Ch 2 · 6.2 Linear Combinations of Random Variables · A Level Mathematics 9709
Topic 2 of 5 in Probability & Statistics 2 — A Level Mathematics 9709 — 73 lessons.
Introduction
A lift's safety limit depends on the total mass of the people in it; a jar of jam "weighs" the jar plus the jam; a race result depends on the difference between two runners' times. Each is a combination of random variables. This lesson gives the rules for the mean and variance of such combinations, and tells you when the result is still normal or Poisson. Proofs are not required. (Syllabus section 6.2, Paper 6.)
Key ideas — A linear function of one variable
E(aX + b) = aE(X) + b Var(aX + b) = a^2 Var(X)
🧑🏫 Teacher's explanation
Imagine you are tracking the heights of students in a classroom. If every student suddenly grows exactly 5cm taller, the average height (the mean) shifts up by 5cm, but the difference between the tallest and shortest student (the spread or variance) stays exactly the same. This is why E(X + b) = E(X) + b and Var(X + b) = Var(X).
📖 Definitions
🔍 Worked example — Easy
A random variable X has E(X) = 10 and Var(X) = 4. Find E(3X + 5) and Var(3X + 5).
🔍 Worked example — Standard
The weight of a bag of flour X follows a normal distribution N(500, 25). A factory machine adds a constant 10g to every bag. Find the mean and variance of the new weight Y = X + 10.
🔍 Worked example — Exam-style
A random variable X has E(X) = 5 and Var(X) = 2. A new variable is defined as Y = 4 - 2X. Find E(Y) and Var(Y).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Combining two variables
E(aX + bY) = aE(X) + bE(Y)
🧑🏫 Teacher's explanation
Imagine you are tracking the weight of two different bags of flour, X and Y. If you want to find the expected total weight of a combined bag, you simply add the expected weights of each individual bag. This is because the average outcome of a sum is always the sum of the averages. This works regardless of whether the bags are related or independent.
📖 Definitions
🔍 Worked example — Easy
A random variable X has E(X) = 10 and Var(X) = 4. Find E(3X + 5) and Var(3X + 5).
🔍 Worked example — Standard
Two independent random variables X and Y have E(X) = 5, Var(X) = 2 and E(Y) = 8, Var(Y) = 3. Find E(2X - 3Y) and Var(2X - 3Y).
🔍 Worked example — Exam-style
Let X sim N(10, 4) and Y sim N(5, 9) be independent normal random variables. Find the distribution of W = 2X + 3Y.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 3X is not the same as X₁ + X₂ + X₃
Read the context: "the total mass of 3 randomly chosen apples" is a sum of three separate variables; "the cost of an item at 3 times its mass" is 3X.
🧑🏫 Teacher's explanation
Imagine you have a single apple that might weigh 100g or 110g. If you triple its weight by some magic process, you are just scaling that one apple's uncertainty. But if you pick three different apples and weigh them together, you are combining three different sources of uncertainty.
📖 Definitions
🔍 Worked example — Easy
A random variable X has E(X) = 10 and Var(X) = 4. Calculate E(3X) and Var(3X).
🔍 Worked example — Standard
Two independent random variables X and Y have E(X) = 5, Var(X) = 2 and E(Y) = 8, Var(Y) = 5. Find E(2X + 3Y) and Var(2X + 3Y).
🔍 Worked example — Exam-style
Three independent random variables X_1, X_2, X_3 are identically distributed with E(X) = μ and Var(X) = σ^2. Find the mean and variance of the sum S = X_1 + X_2 + X_3.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Find the mean and variance of the random variable W = 2X + 3Y. (b) State the distribution of W. [4 marks]
Key ideas — Which distributions are preserved?
🧑🏫 Teacher's explanation
Imagine you are tracking the total weight of a shipment of boxes. If each box's weight is a random variable X, and you want to find the total weight of a boxes plus a fixed weight of b grams, you are performing a linear combination. The 'average' weight scales predictably, but the 'uncertainty' (variance) behaves differently because scaling a value by a spreads out the possible outcomes by a^2.
📖 Definitions
🔍 Worked example — Easy
A random variable X has E(X) = 10 and Var(X) = 4. Find E(3X + 5) and Var(3X + 5).
🔍 Worked example — Standard
Two independent random variables X and Y are normally distributed such that X sim N(μ_1, σ_1^2) and Y sim N(μ_2, σ_2^2). Find the distribution of W = 4X - 2Y.
