IB Maths AA HL
Topic 4 — Statistics & Probability
Paper 1 & 2
~6 min read
Transformation of a Single Variable
If you already know E(X) and Var(X), there’s no need to recompute when X gets scaled or shifted. Two short formulas — E(aX + b) = aE(X) + b and Var(aX + b) = a²·Var(X) — handle every linear transformation. For non-linear transformations, you redo the table by hand.
📘 What you need to know
- Linear mean: E(aX + b) = aE(X) + b.
- Linear variance: Var(aX + b) = a² · Var(X) — the +b drops out.
- Both formulas are in the booklet — read off, don’t memorise.
- Constants shift the mean but never affect the spread (Var(X + b) = Var(X)).
- Multiplication scales spread: SD scales by |a|, variance by a².
- Negative a is fine: a² is always positive, so variance stays non-negative.
- Division: rewrite Xa as (1/a)X to use the formulas.
- Non-linear transformations (X², 1/X, etc.) require a fresh table — the linear shortcuts do NOT apply.
Linear transformations
Mean of aX + b
E(aX + b) = a · E(X) + b
Variance of aX + b
Var(aX + b) = a² · Var(X)
Constant shift
X + b
mean shifts by b; spread unchanged
Constant scale
aX
mean scales by a; variance by a²
Why does +b drop out of variance? Adding a constant to every value shifts the whole distribution sideways without changing how spread out it is. The mean moves with the values, so deviations from the mean stay the same.
Non-linear transformations
If T = f(X) where f is anything other than aX + b — for example T = X², T = X³, T = 1/X — the linear formulas do not apply. You must build a fresh distribution table for T.
🧭 Recipe — non-linear transformation
- Apply f to each value of X to get the values of T.
- Carry the probabilities across unchanged: P(T = f(x)) = P(X = x).
- Combine duplicate T-values: if two x-values map to the same T, add their probabilities.
- Compute E(T) = ∑ t · P(T = t) on the new table.
- Compute Var(T) = E(T²) − [E(T)]² using the new table.
Worked examples
The discrete random variable X has E(X) = 8. Find E(2X + 3).
Identify a = 2, b = 3
Apply E(aX + b) = a·E(X) + b
E(2X + 3) = 2(8) + 3
= 16 + 3 = 19
E(2X + 3) = 19
the constant 3 just shifts the mean — it doesn’t get multiplied by anything
The discrete random variable X has Var(X) = 9. Find Var(4X − 7).
Identify a = 4 (the −7 is irrelevant for variance)
Apply Var(aX + b) = a²·Var(X)
Var(4X − 7) = 4² · 9
= 16 · 9 = 144
Var(4X − 7) = 144
the −7 drops out — constants alone never change spread
WE 3Negative coefficient — both mean and variance
The discrete random variable X has E(X) = 12 and Var(X) = 5. Find E(10 − X) and Var(10 − X).
Rewrite: 10 − X = (−1)X + 10, so a = −1, b = 10
Apply mean formula
E(10 − X) = (−1)(12) + 10 = −12 + 10 = −2
Apply variance formula
Var(10 − X) = (−1)² · 5 = 1 · 5 = 5
E(10 − X) = −2; Var(10 − X) = 5
a negative a squares to positive — variance is unchanged when you flip the sign
WE 4Division — rewrite as multiplication by 1/a
The discrete random variable X has E(X) = 30 and Var(X) = 18. Find E(X/3) and Var(X/3).
Rewrite X/3 as (1/3)X — so a = 1/3, b = 0
Apply mean formula
E(X/3) = (1/3)(30) = 10
Apply variance formula
Var(X/3) = (1/3)² · 18
= (1/9)(18) = 2
E(X/3) = 10; Var(X/3) = 2
division by 3 squared gives division by 9 in the variance
WE 5Practical context — commute time to daily cost
A worker’s daily commute time X (in minutes) has E(X) = 45 and Var(X) = 16. Their daily petrol cost C (in dollars) is given by C = 0.5X + 2 (a fuel component plus a fixed parking charge). Find E(C) and Var(C).
Identify a = 0.5, b = 2 (linear transformation)
Apply mean formula
E(C) = 0.5(45) + 2 = 22.5 + 2 = 24.5
Apply variance formula
Var(C) = 0.5² · 16 = 0.25 · 16 = 4
E(C) = $24.50; Var(C) = 4 (in $²)
remember: variance has units squared — Var(C) is in dollars²
WE 6Non-linear transformation — full table redo
The discrete random variable X has distribution:
| x | −1 | 0 | 1 | 2 |
|---|
| P(X = x) | 0.2 | 0.3 | 0.4 | 0.1 |
|---|
Let T = X². Find E(T) and Var(T).
Step 1: Apply T = X² to each value
X = −1 → T = 1 (prob 0.2)
X = 0 → T = 0 (prob 0.3)
X = 1 → T = 1 (prob 0.4)
X = 2 → T = 4 (prob 0.1)
Step 2: Combine duplicate T-values
P(T = 0) = 0.3
P(T = 1) = 0.2 + 0.4 = 0.6
P(T = 4) = 0.1
Step 3: Compute E(T)
E(T) = 0(0.3) + 1(0.6) + 4(0.1) = 0 + 0.6 + 0.4 = 1.0
Step 4: Compute E(T²)
E(T²) = 0(0.3) + 1(0.6) + 16(0.1) = 0 + 0.6 + 1.6 = 2.2
Step 5: Variance
Var(T) = 2.2 − 1.0² = 1.2
E(T) = 1.0; Var(T) = 1.2
non-linear → can’t shortcut; build the new table and apply standard E/Var formulas
💡 Top tips
- Constants don’t move spread: Var(X + b) = Var(X) regardless of b.
- For division, rewrite X/a as (1/a)X to use the formulas cleanly.
- Negative a squares away — Var(b − X) = Var(X).
- Non-linear transformations always need a fresh table — no shortcut exists.
- Combine duplicate T-values: if two different x map to the same t, add their probabilities.
⚠ Common mistakes
- Writing Var(aX + b) = a·Var(X) + b — both wrong: the b drops, the a gets squared.
- Forgetting to square a in Var(aX + b) = a²·Var(X).
- Applying linear formulas to non-linear transformations like X² or 1/X — they only work for aX + b.
- Mixing variance and standard deviation: if the question gives σ, square it first to get Var.
- Forgetting to combine duplicate T-values on a non-linear redo — leaves probabilities not summing to 1.
That closes the Discrete Random Variables sub-topic. Next up is the Binomial Distribution — a specific named DRV that crops up in nearly every IB Statistics paper, with shortcut formulas for its mean and variance baked into the booklet.
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