Probability distributions: expected value and variance

IB Mathematics AI HL· 6 min read

1. Expected Value of a Random Variable★★☆☆☆⏱ 15 min

The expected value of a random variable $X$ , written $E (X)$ , is the theoretical long-run average value of $X$ over many independent trials. It is also called the expectation or theoretical mean of the distribution.

📘 Definition

Expected Value

$E (X)$ or $μ_{X}$

For a discrete random variable with probability mass function $P (X = x)$ : $E (X) = \sum x \cdot P (X = x)$ . For a continuous random variable with probability density function $f (x)$ : $E (X) = \int_{- \infty}^{\infty} x f (x) d x$

Example:

Expected value of a fair 6-sided die roll is $3.5$

📐 Worked Example

A discrete random variable $X$ has probability distribution: $P (X = 1) = 0.2$ , $P (X = 2) = 0.3$ , $P (X = 3) = 0.5$ . Calculate $E (X)$ .

1
Recall the formula for expected value of a discrete random variable:
2
$E(X) = \sum x \cdot P(X=x)$
3
Substitute the values from the distribution:
4
$(1 \times 0.2) + (2 \times 0.3) + (3 \times 0.5)$
5
Calculate the final result:
6
$0.2 + 0.6 + 1.5 = 2.3$

Exam tip:

Always check that the total probability of the distribution sums to 1 before calculating expected value to catch errors from missing values.

2. Variance and Standard Deviation★★★☆☆⏱ 15 min

Variance measures how spread out the outcomes of a random variable are around the expected value. Standard deviation is the square root of variance, and is measured in the same units as the original random variable.

📘 Definition

Variance

$Var (X)$ or $σ_{X}^{2}$

Variance is the expected value of the squared deviation from the mean: $Var (X) = E [(X - E (X))^{2}] = E (X^{2}) - [E (X)]^{2}$ . The identity on the right is almost always easier for calculations.

Example:

A distribution where all outcomes are equal has variance $0$ .

📐 Worked Example

For the distribution $P (X = 1) = 0.2$ , $P (X = 2) = 0.3$ , $P (X = 3) = 0.5$ , calculate $Var (X)$ and the standard deviation.

1
We already know $E (X) = 2.3$ from the previous example. First calculate $E (X^{2})$ :
2
$E(X^2) = \sum x^2 \cdot P(X=x)$
3
Substitute values:
4
$(1^2 \times 0.2) + (2^2 \times 0.3) + (3^2 \times 0.5) = 0.2 + 1.2 + 4.5 = 5.9$
5
Apply the variance identity:
6
$\text{Var}(X) = E(X^2) - [E(X)]^2 = 5.9 - (2.3)^2 = 0.61$
7
Calculate standard deviation as the square root of variance:
8
$\sigma_X = \sqrt{0.61} \approx 0.781$

Exam tip:

Never forget to square $E (X)$ in the variance formula. This is one of the most common errors marked down in IB exams.

3. Linear Transformation Rules★★★☆☆⏱ 20 min

A linear transformation of a random variable $X$ is a new variable $Y = a X + b$ , where $a$ and $b$ are constants. Simple rules exist for $E (Y)$ and $Var (Y)$ that let you avoid recalculating the entire distribution, a key exam skill.

📘 Definition

Linear Transformation Rules

$Y = a X + b$

For expectation: $E (a X + b) = a E (X) + b$ . For variance: $Var (a X + b) = a^{2} Var (X)$ . Shifting the distribution by a constant does not change its spread.

📐 Worked Example

Daily profit $X$ for a cafe has $E (X) = $120$ and $Var (X) = 250$ . The owner pays 10% tax plus a fixed \$15 daily fee. Find the expected value and variance of net profit.

