE(Y) + c.
  • Variances add for independent variables, even for a difference: Var(X − Y) = Var(X) + Var(Y). Constants multiply variance by their square.
  • 3X is not X₁ + X₂ + X₃. Var(3X) = 9Var(X) (one value tripled); Var(X₁ + X₂ + X₃) = 3Var(X) (three independent values).
  • Normal + normal is normal; Poisson + Poisson is Poisson (when independent). Sums of Poissons add their λ.

By the end you can

  • Use E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
  • Use E and Var for sums and differences of independent variables.
  • Use the facts that linear combinations of independent normal variables are normal, and sums of independent Poisson variables are Poisson.
  • Solve problems such as P(X > Y) or P(total > k).

1Expectation and variance

The rules

E(aX + b) = aE(X) + b ; Var(aX + b) = a²Var(X) FORMULA LIST (MF19)
the constant b shifts but doesn't spread
E(aX ± bY) = aE(X) ± bE(Y) ; Var(aX ± bY) = a²Var(X) + b²Var(Y) FORMULA LIST (MF19)
X and Y independent for the variance rule
Worked example

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

  1. E = 12 − 2 + 1 = 11 B1
  2. Var = 9 × 2 + 4 × 3 = 30 M1 A1
Common mistakeVar(X − Y) = Var(X) − Var(Y). Variances never subtract. This is still the commonest error in this topic.

2Normal combinations

P(X > Y) and totals

Worked example

X ~ N(20, 9) and Y ~ N(15, 4) are independent. Find P(X > Y).

  1. X − Y ~ N(5, 13) M1 A1
  2. P(X − Y > 0) = Φ(5/√13) = Φ(1.387) M1
  3. = 0.917 A1
Worked example

Apple masses are N(150, 10²) grams. Find the probability that a bag of 6 random apples has total mass over 930 g.

  1. T = X₁ + … + X₆ ~ N(900, 6 × 100 = 600) M1 A1
  2. z = 30/√600 = 1.225 M1
  3. 1 − 0.8897 = 0.110 A1
  • Compare: six times one apple's mass, 6X, would have variance 36 × 100 = 3600, which is the wrong model for six different apples.
Worked example

A ~ N(50, 16) and B ~ N(24, 9) are independent. Find P(A > 2B).

  1. A − 2B ~ N(50 − 48, 16 + 4 × 9) = N(2, 52) M1 A1
  2. Φ(2/√52) = Φ(0.277) = 0.609 A1

3Poisson sums

λ values add

Worked example

Accidents at junction A follow Po(1.2) per week and at B, Po(0.8) per week, independently. Find the probability of exactly 3 accidents in total in a week.

  1. total ~ Po(2) B1
  2. e−2 × 2³/3! M1
  3. = 0.180 A1
Common mistakeAdding Poissons that aren't independent, or forming 2X for "two weeks" correctly as Po(2λ) but then 2X for a difference. A difference of Poissons is not Poisson.
Linking questionNormal probabilities are in The normal distribution; the Poisson is in The Poisson distribution; and the distribution of a sample mean is in Sampling and estimation.

Test yourself — cover the answers first

Quick check

  1. Var(X) = 5. Find Var(2X + 7).
  2. Var(X) = 5, Var(Y) = 3, independent. Find Var(X − Y).
  3. X ~ Po(2), Y ~ Po(3), independent. What is X + Y?
  4. E(X) = 10. Find E(4 − 3X).
  5. Var(X) = 4. Find Var(X₁ + X₂) and Var(2X).

Exam-style

  1. Weights of men ~ N(80, 100) and women ~ N(65, 64). A lift holds 4 men and 3 women. Find P(total > 550).
  2. X ~ N(30, 4). Find P(X₁ + X₂ > 3X₃ − 25).
  3. E(X) = 3, Var(X) = 1. Find E(X²).
Show answers
  1. 20
  2. 8
  3. Po(5)
  4. −26
  5. 8 and 16
  6. N(515, 400 + 192 = 592): z = 35/24.33 = 1.438 ⇒ 0.075
  7. X₁ + X₂ − 3X₃ ~ N(−30, 4 + 4 + 36 = 44): P(> −25) = 1 − Φ(5/6.633) = 0.225
  8. Var + mean² = 1 + 9 = 10
Practise this chapter
A Level practice papers
Next chapterContinuous random variables →
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by Cambridge University Press & Assessment. Syllabus codes 9709 and 9702 refer to the Cambridge International AS & A Level syllabuses.