4.5 Statistical distributions (A-level only) - Revision Guide

What you'll learn

How to recognise binomial, Poisson and discrete uniform distributions from context.
How to justify your choice using the assumptions behind each model.
How to identify when a familiar distribution is not appropriate.
How to distinguish between models that can initially appear similar.

Probability models

A random variable is a variable whose value is determined by the outcome of a random process. In this topic, the random variables are discrete, meaning that their possible values can be listed separately.

For example, the number of defective components in a box could be 0, 1, 2, and so on. It cannot be 2.6.

A probability distribution gives the possible values of a random variable together with their probabilities. A distribution is a model: it is a mathematical description of a real situation, based on assumptions.

Key Idea

Choosing a model

Do not choose a distribution just because you notice a word such as “success” or “per hour”. Identify the random variable, then check every assumption of the proposed model against the context.

The discrete uniform distribution

A discrete uniform distribution models a finite set of possible values when every value is equally likely.

If a random variable $X$ can take $k$ equally likely values, then each value has probability

P(X=x)=1k.P(X=x)=\frac{1}{k}.

Typical contexts include:

the score from a fair die;
a number selected at random from a finite list;
the sector on which a fair spinner lands, provided its sectors have equal areas;
a randomly selected day of the week, if every day is equally likely.

Definition

Discrete uniform distribution

A random variable has a discrete uniform distribution when it has a finite or countable set of possible values and all the relevant values are equally likely.

Example

Choosing a model for a random number

A computer selects one integer at random from 1 to 20 inclusive. Let $X$ be the selected integer.

The possible values are the finite set $1,2,…,201,2,\ldots,20$ .
The phrase “at random” indicates that each of these 20 integers has the same probability, assuming the computer’s selection process is fair.
Therefore, a discrete uniform distribution is appropriate, with $P(X=x)=120P(X=x)=\frac{1}{20}$ for each possible value.

When discrete uniform is inappropriate

The fact that the outcomes form a neat list does not mean they are equally likely.

For example, the possible totals from rolling two fair dice range from 2 to 12, but these totals are not equally likely. A total of 7 can occur in six ways, whereas a total of 2 can occur in only one way.

Common Mistake

Assuming listed outcomes are equally likely

Check the underlying process, not just the list of possible values. A discrete uniform model requires equal probabilities, not merely evenly spaced numerical values.

The binomial distribution

The binomial distribution models the number of successes in a fixed number of repeated trials.

A trial must have exactly two relevant outcomes:

success, meaning the outcome being counted;
failure, meaning any other outcome.

The words success and failure do not imply something desirable or undesirable. A “success” could be a defective item if defects are what the question counts.

We write

X∼B(n,p),X\sim B(n,p),

where:

$n$ is the fixed number of trials;
$p$ is the probability of success on each trial.

Conditions for a binomial model

You should check all four conditions:

There is a fixed number, $n$ , of trials.
Each trial has only two relevant outcomes.
The trials are independent, so one result does not affect another.
The probability of success, $p$ , is constant for every trial.

Key Idea

Binomial checklist

Think: fixed number of independent trials, two outcomes per trial, constant probability of success.

Example

Modelling faulty components

A factory produces components independently, and each component has probability 0.03 of being faulty. Let $X$ be the number of faulty components among the next 50 produced.

There is a fixed number of trials: 50 components are inspected.
Each component has two relevant outcomes: faulty or not faulty.
Production outcomes are stated to be independent, and the fault probability remains constant at 0.03.
Therefore, a binomial model is appropriate:

X∼B(50,0.03).X\sim B(50,0.03).

When binomial is inappropriate

A binomial model may fail when:

the number of trials is not fixed;
there are more than two relevant outcomes;
the trials influence one another;
the success probability changes between trials.

Example

Sampling without replacement

A bag contains 6 red counters and 4 blue counters. Three counters are selected without replacement, and $X$ is the number of red counters selected.

There is a fixed number of selections and each selection can be classified as red or not red.
However, the selections are not independent. After a counter is removed, the contents of the bag change.
The probability of selecting red is therefore not constant: it begins at $610\frac{6}{10}$ but changes after the first selection.
An exact binomial model is not appropriate.

Tip

Replacement matters

Sampling with replacement can preserve independence and a constant probability. Sampling without replacement usually destroys both conditions.

The Poisson distribution

The Poisson distribution models the number of times an event occurs in a fixed interval or region.

The interval might measure:

time, such as calls per minute;
length, such as defects per metre of cable;
area, such as plants per square metre;
volume, such as particles in a sample.

We write

X∼Po⁡(λ),X\sim \operatorname{Po}(\lambda),

where $λ\lambda$ is the mean number of events in the specified interval.

