Contents · 0% complete

Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

LearnYMA01WST01 · Statistics 1Discrete random variablesProbability distributions and the CDF

Discrete random variables · Subtopic 1 of 2

Probability Distributions and the CDF

A discrete random variable takes separate values, each with a probability. All the probabilities add to $1$ , which lets you find unknown constants. The cumulative distribution function $F(x)=P(X\le x)$ is a running total; subtract consecutive values to get back $P(X=x)$ .

Edexcel YMA01 WST01≈ 30 minRevision 5 · updated 2026-10-27

Practise this topicTimed test Ask Salam Tutor

Section 1 of 14

Learning objectives

Write a probability distribution as a table or a function.
Use $\sum P(X=x)=1$ to find unknown constants.
Find probabilities such as $P(X>2)$ from a distribution.
Use and find the cumulative distribution function $F(x)$ .

Pause and think

Toss two coins and let $X$ be the number of heads. What values can $X$ take, and how likely is each?

Section 2 of 14

Core idea

Start hereA table of chances

Count the number of heads in 3 coin tosses: it can be 0, 1, 2 or 3, each with a known probability. A quantity like this is a discrete random variable, and its table of probabilities is its probability function.

The big idea

$P(X=x)$ for each value, with $\sum P(X=x)=1$ . Cumulative $F(x)=P(X\le x)$ ; $P(X=x)=F(x)-F(x-1)$ .

1. Probability function and CDF

Probability function (bars) and cumulative distribution (steps)

Each jump in

F

is the height of a bar.

F(3)=0.1+0.3+0.4=0.8

Worked example $P(X=x)=0.1,0.3,0.4,0.2$ for $x=1,2,3,4$

Find $F$ and $P(2\le X\le3)$ .

Goal: CDF; probability.
Strategy: Running totals.

$F(1)=0.1,\ F(2)=0.4,\ F(3)=0.8,\ F(4)=1$
Add as you go.
$P(2\le X\le3)=F(3)-F(1)=0.7$
Remove values below 2.

Answer

0.7

✓ Check

0.3+0.4=0.7

✓.

Worked exampleFrom F to P

$F(x)=\frac{x^2}{16}$ for $x=1,2,3,4$ . $P(X=3)$ ?

Goal: Probability.
Strategy: Difference of consecutive $F$ .

$F(3)=\tfrac9{16},\ F(2)=\tfrac4{16}$
Evaluate.
$P(X=3)=\tfrac5{16}$
Subtract.

Answer

\frac5{16}

✓ Check

F(4)=1

✓.

Worked exampleFinding k

$P(X=x)=k(5-x)$ , $x=1,2,3,4$ .

Goal: $k$ .
Strategy: Sum to 1.

$k(4+3+2+1)=10k$
Add.
$10k=1\Rightarrow k=0.1$
Total probability.

Answer

k=0.1

✓ Check

0.4+0.3+0.2+0.1=1

✓.

Common misconception $P(X<3)$ vs $P(X\le3)$ For integers,

P(X<3)=F(2)

Summary

$\sum P=1$ .
$F(x)=P(X\le x)$ .

Cumulative distribution function: $F(x)=P(X\le x)$ .

HistoryDice and the first distributions. In the 1600s Galileo was asked by gamblers why a total of

10

with three dice comes up more often than

9

, even though both can be made in six ways. By listing all

216

outcomes he found

27

ways for

10

and

25

for

9

: an early probability distribution.

Fun factYour phone is full of distributions. Predictive text keeps a probability distribution over the next word you might type and shows the most likely few. Each suggestion is the value with the biggest

P(X=x)

Section 3 of 14

Key formulae

Total probability

\sum P(X=x)=1

CDF

F(x)=P(X\le x)

From F to P

P(X=x)=F(x)-F(x-1)

TrickCheck with the total. After finding a constant, add up all the probabilities to check they really make

1

. It takes seconds and catches most slips.

Section 4 of 14

Worked examples

Example 1 · Finding k0 / 2 steps

P(X=x)=k(x+1)

for

x=0,1,2,3

. Find

k

Example 2 · From the CDF0 / 2 steps

F(x)=\dfrac{x^2}{16}

for

x=1,2,3,4

. Find

P(X=3)

Section 5 of 14

Explore the graph

Making the probabilities add to 1

Loading the interactive graph…

Try this

The bars show $P(X=x)=kx$ for $x=1,2,3,4$ . The value is the total $10k$ .
Adjust $k$ until the total is exactly $1$ . That is the only valid distribution.
Which value of $X$ is most likely?

Only one value of $k$ makes a genuine probability distribution.

