IB Maths AA HL · Unit 4: Statistics and Probability
IB Maths AA HL Advanced Random Variables Questions
Exam-style IB Maths AA HL advanced random variables 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.
- 35 questions
- Paper 1: 27
- Paper 2: 8
- 7 easy
- 14 medium
- 8 hard
- 6 starter
- 3 worked examples
Practise Advanced Random Variables questions →
AA HL formula booklet
What's examined in AA HL advanced random variables
The question bank covers these advanced random variables question types (number of questions in brackets):
- Defining Probability Functions (18)
- Moments of Random Variables (12)
- Transformations & Normal Dist (5)
Key formulas
- Variance (continuous)
- \(\mathrm{Var}(X) = E(X^2) - \bigl[E(X)\bigr]^2\)
In the same notation as the IB formula booklet. All AA HL formulas →
Advanced Random Variables worked examples
Worked example 1: Linear transformations of expected value and variance · easy
A discrete random variable $X$ has an expected value $E(X) = 4$ and variance $\text{Var}(X) = 3$. Find the exact values of $E(5X - 2)$ and $\text{Var}(5X - 2)$.
1. Identify the rule for the expected value of a linear transformation: $E(aX + b) = aE(X) + b$.
2. Substitute the given expected value into the formula: $E(5X - 2) = 5(4) - 2$.
3. Evaluate the expected value: $20 - 2 = 18$.
4. Identify the rule for the variance of a linear transformation: $\text{Var}(aX + b) = a^2\text{Var}(X)$.
5. Substitute the given variance into the formula: $\text{Var}(5X - 2) = 5^2(3) = 25(3)$.
6. Evaluate to find the final exact variance: $\mathbf{E(5X - 2) = 18}$ and $\mathbf{\text{Var}(5X - 2) = 75}$.
Examiner tip: A very common mistake is incorrectly subtracting the constant $b$ from the variance or forgetting to square the coefficient $a$ when applying the variance transformation rule.
Worked example 2: Finding constants for continuous probability density functions · medium
A continuous random variable $X$ has a probability density function $f(x) = k(4x - x^2)$ for $0 \le x \le 4$, and $f(x)=0$ otherwise. Find the exact value of the constant $k$.
1. State the fundamental property of continuous probability density functions: the total area under the valid curve must equal $1$, so $\int_0^4 f(x) \,dx = 1$.
2. Set up the definite integral with the constant $k$ factored out: $k \int_0^4 (4x - x^2) \,dx = 1$.
3. Integrate the polynomial terms inside the bracket with respect to $x$: $k \left[ 2x^2 - \frac{x^3}{3} \right]_0^4 = 1$.
4. Substitute the upper limit (since the lower limit evaluates to $0$): $k \left( 2(16) - \frac{64}{3} \right) = 1$.
5. Simplify the algebraic expression inside the bracket: $k \left( 32 - \frac{64}{3} \right) = k \left( \frac{96 - 64}{3} \right) = k \left( \frac{32}{3} \right) = 1$.
6. Solve the resulting equation for the constant: $\mathbf{k = \frac{3}{32}}$.
Examiner tip: Students often forget that the total area under a continuous PDF must equal exactly 1, or they make arithmetic errors when subtracting fractions after substituting the integral limits.
Worked example 3: Calculating variance from a discrete probability distribution · hard
A discrete random variable $X$ has a probability distribution given by $P(X=x) = \frac{cx}{4}$ for $x \in \{1, 2, 3\}$. Find the exact value of the constant $c$, and hence calculate $\text{Var}(X)$.
1. Set the sum of probabilities equal to $1$ to find $c$: $\frac{c}{4} + \frac{2c}{4} + \frac{3c}{4} = 1 \implies \frac{6c}{4} = 1 \implies c = \frac{2}{3}$.
2. Rewrite the probability distribution explicitly: $P(X=1) = \frac{2}{12}$, $P(X=2) = \frac{4}{12}$, and $P(X=3) = \frac{6}{12}$.
3. Calculate the expected value $E(X) = \sum x P(X=x)$: $1\left(\frac{2}{12}\right) + 2\left(\frac{4}{12}\right) + 3\left(\frac{6}{12}\right) = \frac{2 + 8 + 18}{12} = \frac{28}{12} = \frac{7}{3}$.
4. Calculate the expected value of the squares $E(X^2) = \sum x^2 P(X=x)$: $1^2\left(\frac{2}{12}\right) + 2^2\left(\frac{4}{12}\right) + 3^2\left(\frac{6}{12}\right) = \frac{2 + 16 + 54}{12} = \frac{72}{12} = 6$.
5. Apply the discrete variance formula: $\text{Var}(X) = E(X^2) - [E(X)]^2$.
6. Evaluate the final exact variance: $6 - \left(\frac{7}{3}\right)^2 = \frac{54}{9} - \frac{49}{9} = \mathbf{\frac{5}{9}}$.
Examiner tip: A frequent trap is calculating the sum of the squares $E(X^2)$ and stopping there, forgetting to subtract the square of the expected value $[E(X)]^2$ to find the actual variance.
Try these IB Maths AA HL advanced random variables questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 2 marks · Paper 2
A continuous random variable \(T\) follows a uniform distribution with probability density function \(f(t) = 0.5\) for \(2 \le t \le 4\).
Find \(P(T \ge 3.5)\).
Question 2 · medium · 6 marks · Paper 2
A continuous random variable \(X\) has the probability density function \(f(x) = \frac{3}{8}x^2\) for \(0 \le x \le 2\).
Calculate the expected value \(\text{E}(X)\) and the variance \(\text{Var}(X)\).
Question 3 · hard · 6 marks · Paper 1
A continuous random variable \(X\) has the probability density function \(f(x) = \frac{1}{2}\sin x\) for \(0 \le x \le \pi\).
Using integration by parts, prove that the expected value \(\text{E}(X) = \frac{\pi}{2}\).
All 35 advanced random variables questions with mark schemes →
FAQ
How many IB Maths AA HL advanced random variables questions are there?
There are 35 exam-style advanced random variables questions in the AA HL question bank (Paper 1: 27 · Paper 2: 8), graded 7 easy, 14 medium, 8 hard, 6 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is advanced random variables on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 27 · Paper 2: 8. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA 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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