Conditions for a Poisson model

A Poisson model assumes that:

events occur independently;
events occur at a constant mean rate;
events occur singly, so two events do not occur at exactly the same point or instant;
the mean number of events is proportional to the size of the interval.

Key Idea

Poisson checklist

Think: counting independent events in a fixed interval at a constant average rate.

Example

Modelling calls to a helpline

A helpline receives calls independently at a constant mean rate of 6 calls every 10 minutes. Let $X$ be the number of calls received in a particular 10-minute period.

The random variable counts events: the number of incoming calls.
The events are counted over a fixed interval of 10 minutes.
The calls are stated to arrive independently at a constant mean rate.
The mean number in the chosen interval is 6, so a Poisson model is appropriate:

X∼Po⁡(6).X\sim \operatorname{Po}(6).

Adjusting the interval

The value of $λ\lambda$ must correspond to the interval in the question. If the interval changes, scale the mean proportionally.

Example

Changing the Poisson time interval

Customers enter a shop at a constant mean rate of 12 per hour. Let $Y$ be the number entering during a 15-minute period.

Fifteen minutes is $14\frac{1}{4}$ of an hour.
Scale the hourly mean by the same factor:

λ=12×14=3.\lambda=12\times\frac{1}{4}=3.

Provided arrivals are independent and the rate is constant, the model is

Y∼Po⁡(3).Y\sim\operatorname{Po}(3).

When Poisson is inappropriate

A Poisson model may be unsuitable if events influence one another or the event rate changes substantially.

For example, customers arriving at a railway station may have a much higher rate just before a train departs. Earthquakes may produce dependent aftershocks, and cars may arrive together because traffic lights release them in groups.

Example

Recognising a changing event rate

A café receives an average of 30 customers per hour across the whole day. A manager proposes $X∼Po⁡(30)X\sim\operatorname{Po}(30)$ for the number arriving between 12:00 and 13:00.

The proposed model treats 30 as the mean for the particular one-hour interval.
However, a whole-day average does not establish a constant arrival rate. Lunchtime may be substantially busier than other periods.
Unless there is evidence that the rate is constant throughout the day, the parameter 30 is not justified for 12:00 to 13:00.
The proposed Poisson model may therefore be inappropriate.

Distinguishing binomial from Poisson

Both distributions count events, so focus on what is fixed.

Binomial: the number of trials is fixed, and you count successes.
Poisson: the interval is fixed, and you count how many events occur.
Discrete uniform: the possible values are all equally likely.

For example, “the number of faulty bulbs among 40 bulbs” suggests binomial because 40 trials are fixed. “The number of faults found in 100 metres of cable” suggests Poisson because a length is fixed and events are counted within it.

Common Mistake

Treating every count as Poisson

A count is not automatically Poisson. If it counts successes among a fixed number of trials, check the binomial conditions first.

Judging assumptions from context

Real data rarely obey a model perfectly. Your task is to decide whether the assumptions are reasonable enough for the model to be useful.

Look for contextual evidence:

“independently” supports binomial or Poisson modelling;
“constant probability” supports a binomial model;
“constant mean rate” supports a Poisson model;
“fair” or “chosen at random” may support a discrete uniform model;
“without replacement”, “rush hour”, “seasonal” or “in batches” may challenge the assumptions.

Do not merely state that a model is inappropriate. Name the failed condition and connect it to the context.

Exam technique

In the exam

Define the random variable clearly: say exactly what is being counted or selected.
State the proposed distribution and its parameters, such as $X∼B(n,p)X\sim B(n,p)$ or $X∼Po⁡(λ)X\sim\operatorname{Po}(\lambda)$ .
Justify the choice using contextual conditions: fixed trials, independence, constant probability, constant rate or equal likelihood.
If rejecting a model, identify the specific failed assumption and explain why it fails in that situation.
Check that $p$ is a probability for one trial, while $λ\lambda$ is a mean number of events for the exact interval being modelled.

Self review

Check yourself

Which conditions must hold before you use a binomial distribution?
Why might customer arrivals over an entire day fail to follow one Poisson model?
Are the totals from two fair dice discrete uniform? Explain your reasoning.

What you'll learn

How to recognise binomial, Poisson and discrete uniform distributions from context.
How to justify your choice using the assumptions behind each model.
How to identify when a familiar distribution is not appropriate.
How to distinguish between models that can initially appear similar.

Probability models

For example, the number of defective components in a box could be 0, 1, 2, and so on. It cannot be 2.6.

Key Idea

Choosing a model

The discrete uniform distribution

A discrete uniform distribution models a finite set of possible values when every value is equally likely.

