Discrete probability distributions, expectation and variance

IB Mathematics: Analysis and Approaches SL· Unit 4: Statistics & Probability, Topic 6· 15 min read

1. Discrete Random Variables and Valid Distributions★★☆☆☆⏱ 4 min

📘 Definition

Discrete random variable

$X$

A variable that takes a countable set of distinct values, where each value corresponds to a random outcome.

Example:

Number of heads in 3 coin flips, number of customers entering a shop per hour

A discrete probability distribution is described by a probability mass function (PMF) $p (x) = P (X = x)$ , which gives the probability of the random variable taking each value $x$ . For a PMF to be valid, it must satisfy two core conditions:

All probabilities satisfy $0 \leq p (x) \leq 1$
The sum of all probabilities equals 1: $\sum_{x} p (x) = 1$

📐 Worked Example

A random variable $X$ has PMF $P (X = x) = k x$ for $x = 1, 2, 3$ , and 0 otherwise. Find the value of $k$ that makes this a valid distribution.

1
Use the condition that the sum of all probabilities must equal 1:
2
$P(X=1) + P(X=2) + P(X=3) = 1$
3
Substitute the PMF into the equation:
4
$k(1) + k(2) + k(3) = 1 \implies 6k = 1$
5
Solve for $k$ , then check all probabilities are between 0 and 1:
6
$k = \frac{1}{6}, \quad \frac{1}{6}, \frac{2}{6}, \frac{3}{6} \in [0,1] \text{ so valid}$

Exam tip:

Always check both conditions (sum to 1, all probabilities between 0 and 1) when asked to find a constant $k$ in an exam.

2. Expectation of a Discrete Distribution★★☆☆☆⏱ 5 min

📘 Definition

Expected Value (Expectation)

$E [X]$ or $μ$

The long-run average value of the random variable over many repeated trials, calculated as a weighted average of outcomes weighted by their probability.

For any discrete random variable $X$ , expectation is calculated with the formula:

E[X] = \sum_{x} x \cdot P(X=x)

📐 Worked Example

For the valid distribution $P (X = x) = \frac{x}{6}$ for $x = 1, 2, 3$ , calculate the expected value of $X$ .

1
Substitute into the expectation formula:
2
$E[X] = (1 \cdot \frac{1}{6}) + (2 \cdot \frac{2}{6}) + (3 \cdot \frac{3}{6})$
3
Simplify the sum:
4
$E[X] = \frac{1 + 4 + 9}{6} = \frac{14}{6} = \frac{7}{3} \approx 2.33$

3. Variance and Standard Deviation★★★☆☆⏱ 5 min

📘 Definition

Variance

$Var (X)$ or $σ^{2}$

A measure of the spread of the distribution around the expected value, equal to the expected value of the squared deviation from the mean.

For IB exams, the computational formula below is almost always faster and less error-prone than the definition formula:

\text{Var}(X) = E[X^2] - (E[X])^2, \quad E[X^2] = \sum_{x} x^2 P(X=x)

Standard deviation $σ = Var (X)$ is also a measure of spread, in the same units as the original variable $X$ .

📐 Worked Example

Given $E [X] = \frac{7}{3}$ for $P (X = x) = \frac{x}{6}$ for $x = 1, 2, 3$ , calculate the variance of $X$ .

1
First calculate $E [X^{2}]$ , the expected value of $X^{2}$ :
2
$E[X^2] = (1^2 \cdot \frac{1}{6}) + (2^2 \cdot \frac{2}{6}) + (3^2 \cdot \frac{3}{6}) = \frac{1 + 8 + 27}{6} = 6$
3
Substitute into the variance formula:
4
$\text{Var}(X) = 6 - \left(\frac{7}{3}\right)^2 = 6 - \frac{49}{9} = \frac{5}{9} \approx 0.56$

Exam tip:

If you are running low on time, use the computational formula to avoid mistakes with squared deviations.

4. Properties of Expectation and Variance★★★☆☆⏱ 5 min

For any constants $a$ and $b$ , the following linear properties hold for transformed random variables $Y = a X + b$ , and are heavily tested in IB exams:

$E [a X + b] = a E [X] + b$
$Var (a X + b) = a^{2} Var (X)$

📐 Worked Example

Given $X$ has $E [X] = 3$ and $Var (X) = 2$ , find $E [Y]$ and $Var (Y)$ for $Y = 2 X - 4$ .

1
Calculate expectation using the linear property:
2
$E[2X - 4] = 2E[X] - 4 = 2(3) - 4 = 2$
3
Calculate variance, note the constant term disappears:
4
$\text{Var}(2X - 4) = 2^2 \text{Var}(X) = 4(2) = 8$

5. Common Pitfalls

Wrong move:

Forgetting to square the scaling factor $a$ when calculating $Var (a X + b)$

Why:

Variance measures squared spread, so scaling by $a$ changes variance by $a^{2}$ , not $a$

Correct move:

Always use $Var (a X + b) = a^{2} Var (X)$ , the constant term $b$ is removed completely

Wrong move:

Only checking that probabilities sum to 1 when finding a constant $k$ , ignoring the $0 \leq p (x) \leq 1$ condition

Why:

Solving for $k$ can give a value that produces negative probabilities or probabilities greater than 1

Correct move:

Always verify that all individual probabilities are between 0 and 1 after solving for $k$

Wrong move:

Calculating expectation as an unweighted average of $x$ values, ignoring their probabilities

Why:

Confusing the sample mean of data with the expected value of a distribution

Correct move:

Always multiply each $x$ by its probability before summing to get expectation

Wrong move:

Mixing up the variance formula to get $Var (X) = (E [X^{2}] - E [X])^{2}$

Why:

Misremembering where the square goes in the computational formula

Correct move:

The square is only on the second term: $Var (X) = E [X^{2}] - (E [X])^{2}$

Wrong move:

Expecting expectation to always be one of the possible values of $X$

Why:

Thinking expectation is a possible outcome, not a long-run average

Correct move:

It is completely acceptable for expectation to be a non-integer or value not in the sample space of $X$

6. Quick Reference Cheatsheet

Concept	Formula	Exam Note
Valid PMF	$0 \leq P (X = x) \leq 1$ , $\sum P (X = x) = 1$	Check both conditions
Expectation	$E [X] = \sum x P (X = x)$	Weighted average of outcomes
Variance	$Var (X) = E [X^{2}] - (E [X])^{2}$	Use this formula for exams
$E [X^{2}]$	$E [X^{2}] = \sum x^{2} P (X = x)$	Intermediate step for variance
Linear Expectation	$E [a X + b] = a E [X] + b$	Always holds for any constants
Linear Variance	$Var (a X + b) = a^{2} Var (X)$	Constant $b$ does not change variance

7. Frequently Asked

Do I need to calculate expectation and variance for continuous distributions in AA SL?

No, for IB AA SL you are only required to calculate expectation and variance explicitly for discrete probability distributions.

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.

2023 · 1
6 mark variance calculation question
2022 · 2
5 mark expectation of transformed X
2021 · 1
4 mark valid distribution check

Going deeper

syllabusOfficial IB AA SL SyllabusFull topic 4 learning outcomes

What's Next

This sub-topic lays the foundation for all named discrete probability distributions you will study next in IB AA SL, including the binomial and Poisson distributions. The expectation and variance rules you learned here apply directly to these distributions, and you can use them to derive the standard formulas for named distributions. These concepts also form the basis for statistical inference, where you use sample statistics to estimate unknown population parameters like the true population mean. Mastery of these calculation methods is required for almost all probability questions on both Paper 1 and Paper 2 of the IB AA SL exam.