# Discrete probability distributions

> CIE A-Level Further Mathematics · Unit 4: Further Probability & Statistics
> Source: https://www.owlsprep.com/study/cie-9231-u4-discrete-probability-distributions/

This module covers core properties, expectation, variance and probability generating functions for discrete probability distributions, aligned to CIE A-Level Further Mathematics 9231 exam requirements. You will learn to solve standard exam questions on this foundational topic.

**Prerequisites:** [Basic probability concepts and discrete random variables (A-Level Mathematics)](https://www.owlsprep.com/study/cie-9709-probability-discrete-random-variables/)

## Learning objectives

- State and apply core properties of discrete probability distributions
- Calculate expectation and variance for any discrete distribution
- Define and use probability generating functions for discrete random variables
- Recall standard results for common discrete distributions

## Core Definitions & Properties

**Discrete Probability Distribution** — A distribution that describes all possible probabilities for a discrete random variable, which can only take a countable set of distinct values.

*Notation:* Described by a probability mass function (PMF) $p(x) = P(X=x)$

- All probabilities satisfy $0 \leq P(X=x_i) \leq 1$ for all outcomes $x_i$
- The sum of all probabilities equals 1: $\sum_{i} P(X=x_i) = 1$
- The cumulative distribution function (CDF) $F(x) = P(X \leq x)$ is non-decreasing

**Worked example:** A discrete random variable $X$ has PMF $P(X=x) = kx$ for $x = 1, 2, 3$, and 0 otherwise. Find the value of $k$.

1. Use the core property that the sum of all probabilities equals 1:
2. $$P(X=1) + P(X=2) + P(X=3) = 1 \\ k(1) + k(2) + k(3) = 1$$
3. Simplify and solve for $k$:
4. $$6k = 1 \implies k = \frac{1}{6}$$

> **Exam tip:** Always check that your probabilities sum to 1 after finding an unknown constant, this catches simple arithmetic errors.

## Expectation and Variance

**Expected Value & Variance** — Expected value is the long-run average of the distribution. Variance is a measure of spread around the expected value.

*Notation:* $E[X]$, $Var(X)$

*Example:* For discrete distributions: $E[X] = \sum x P(X=x)$, $Var(X) = E[X^2] - (E[X])^2$

**Worked example:** For the distribution $P(X=x) = \frac{x}{6}$ for $x=1,2,3$, find $E[X]$ and $Var(X)$.

1. Calculate $E[X]$ by summing $x \cdot P(X=x)$:
2. $$E[X] = \left(1 \cdot \frac{1}{6}\right) + \left(2 \cdot \frac{2}{6}\right) + \left(3 \cdot \frac{3}{6}\right) = \frac{1 + 4 + 9}{6} = \frac{7}{3}$$
3. Calculate $E[X^2]$ for the variance shortcut formula:
4. $$E[X^2] = \left(1^2 \cdot \frac{1}{6}\right) + \left(2^2 \cdot \frac{2}{6}\right) + \left(3^2 \cdot \frac{3}{6}\right) = \frac{36}{6} = 6$$
5. Substitute into the variance formula:
6. $$Var(X) = E[X^2] - (E[X])^2 = 6 - \left(\frac{7}{3}\right)^2 = \frac{5}{9}$$

> **Exam tip:** Always use the shortcut formula $E[X^2] - (E[X])^2$ for variance, it is much faster than expanding $E[(X-E[X])^2]$ in exams.

## Probability Generating Functions (PGFs)

**Probability Generating Function** — A function that encodes the full probability distribution of a non-negative integer-valued discrete random variable.

*Notation:* $G_X(t) = E[t^X] = \sum_{x=0}^\infty P(X=x) t^x$

- $G_X(1) = 1$
- $E[X] = G_X'(1)$ (first derivative at $t=1$)
- $Var(X) = G_X''(1) + G_X'(1) - (G_X'(1))^2$
- For independent $X,Y$: $G_{X+Y}(t) = G_X(t)G_Y(t)$

**Worked example:** Find the PGF of the distribution $X$ from the previous example, and use it to confirm $E[X]$.

1. Write the PGF using the definition:
2. $$G_X(t) = P(X=1)t^1 + P(X=2)t^2 + P(X=3)t^3 = \frac{t + 2t^2 + 3t^3}{6}$$
3. Differentiate $G_X(t)$ with respect to $t$:
4. $$G_X'(t) = \frac{1 + 4t + 9t^2}{6}$$
5. Evaluate at $t=1$ to get $E[X]$:
6. $$E[X] = G_X'(1) = \frac{1 + 4 + 9}{6} = \frac{14}{6} = \frac{7}{3}$$
7. This matches our earlier calculation, so the result is confirmed.