1
Write net profit $Y$ as a linear transformation of $X$ (the owner keeps 90% of profit minus the fixed fee):
2
$Y = 0.9X - 15$
3
Calculate expected net profit using the expectation rule:
4
$E(Y) = 0.9E(X) - 15 = 0.9(120) - 15 = 93$
5
Calculate variance, remembering to square the coefficient 0.9:
6
$\text{Var}(Y) = (0.9)^2 \text{Var}(X) = 0.81(250) = 202.5$

Exam tip:

Only the scaling factor affects variance. Adding or subtracting a constant never changes variance, because it just shifts all outcomes without changing how spread out they are.

4. Interpretation in Context★★☆☆☆⏱ 10 min

IB exams frequently ask for interpretation of calculated expected value or variance/standard deviation in the context of the problem, not just for a numerical answer. Full marks require a correct contextual interpretation.

📐 Worked Example

Interpret the expected net profit of \$93 from the cafe example in context.

1
Correct interpretation (full marks): Over a large number of days, the average daily net profit for the cafe owner is approximately \$93.
2
Incorrect interpretation (no marks): The cafe owner will make \$93 every day. This is wrong because expected value is a long-run average, not a guaranteed outcome for one day.

Exam tip:

Always mention "long run" or "over many repeated trials" when interpreting expected value to get full marks.

5. Common Pitfalls

Wrong move:

Forgetting to square $E (X)$ in the variance formula, writing $Var (X) = E (X^{2}) - E (X)$

Why:

Variance is defined as the expected squared deviation from the mean, so the entire mean term must be squared

Correct move:

Always write the formula explicitly as $Var (X) = E (X^{2}) - [E (X)]^{2}$ to remember the square

Wrong move:

Forgetting to square the coefficient $a$ when calculating $Var (a X + b)$ , writing $a Var (X)$

Why:

Variance depends on squared deviations, so scaling $X$ by $a$ scales deviations by $a$ and squared deviations by $a^{2}$

Correct move:

Memorize: $Var (a X + b) = a^{2} Var (X)$

Wrong move:

Including the constant $b$ in variance: $Var (a X + b) = a^{2} Var (X) + b^{2}$

Why:

Confusing the rules for expectation and variance

Correct move:

Shifting the entire distribution by a constant does not change spread, so $b$ does not affect variance

Wrong move:

Interpreting expected value as the most likely outcome or the outcome of a single trial

Why:

Confusion between expected value and mode, misunderstanding the definition of expectation

Correct move:

Always interpret expected value as the long-run average over many repeated trials

Wrong move:

Using summation to calculate expected value for a continuous distribution

Why:

Over-practice with discrete distributions leads to automatic use of summation

Correct move:

For continuous random variables, use integration: $E (X) = \int x f (x) d x$

6. Quick Reference Cheatsheet

Concept	Formula	Key Note
Discrete $E (X)$	$E (X) = \sum x P (X = x)$	Sum over all outcomes
Continuous $E (X)$	$E (X) = \int_{- \infty}^{\infty} x f (x) d x$	Integrate over domain
Variance	$Var (X) = E (X^{2}) - [E (X)]^{2}$	Always use this identity
Expectation (linear transform)	$E (a X + b) = a E (X) + b$	Works for all distributions
Variance (linear transform)	$Var (a X + b) = a^{2} Var (X)$	$b$ does not affect variance
Standard Deviation	$σ_{X} = Var (X)$	Same units as $X$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Calculate E(X) and Var(X) for discrete distribution
2024 · 2
Linear transformation of net profit expectation
2023 · 1
Interpret expected value in context

What's Next

Expected value and variance are foundational concepts for all further statistical work in IB AI HL. These descriptors underpin all common named probability distributions you will study next, including binomial, Poisson, and normal distributions, and are core to inferential statistics topics like confidence intervals and hypothesis testing. The linear transformation rules you learned here also apply to coded data in descriptive statistics and are used repeatedly for probability problem solving. Mastery of this sub-topic is essential for success on both Paper 1 and Paper 2 exams.