IB Maths AI HL · Unit 4: Statistics and Probability
IB Maths AI HL Linear Combinations of Random Variables & the Central Limit Theorem Questions
Exam-style IB Maths AI HL linear combinations of random variables & the central limit theorem questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 12 questions
- Paper 1: 12
- 12 starter
- 3 worked examples
Practise Linear Combinations of Random Variables & the Central Limit Theorem questions →
AI HL formula booklet
What's examined in AI HL linear combinations of random variables & the central limit theorem
The question bank covers these linear combinations of random variables & the central limit theorem question types (number of questions in brackets):
- Single Random Variable Transform (6)
- Sums of Independent Variables (6)
Key formulas
- Mean & variance of Poisson
- \(E(X) = \lambda,\ \mathrm{Var}(X) = \lambda\)
- Confidence interval for a mean (large n)
- \(\bar{x} \pm z^* \cdot \dfrac{\sigma}{\sqrt{n}}\)
In the same notation as the IB formula booklet. All AI HL formulas →
Linear Combinations of Random Variables & the Central Limit Theorem worked examples
Worked example 1: Transforming a Single Random Variable · easy
A random variable $X$ has an expected value $E(X) = 8$ and a variance $\text{Var}(X) = 5$. A new random variable is defined by the linear transformation $Y = 3X - 4$. Calculate the exact values of $E(Y)$ and $\text{Var}(Y)$.
1. Identify the rule for expected values: $E(aX + b) = aE(X) + b$.
2. Calculate $E(Y)$: $E(3X - 4) = 3(8) - 4 = 24 - 4 =$ $20$.
3. Identify the rule for variance: $\text{Var}(aX + b) = a^2\text{Var}(X)$. (Note that the shift $-4$ does not affect the spread).
4. Calculate $\text{Var}(Y)$: $\text{Var}(3X - 4) = 3^2(5) = 9 \times 5 =$ $45$.
Examiner tip: A common error is subtracting the constant $b$ from the variance. Remember that shifting a dataset left or right does absolutely nothing to its variance or standard deviation.
Worked example 2: Summing Independent Random Variables · medium
The weights of red apples are normally distributed with a mean of $150\text{ g}$ and a variance of $10\text{ g}^2$. The weights of green apples are normally distributed with a mean of $130\text{ g}$ and a variance of $8\text{ g}^2$. If one red apple and one green apple are chosen at random, find the probability that their combined weight is greater than $290\text{ g}$.
1. Define the combined random variable: $W = R + G$.
2. Calculate the expected combined weight: $E(W) = E(R) + E(G) = 150 + 130 = 280\text{ g}$.
3. Calculate the combined variance (since they are independent): $\text{Var}(W) = \text{Var}(R) + \text{Var}(G) = 10 + 8 = 18\text{ g}^2$.
4. Determine the combined distribution: $W \sim N(280, 18)$. The standard deviation is $\sqrt{18} \approx 4.2426\text{ g}$.
5. Set up the probability statement: $P(W > 290)$.
6. Evaluate using the GDC Normal CDF with lower bound $290$, upper bound $10^{99}$, $\mu = 280$, $\sigma = \sqrt{18}$. The result is $0.00921$ (or $0.921\%$).
Examiner tip: When combining variables, you must add their variances, never their standard deviations. If standard deviations are given in the prompt, square them first before adding.
Worked example 3: Applying the Central Limit Theorem · hard
The volume of liquid in a certain brand of energy drink follows a highly skewed distribution with a population mean of $\mu = 250\text{ ml}$ and a population standard deviation of $\sigma = 12\text{ ml}$. A quality control inspector takes a random sample of $36$ cans. Find the probability that the mean volume of this sample, $\bar{X}$, is less than $245\text{ ml}$.
1. Recognize that because the sample size is large ($n = 36 > 30$), the Central Limit Theorem states that the distribution of the sample mean $\bar{X}$ is approximately Normal, regardless of the population's underlying skew.
2. Define the distribution of the sample mean: $\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$.
3. Calculate the parameters for $\bar{X}$: Mean $= 250$, Variance $= \frac{12^2}{36} = \frac{144}{36} = 4$. The standard error (standard deviation of the sample mean) is $\sqrt{4} = 2$.
4. Set up the probability statement: $P(\bar{X} < 245)$.
5. Evaluate using Normal CDF on the GDC with lower bound $-10^{99}$, upper bound $245$, $\mu = 250$, and $\sigma = 2$.
6. The probability is $0.006209\dots$, giving exactly $0.00621$.
Examiner tip: The Central Limit Theorem applies to the distribution of the sample mean. The original population is still skewed; it is only the means of the large samples that form a normal bell curve.
Try these IB Maths AI HL linear combinations of random variables & the central limit theorem questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · starter · 4 marks · Paper 1
A random variable \(X\) follows a normal distribution \(X \sim N(50, 4^2)\). A new variable \(Y\) is defined by the linear transformation \(Y = 3X - 10\). Find the exact expected value and variance of \(Y\).
Question 2 · starter · 3 marks · Paper 1
The weights of regular apples \(A\) are normally distributed with variance 15. The weights of large apples \(B\) are normally distributed with variance 25. An order contains 4 independently chosen regular apples and 2 independently chosen large apples. Find the exact variance of the total combined weight of the order.
Question 3 · starter · 4 marks · Paper 1
A random variable \(X\) follows a normal distribution \(X \sim N(30, 5^2)\). A new variable \(Y\) is defined by the linear transformation \(Y = 2X + 7\). Find the exact expected value and variance of \(Y\).
All 12 linear combinations of random variables & the central limit theorem questions with mark schemes →
FAQ
How many IB Maths AI HL linear combinations of random variables & the central limit theorem questions are there?
There are 12 exam-style linear combinations of random variables & the central limit theorem questions in the AI HL question bank (Paper 1: 12), graded 12 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is linear combinations of random variables & the central limit theorem on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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