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 $0.921\%$).

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 $0.00621$ .

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IB Maths AI HL · Unit 4: Statistics and Probability

IB Maths AI HL Linear Combinations of Random Variables & the Central Limit Theorem Questions

Q: How many IB Maths AI HL linear combinations of random variables & the central limit theorem questions are there?

There are 22 exam-style linear combinations of random variables & the central limit theorem questions in the AI HL question bank (Paper 1: 16 · Paper 2: 6), graded 9 medium, 9 easy, 1 starter, 3 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is linear combinations of random variables & the central limit theorem on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 16 · Paper 2: 6. Practise with your GDC — AI papers expect calculator methods throughout.

Q: 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.

Updated October 2026

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, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

22 questions
Paper 1: 16
Paper 2: 6
9 easy
9 medium
3 hard
1 starter
3 worked examples

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Start with one question on linear combinations of random variables. Stuck? Take a hint. No sign-in needed.

E (X) = 3

and

Var (X) = 2

. Find

Var (5 X - 4)

Choose your answer · keys 1–4

11

46

10

50

Find

E (X)

for the distribution in the table.

$x$	1	3	5	7
$P (X = x)$	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$

Choose your answer · keys 1–4

19

1

4

16

Find

E (X)

for the distribution in the table.

$x$	10	20	50
$P (X = x)$	$\frac{7}{10}$	$\frac{1}{5}$	$\frac{1}{10}$

Choose your answer · keys 1–4

400

\frac{80}{3}

1

16

What you need to know

HL AI expects you to plot the residuals, check for randomness, and interpret r² as a measure of model fit. Linear regression and residual analysis overview →

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 (7)
Sums of Independent Variables (7)
Linear Combinations of Normal Variables (3)
Central Limit Theorem (3)
Differences of Independent Variables (2)

Key formulas

Mean & variance of Poisson: $E (X) = λ, Var (X) = λ$
Confidence interval for a mean (large n): $\bar{x} \pm z^{*} \cdot \frac{σ}{\sqrt{n}}$

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 $Var (X) = 5$ . A new random variable is defined by the linear transformation $Y = 3 X - 4$ . Calculate the exact values of $E (Y)$ and $Var (Y)$ .

Solution

4 steps

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 $150 g$ and a variance of $10 g^{2}$ . The weights of green apples are normally distributed with a mean of $130 g$ and a variance of $8 g^{2}$ . If one red apple and one green apple are chosen at random, find the probability that their combined weight is greater than $290 g$ .

Solution

6 steps · 1 to fill in

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 $μ = 250 ml$ and a population standard deviation of $σ = 12 ml$ . 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 $245 ml$ .

Solution

6 steps · 1 to fill in

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. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 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.

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Question 2 · medium · 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 $Y = 3 X - 10$ . Find the exact expected value and variance of $Y$ .

Full worked solution Free, no sign-in

Apply the expectation rule: $E (Y) = 3 E (X) - 10 = 3 (50) - 10$ M1
$E (Y) = 140$ A1
Apply the variance rule (the constant does not affect variance): $Var (Y) = 3^{2} Var (X) = 9 \times 4^{2}$ M1
$Var (Y) = 144$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 7 marks · Paper 2

The mass of a chocolate bar, $X$ g, is distributed as $X \sim N (45, 2^{2})$ . A gift box contains $6$ randomly chosen bars, and the empty packaging has mass $P$ g, where $P \sim N (20, {1.5}^{2})$ . All masses are independent.

Find the probability that the total mass of a full gift box is between $285$ g and $295$ g.
Two full gift boxes are chosen at random. Find the probability that the first box is more than $5$ g heavier than the second box.

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FAQ

How many IB Maths AI HL linear combinations of random variables & the central limit theorem questions are there?

There are 22 exam-style linear combinations of random variables & the central limit theorem questions in the AI HL question bank (Paper 1: 16 · Paper 2: 6), graded 9 medium, 9 easy, 1 starter, 3 hard. 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?

Both. In the bank, Paper 1: 16 · Paper 2: 6. Practise with your GDC — AI papers expect calculator methods throughout.

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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Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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IB Maths AI HL Linear Combinations of Random Variables & the Central Limit Theorem Questions

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What you need to know

What's examined in AI HL linear combinations of random variables & the central limit theorem

Key formulas

Linear Combinations of Random Variables & the Central Limit Theorem worked examples

Worked example 1: Transforming a Single Random Variable · easy

Solution

Worked example 2: Summing Independent Random Variables · medium

Solution

Worked example 3: Applying the Central Limit Theorem · hard

Solution

Try these IB Maths AI HL linear combinations of random variables & the central limit theorem questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 4 marks · Paper 1

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Question 3 · hard · 7 marks · Paper 2

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