AS & A Level · Practice questions · Linear combinations of random variables

Add the means, add the variances.

Six original Cambridge-style Paper 6 questions on linear combinations of random variables, with full worked solutions.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Write the combination, then its mean and variance.

Define the combination, e.g. \(D = X_1 + X_2 + X_3 - 2Y\), so the question becomes \(\text{P}(D > 0)\).
Mean: apply the coefficients directly.
Variance: square every coefficient and ADD, for sums and differences alike.
Separate items or one item scaled? \(n\) items: \(n\sigma^{2}\); one item times \(n\): \(n^{2}\sigma^{2}\).
Shape: normal stays normal; Poisson sums stay Poisson.
Standardise with the square root of the variance.

Recall

[2 marks]

\(\text{E}(X) = 6\) and \(\text{Var}(X) = 2.5\). Find \(\text{E}(4X - 3)\) and \(\text{Var}(4X - 3)\).

\(\text{E}(4X - 3) = 4 \times 6 - 3 = 21\). ✓
\(\text{Var}(4X - 3) = 16 \times 2.5 = 40\). ✓

Calculation

[3 marks]

\(X \sim N(30, 16)\) and \(Y \sim N(20, 9)\) are independent. Find \(\text{P}(X + Y > 55)\).

\(X + Y \sim N(50, 25)\). ✓
\(z = \dfrac{55 - 50}{5} = 1\). ✓
\(\text{P} = 1 - \Phi(1) = 1 - 0.8413 = 0.159\) (3 s.f.). ✓

Analysis

[4 marks]

\(X \sim N(10, 4)\). \(X_1\) and \(X_2\) are two independent observations of \(X\).

(a) State the distributions of \(2X\) and of \(X_1 + X_2\).

(b) Find \(\text{P}(X_1 + X_2 > 23)\).

(a) \(2X \sim N(20, 16)\); \(X_1 + X_2 \sim N(20, 8)\). ✓
(b) \(z = \dfrac{23 - 20}{\sqrt{8}} = 1.061\). ✓
\(\text{P} = 1 - 0.8556 = 0.144\) (3 s.f.). (Using \(2X\) wrongly would give \(z = 0.75\) and \(0.227\).) ✓

Calculation

[3 marks]

Machine A breaks down at random at a mean rate of 0.4 per week and, independently, machine B at a mean rate of 0.6 per week. Find the probability of at least 3 breakdowns in total in a 2-week period.

Total in 2 weeks: \(\text{Po}(2 \times 0.4 + 2 \times 0.6) = \text{Po}(2)\). ✓
\(\text{P}(T \ge 3) = 1 - e^{-2}(1 + 2 + 2) = 1 - 5e^{-2}\). ✓
\(= 1 - 0.6767 = 0.323\) (3 s.f.). ✓

Analysis

[4 marks]

Men's masses are \(N(78, 10^{2})\) kg and women's masses are \(N(64, 8^{2})\) kg. A lift carries 4 men and 4 women, chosen at random. Find the probability that their total mass exceeds 600 kg.

\(T = M_1 + \cdots + M_4 + W_1 + \cdots + W_4\). ✓
\(\text{E}(T) = 4 \times 78 + 4 \times 64 = 568\); \(\text{Var}(T) = 4 \times 100 + 4 \times 64 = 656\). ✓
\(z = \dfrac{600 - 568}{\sqrt{656}} = 1.249\). ✓
\(\text{P}(T > 600) = 1 - 0.8942 = 0.106\) (3 s.f.). ✓

Analysis

[6 marks]

A café pours cups of coffee with volumes \(C \sim N(250, 9)\) ml. A jug holds a volume \(J \sim N(2040, 400)\) ml. All volumes are independent.

(a) Find the probability that a cup contains more than 255 ml.

(b) Find the probability that one jug is enough to fill 8 cups.

(a) \(z = \dfrac{255 - 250}{3} = 1.667\), so \(\text{P} = 1 - 0.9522 = 0.0478\) (3 s.f.). ✓
(b) Let \(D = J - (C_1 + \cdots + C_8)\). We need \(\text{P}(D > 0)\). ✓
\(\text{E}(D) = 2040 - 8 \times 250 = 40\); \(\text{Var}(D) = 400 + 8 \times 9 = 472\). ✓
\(z = \dfrac{0 - 40}{\sqrt{472}} = -1.841\), so \(\text{P}(D > 0) = \Phi(1.841) = 0.967\) (3 s.f.). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.