If a random variable $X$ can take $k$ equally likely values, then each value has probability

P(X=x)=1k.P(X=x)=\frac{1}{k}.

Typical contexts include:

the score from a fair die;
a number selected at random from a finite list;
the sector on which a fair spinner lands, provided its sectors have equal areas;
a randomly selected day of the week, if every day is equally likely.

Definition

Discrete uniform distribution

A random variable has a discrete uniform distribution when it has a finite or countable set of possible values and all the relevant values are equally likely.

Example

Choosing a model for a random number

A computer selects one integer at random from 1 to 20 inclusive. Let $X$ be the selected integer.

The possible values are the finite set $1,2,…,201,2,\ldots,20$ .
The phrase “at random” indicates that each of these 20 integers has the same probability, assuming the computer’s selection process is fair.
Therefore, a discrete uniform distribution is appropriate, with $P(X=x)=120P(X=x)=\frac{1}{20}$ for each possible value.

When discrete uniform is inappropriate

The fact that the outcomes form a neat list does not mean they are equally likely.

Common Mistake

Assuming listed outcomes are equally likely

Check the underlying process, not just the list of possible values. A discrete uniform model requires equal probabilities, not merely evenly spaced numerical values.

The binomial distribution

The binomial distribution models the number of successes in a fixed number of repeated trials.

A trial must have exactly two relevant outcomes:

success, meaning the outcome being counted;
failure, meaning any other outcome.

The words success and failure do not imply something desirable or undesirable. A “success” could be a defective item if defects are what the question counts.

We write

X∼B(n,p),X\sim B(n,p),

where:

$n$ is the fixed number of trials;
$p$ is the probability of success on each trial.

Conditions for a binomial model

You should check all four conditions:

There is a fixed number, $n$ , of trials.
Each trial has only two relevant outcomes.
The trials are independent, so one result does not affect another.
The probability of success, $p$ , is constant for every trial.

Key Idea

Binomial checklist

Think: fixed number of independent trials, two outcomes per trial, constant probability of success.

Example

Modelling faulty components

A factory produces components independently, and each component has probability 0.03 of being faulty. Let $X$ be the number of faulty components among the next 50 produced.

There is a fixed number of trials: 50 components are inspected.
Each component has two relevant outcomes: faulty or not faulty.
Production outcomes are stated to be independent, and the fault probability remains constant at 0.03.
Therefore, a binomial model is appropriate:

X∼B(50,0.03).X\sim B(50,0.03).

When binomial is inappropriate

A binomial model may fail when:

the number of trials is not fixed;
there are more than two relevant outcomes;
the trials influence one another;
the success probability changes between trials.

Example

Sampling without replacement

A bag contains 6 red counters and 4 blue counters. Three counters are selected without replacement, and $X$ is the number of red counters selected.

There is a fixed number of selections and each selection can be classified as red or not red.
However, the selections are not independent. After a counter is removed, the contents of the bag change.
The probability of selecting red is therefore not constant: it begins at $610\frac{6}{10}$ but changes after the first selection.
An exact binomial model is not appropriate.

Tip

Replacement matters

Sampling with replacement can preserve independence and a constant probability. Sampling without replacement usually destroys both conditions.

The Poisson distribution

The Poisson distribution models the number of times an event occurs in a fixed interval or region.

The interval might measure:

time, such as calls per minute;
length, such as defects per metre of cable;
area, such as plants per square metre;
volume, such as particles in a sample.

We write

X∼Po⁡(λ),X\sim \operatorname{Po}(\lambda),

where $λ\lambda$ is the mean number of events in the specified interval.

Conditions for a Poisson model

A Poisson model assumes that:

events occur independently;
events occur at a constant mean rate;
events occur singly, so two events do not occur at exactly the same point or instant;
the mean number of events is proportional to the size of the interval.

Key Idea

Poisson checklist

Think: counting independent events in a fixed interval at a constant average rate.

Example

Modelling calls to a helpline

A helpline receives calls independently at a constant mean rate of 6 calls every 10 minutes. Let $X$ be the number of calls received in a particular 10-minute period.

The random variable counts events: the number of incoming calls.
The events are counted over a fixed interval of 10 minutes.
The calls are stated to arrive independently at a constant mean rate.
The mean number in the chosen interval is 6, so a Poisson model is appropriate:

X∼Po⁡(6).X\sim \operatorname{Po}(6).

Adjusting the interval

The value of $λ\lambda$ must correspond to the interval in the question. If the interval changes, scale the mean proportionally.

Example

Changing the Poisson time interval

Customers enter a shop at a constant mean rate of 12 per hour. Let $Y$ be the number entering during a 15-minute period.