> **Exam tip:** PGFs only work for non-negative integer-valued discrete random variables. Do not use them for variables that can take negative values.

## Common Discrete Distributions

| Distribution | PMF | $E[X]$ | $Var(X)$ | PGF |
| --- | --- | --- | --- | --- |
| Discrete Uniform $(1..n)$ | $P(X=k) = \frac{1}{n}$ | $\frac{n+1}{2}$ | $\frac{n^2-1}{12}$ | $\frac{t(1-t^n)}{n(1-t)}$ |
| Bernoulli $(p)$ | $P(1)=p, P(0)=1-p$ | $p$ | $p(1-p)$ | $1-p + pt$ |
| Binomial $(n,p)$ | $\binom{n}{k}p^k(1-p)^{n-k}$ | $np$ | $np(1-p)$ | $(1-p+pt)^n$ |
| Poisson $(\lambda)$ | $\frac{e^{-\lambda}\lambda^k}{k!}$ | $\lambda$ | $\lambda$ | $e^{\lambda(t-1)}$ |

> **tip**
>
> Unless a question explicitly asks you to derive these standard results, you can use them directly in your exam answers to save time.

## Common pitfalls

- **Wrong:** Forgetting to check that the sum of probabilities equals 1 after finding an unknown constant
  - Why it fails: Arithmetic errors are common when solving for constants, and skipping the check leads to unnecessary lost marks
  - Correct: Always add up your final probabilities to confirm they sum to 1, this takes 10 seconds and catches simple mistakes
- **Wrong:** Writing the variance formula as $Var(X) = E[X^2] - E[X]$ instead of $E[X^2] - (E[X])^2$
  - Why it fails: This is a common algebraic slip when writing formulas quickly under exam pressure
  - Correct: Memorize the formula as 'expectation of X squared minus (expectation of X) all squared' and double-check the exponent
- **Wrong:** Using a probability generating function for a discrete random variable that takes negative values
  - Why it fails: PGFs are only defined for non-negative integer-valued random variables, so results will be incorrect
  - Correct: Use direct calculation or moment generating functions for discrete variables that can take negative values
- **Wrong:** Multiplying PGFs for dependent random variables to get the PGF of their sum
  - Why it fails: The product rule for PGFs of sums only holds for independent random variables
  - Correct: Only use $G_{X+Y}(t) = G_X(t)G_Y(t)$ if the question confirms X and Y are independent, otherwise calculate directly
- **Wrong:** Confusing the probability mass function (PMF) with the cumulative distribution function (CDF)
  - Why it fails: Students often mix these up when a question asks for one specifically
  - Correct: Remember PMF = $P(X=x)$, CDF = $P(X \leq x)$, label your answer clearly to avoid confusion

## Cheatsheet

| Concept | Key Formula |
| --- | --- |
| Core Distribution Property | $\sum P(X=x_i) = 1$ |
| Expectation | $E[X] = \sum x P(X=x)$ |
| Variance | $Var(X) = E[X^2] - (E[X])^2$ |
| PGF Definition | $G_X(t) = \sum P(X=x) t^x$ |
| PGF Expectation | $E[X] = G_X'(1)$ |
| PGF Variance | $Var(X) = G_X''(1) + G_X'(1) - (G_X'(1))^2$ |
| Sum of Independent Variables | $G_{X+Y}(t) = G_X(t)G_Y(t)$ |

## What's next

Discrete probability distributions are the foundation for all further topics in CIE 9231 Further Probability & Statistics. The core concepts of expectation, variance and generating functions you learned here are tested in nearly every exam paper, and are required for all subsequent topics in statistics. PGF properties for sums of independent variables are especially common in combined exam questions that connect multiple concepts. Mastering the core rules here will make learning named discrete distributions, continuous distributions and statistical inference much more straightforward, and help you secure easy marks on foundational exam questions.

- [Continuous Probability Distributions](https://www.owlsprep.com/study/cie-9231-u4-continuous-probability-distributions/)
- [Statistical Inference](https://www.owlsprep.com/study/cie-9231-u4-statistical-inference/)
- [Generating Functions](https://www.owlsprep.com/study/cie-9231-u4-generating-functions/)

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