Probability Distributions: Binomial, Poisson & Normal

What Is a Probability Distribution?

A probability distribution describes how the probabilities of outcomes are spread across the possible values of a random variable. It tells you what values are possible and how likely each one is.

Distributions are classified as discrete (countable outcomes) or continuous (any value in an interval).

Learning Goals: By the end of this guide, you should be able to:

Identify when to use binomial, Poisson or normal distributions.

Calculate probabilities using each distribution's formula.

Find expected values and variances.

Use the normal approximation where appropriate.

Binomial Distribution

The binomial distribution models the number of successes in $n$ independent trials, each with probability $p$ of success.

X \sim B(n, p)

Probability Formula

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Key Properties

Property	Formula
Mean	$E(X) = np$
Variance	$\text{Var}(X) = np(1-p)$
Standard deviation	$\sigma = \sqrt{np(1-p)}$

When to Use

Fixed number of trials $n$
Each trial has exactly two outcomes (success/failure)
Constant probability $p$ for each trial
Trials are independent

Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen at a constant average rate.

X \sim \text{Po}(\lambda)

Probability Formula

P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}

Key Properties

Property	Formula
Mean	$E(X) = \lambda$
Variance	$\text{Var}(X) = \lambda$

Defining feature: In a Poisson distribution, the mean equals the variance.

When to Use

Events occur independently
Events occur at a constant average rate $\lambda$
Two events cannot occur at exactly the same instant
Examples: number of emails per hour, customers entering a shop per day

Normal Distribution

The normal (Gaussian) distribution is the most important continuous distribution. Its bell-shaped curve describes many natural phenomena.

X \sim N(\mu, \sigma^2)

Probability Density Function

f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

Key Properties

Property	Value
Mean	$\mu$ (centre of the bell)
Variance	$\sigma^2$
Symmetry	Symmetric about $\mu$
68-95-99.7 rule	68% within $\pm 1\sigma$ , 95% within $\pm 2\sigma$ , 99.7% within $\pm 3\sigma$

Standard Normal Distribution

The standard normal $Z \sim N(0, 1)$ is obtained by standardising:

Z = \frac{X - \mu}{\sigma}

Use $Z$ -tables or a calculator to find probabilities.

Relationships Between Distributions

Approximation	Condition
$B(n,p) \approx \text{Po}(\lambda)$	When $n$ is large and $p$ is small ( $\lambda = np$ )
$B(n,p) \approx N(np, np(1-p))$	When $np > 5$ and $n(1-p) > 5$
$\text{Po}(\lambda) \approx N(\lambda, \lambda)$	When $\lambda > 15$ (approximately)

Continuity correction: When approximating a discrete distribution with a normal distribution, use continuity corrections: $P(X \leq k) \approx P(Y \leq k + 0.5)$ .

Worked Example

Question: A factory produces light bulbs with a 3% defect rate. In a sample of 100 bulbs, find the probability that at most 5 are defective.

Model: $X \sim B(100, 0.03)$ . Since $n$ is large and $p$ is small, use Poisson approximation with $\lambda = np = 3$ .

P(X \leq 5) = \sum_{k=0}^{5} \frac{e^{-3} 3^k}{k!} = e^{-3}\left(1 + 3 + 4.5 + 4.5 + 3.375 + 2.025\right) \approx 0.916

Common Mistakes

Using the wrong distribution — Check the conditions carefully. Binomial needs fixed $n$ ; Poisson needs a rate in an interval; Normal is for continuous data.
Forgetting the continuity correction — When using a normal approximation for a discrete distribution, always apply the $\pm 0.5$ correction.
Confusing $P(X = k)$ with $P(X \leq k)$ — For continuous distributions like the normal, $P(X = k) = 0$ . You always calculate intervals.

Exam Tips (A-Level / AP / IB)

For binomial calculations with small $n$ and manageable $p$ , compute directly. For large $n$ , switch to a Poisson or normal approximation.
Memorise the 68-95-99.7 rule — it's enough for quick estimates and sanity checks.
When the question says "on average 5 per hour", think Poisson. When it says "10 trials with 0.3 probability", think binomial.

Frequently Asked Questions

What is the difference between a probability mass function and a probability density function?

A PMF gives exact probabilities for discrete random variables: $P(X = k)$ . A PDF gives probability density for continuous variables — you must integrate over an interval to get a probability: $P(a < X < b) = \int_a^b f(x)\,dx$ .

Why is the normal distribution so common?

The Central Limit Theorem states that the sum (or average) of many independent random variables tends to follow a normal distribution, regardless of the original distribution. This is why it appears so frequently in nature and statistics.

Linear Regression — Regression assumes normally distributed residuals.
Sequences & Series — The Poisson formula involves a series expansion of $e^x$ .
Derivative Applications — Finding the mode of a distribution is an optimisation problem.

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What Is a Probability Distribution?

Binomial Distribution

Probability Formula

Key Properties

When to Use

Poisson Distribution

Probability Formula

Key Properties

When to Use

Normal Distribution

Probability Density Function

Key Properties

Standard Normal Distribution

Relationships Between Distributions

Worked Example

Common Mistakes

Exam Tips (A-Level / AP / IB)

Frequently Asked Questions

What is the difference between a probability mass function and a probability density function?

Why is the normal distribution so common?

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