In the real worldStock control. A shop models the number of a product sold each day as a discrete random variable. The CDF tells the manager the probability that demand is at most the stock held, which decides how much to order.

Section 6 of 14

Common mistakes

Incorrect

P(X=3)=F(3)

Correct

F(3)-F(2)

F(3)

includes

1

2

and

3

Incorrect

P(X>2)=1-F(3)

Correct

1-F(2)

X>2

excludes

2

only.

Incorrect

k\times4=1

for

kx

x=1

4

Correct

k(1+2+3+4)=1

Each value has a different probability.

IncorrectProbabilities summing to more than

1

CorrectExactly

1

Recheck the constant.

Expectation

$F(3)=P(X=3)$

Reality

$F(3)=P(X\le3)$ : cumulative means everything up to and including

Section 7 of 14

Exam tip

Exam tipWrite F(x) as a table. When asked for the cumulative distribution function, give a full table of

x

and

F(x)

, ending with

F=1

. Missing the last column, or not reaching

1

, loses the mark.

Section 8 of 14

Guided practice

SA-IAL-S1-DR-G012 marksL1 FoundationCalculatorProbability function

P(X=x)=kx

for

x=1,2,3,4

. Find

k

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-IAL-S1-DR1-0011 markL1 FoundationNon-calculatorProbability function

P(X=x)=k

for

x=1,2,3,4,5

. Find

k

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-DR1-0021 markL1 FoundationNon-calculatorProbability function

X

takes the values

1,2,3,4

with probabilities

0.3

0.25

p

and

0.15

. Find

p

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-DR1-0031 markL1 FoundationNon-calculatorProbability function

X

takes the values

1,2,3,4

with probabilities

0.1

0.4

0.3

0.2

. Find

P(X\ge2)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-DR1-0052 marksL2 StandardNon-calculatorProbability function

P(X=x)=\dfrac x{15}

for

x=1,2,3,4,5

. Find

P(X>3)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-DR1-0041 markL1 FoundationNon-calculatorCDF

X

takes the values

1,2,3,4

with probabilities

0.1

0.2

0.3

0.4

. Find

F(3)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-DR-0032 marksL1 FoundationCalculatorCDF

The cumulative distribution function of

X

F(1)=0.1

F(2)=0.35

F(3)=0.7

F(4)=1

. Find

P(X=3)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-DR-0042 marksL2 StandardCalculatorCDF

The cumulative distribution function of

X

F(x)=\dfrac{x^2}{16}

for

x=1,2,3,4

. Find

P(X=3)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-DR1-0062 marksL2 StandardNon-calculatorCDF

The cumulative distribution function of

X

F(x)=\dfrac{x+k}8

for

x=1,2,3,4

. Find

k

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-IAL-S1-DR-0074 marksL2 StandardCalculatorProbability function

P(X=x)=k(x+1)

for

x=0,1,2,3

. Find (i)

k

and (ii)

E(X)

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

A distribution lists $P(X=x)$ for every value.
$\sum P(X=x)=1$ finds unknowns.
$F(x)=P(X\le x)$ , ending at $1$ .
$P(X=x)=F(x)-F(x-1)$ ; $P(X>x)=1-F(x)$ .

Section 12 of 14

Quick quiz

Q1 · SA-IAL-S1-DR-Q011 markL1 FoundationCalculatorCDF

F(x)

, the cumulative distribution function, means

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-IAL-S1-DR1-0071 markL1 FoundationNon-calculatorProbability function

Two fair coins are tossed and

X

is the number of heads. Find

P(X=1)

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics, unit WST01 (Statistics 1)
Where: S1 · 5 Discrete random variables
Expected: The probability function and cumulative distribution function of a discrete random variable; probabilities sum to 1; finding unknown constants; P(X ≤ x) from F(x).
Calculator: Calculators are permitted, but exact answers (surds, fractions, π) are expected where the question asks.
Typical style: "Given that P(X = x) = kx for x = 1, 2, 3, 4, find k" then "find F(2)".
Open in: YMA01 · WST01

Section 14 of 14

Past paper questions

3 questions (36 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WST01/01 · June 2025Question 19 marks
Discrete distribution: cumulative probability, expectation, variance and a linear change.
Question paper, page 2 Mark scheme
WST01/01 · January 2025Question 115 marks
Two four-sided dice: probabilities, expectation, variance and a missing value in a cumulative distribution.
Question paper, page 2 Mark scheme
WST01/01 · June 2024Question 212 marks
Discrete distribution, E(X) and Var(X), and linear transformations.
Question paper, page 6 Mark scheme

All YMA01 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).