🔍 Worked example — Exam-style
Two independent Poisson random variables X and Y have parameters λ = 3 and μ = 5 respectively. Find the distribution of Z = X + Y and state its parameter.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Comparing two variables
"Find P(X > Y)" becomes "find P(X - Y > 0)": form D = X - Y, find its mean and variance, then use the normal distribution.
🧑🏫 Teacher's explanation
Imagine you are measuring the weight of two different bags of flour. Each bag has a slight variation due to the machine filling them. If you want to know the weight of both bags combined, or the difference between them, you aren't just adding or subtracting the average weights; you are also combining the uncertainty (the variance) of both bags.
📖 Definitions
🔍 Worked example — Easy
A random variable X has mean E(X) = 10 and variance Var(X) = 4. Find E(3X + 5) and Var(3X + 5).
🔍 Worked example — Standard
Two independent random variables X and Y are normally distributed with X sim N(20, 4) and Y sim N(15, 9). Find the distribution of W = X - Y.
🔍 Worked example — Exam-style
Let X sim N(10, 4) and Y sim N(12, 9) be independent random variables. Find P(X > Y).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: which is bigger?
X sim N(50, 9) and Y sim N(40, 16) are independent. Find P(X > Y).
🔍 Worked example — Easy
X sim N(50, 9) and Y sim N(40, 16) are independent. Find P(X > Y).
🔍 Worked example — Standard
If X sim N(10, 4) and Y sim N(15, 9) are independent, find the mean and variance of W = 3X - 2Y.
🔍 Worked example — Exam-style
Two independent variables X and Y follow Poisson distributions such that X sim Po(λ_1) and Y sim Po(λ_2). Find the distribution of Z = X + Y.
⚠️ Common errors
Worked examples — Example 2: a total of several items
Apples have masses N(150, 10^2) grams. Find the probability that 4 randomly chosen apples have total mass more than 620 g.
🔍 Worked example — Easy
A single component has a mass X sim N(150, 10^2) grams. Find the mean and variance of the total mass of 4 such components.
🔍 Worked example — Standard
Apples have masses N(150, 10^2) grams. Find the probability that 4 randomly chosen apples have total mass more than 620 g.
🔍 Worked example — Exam-style
Two independent random variables X and Y follow N(10, 4) and N(20, 9) respectively. Find the probability that 2X + 3Y < 85.
⚠️ Common errors
Worked examples — Example 3: 2X versus X₁ + X₂
A lap time is X sim N(12, 1.5^2) minutes. Compare P(2X > 26) and P(X_1 + X_2 > 26).
🔍 Worked example — Easy
A single lap time is X sim N(μ, σ^2). Find the mean and variance of 3X + 5.
🔍 Worked example — Standard
Let X sim N(12, 1.5^2) and Y sim N(12, 1.5^2) be independent. Find the distribution of W = X + Y.
🔍 Worked example — Exam-style
A lap time is X sim N(12, 1.5^2) minutes. Compare P(2X > 26) and P(X_1 + X_2 > 26) where X_1, X_2 are independent laps with the same distribution.
⚠️ Common errors
Worked examples — Example 4: sum of Poisson variables
Calls to line A follow Po(1.2) per minute and to line B Po(0.8) per minute, independently. In 2 minutes the total is Po(2 × 2) = Po(4): P(at least 3) = 1 - e^-4(1 + 4 + 8) = 0.762.
🔍 Worked example — Easy
Two independent Poisson random variables, X sim Po(λ=3) and Y \sim Po(λ=5), represent the number of emails received by two different departments in an hour. Find the distribution of the total number of emails W = X + Y.
🔍 Worked example — Standard
Calls to line A follow Po(1.2) per minute and to line B follow Po(0.8) per minute, independently. Find the probability that in 2 minutes, the total number of calls to both lines is at least 3.
🔍 Worked example — Exam-style
Let X sim Po(2) and Y sim Po(3) be independent. Find the variance of the random variable Z = 3X + 2Y.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 17; 32. (2) D sim N(5, 13): P (Z < -2/√(13) ) = P(Z < -0.555) = 0.290. (3) T sim N(5.05, 0.002): P (Z > 0.05/0.0447 ) = P(Z > 1.118) = 0.132. (4) E = 40 - 30 = 10; Var = 4 × 9 + 9 × 4 = 72. (5) Po(4): e^-44^3/6 = 0.195.