Fifteen minutes is $14\frac{1}{4}$ of an hour.
Scale the hourly mean by the same factor:

λ=12×14=3.\lambda=12\times\frac{1}{4}=3.

Provided arrivals are independent and the rate is constant, the model is

Y∼Po⁡(3).Y\sim\operatorname{Po}(3).

When Poisson is inappropriate

A Poisson model may be unsuitable if events influence one another or the event rate changes substantially.

Example

Recognising a changing event rate

A café receives an average of 30 customers per hour across the whole day. A manager proposes $X∼Po⁡(30)X\sim\operatorname{Po}(30)$ for the number arriving between 12:00 and 13:00.

The proposed model treats 30 as the mean for the particular one-hour interval.
However, a whole-day average does not establish a constant arrival rate. Lunchtime may be substantially busier than other periods.
Unless there is evidence that the rate is constant throughout the day, the parameter 30 is not justified for 12:00 to 13:00.
The proposed Poisson model may therefore be inappropriate.

Distinguishing binomial from Poisson

Both distributions count events, so focus on what is fixed.

Binomial: the number of trials is fixed, and you count successes.
Poisson: the interval is fixed, and you count how many events occur.
Discrete uniform: the possible values are all equally likely.

Common Mistake

Treating every count as Poisson

A count is not automatically Poisson. If it counts successes among a fixed number of trials, check the binomial conditions first.

Judging assumptions from context

Real data rarely obey a model perfectly. Your task is to decide whether the assumptions are reasonable enough for the model to be useful.

Look for contextual evidence:

“independently” supports binomial or Poisson modelling;
“constant probability” supports a binomial model;
“constant mean rate” supports a Poisson model;
“fair” or “chosen at random” may support a discrete uniform model;
“without replacement”, “rush hour”, “seasonal” or “in batches” may challenge the assumptions.

Do not merely state that a model is inappropriate. Name the failed condition and connect it to the context.

Exam technique

In the exam

Define the random variable clearly: say exactly what is being counted or selected.
State the proposed distribution and its parameters, such as $X∼B(n,p)X\sim B(n,p)$ or $X∼Po⁡(λ)X\sim\operatorname{Po}(\lambda)$ .
Justify the choice using contextual conditions: fixed trials, independence, constant probability, constant rate or equal likelihood.
If rejecting a model, identify the specific failed assumption and explain why it fails in that situation.
Check that $p$ is a probability for one trial, while $λ\lambda$ is a mean number of events for the exact interval being modelled.

Self review

Check yourself

Which conditions must hold before you use a binomial distribution?
Why might customer arrivals over an entire day fail to follow one Poisson model?
Are the totals from two fair dice discrete uniform? Explain your reasoning.

2.4.3 Statistical distributions

What you'll learn

Probability models

The discrete uniform distribution

When discrete uniform is inappropriate

The binomial distribution

Conditions for a binomial model

When binomial is inappropriate

The Poisson distribution

Conditions for a Poisson model

Adjusting the interval

When Poisson is inappropriate

Distinguishing binomial from Poisson

Judging assumptions from context

Review 4.5 Statistical distributions (A-level only) by teaching Genie

2.4.3 Statistical distributions

What you'll learn

Probability models

The discrete uniform distribution

When discrete uniform is inappropriate

The binomial distribution

Conditions for a binomial model

When binomial is inappropriate

The Poisson distribution

Conditions for a Poisson model

Adjusting the interval

When Poisson is inappropriate

Distinguishing binomial from Poisson

Judging assumptions from context

Review 4.5 Statistical distributions (A-level only) by teaching Genie

2.4.3 Statistical distributions

What you'll learn

Probability models

The discrete uniform distribution

When discrete uniform is inappropriate

The binomial distribution

Conditions for a binomial model

When binomial is inappropriate

The Poisson distribution

Conditions for a Poisson model

Adjusting the interval

When Poisson is inappropriate

Distinguishing binomial from Poisson

Judging assumptions from context

Review 4.5 Statistical distributions (A-level only) by teaching Genie

4.5 Statistical distributions (A-level only) Revision Guide

2.4.3 Statistical distributions

What you'll learn

Probability models

The discrete uniform distribution

When discrete uniform is inappropriate

The binomial distribution

Conditions for a binomial model

When binomial is inappropriate

The Poisson distribution

Conditions for a Poisson model

Adjusting the interval

When Poisson is inappropriate

Distinguishing binomial from Poisson

Judging assumptions from context

Review 4.5 Statistical distributions (A-level only) by teaching Genie

4.5 Statistical distributions (A-level only) Revision Guide