# Binomial, Poisson and normal distributions

> IB Mathematics: Applications and Interpretation HL · Unit 5: Statistics and probability
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u5-binomial-poisson-and-normal-distributions/

This subtopic covers three of the most widely used probability distributions for statistical modelling: discrete binomial and Poisson, and continuous normal. You will learn distribution conditions, probability calculations and when to use approximations between distributions.

**Prerequisites:** [Basic probability rules](https://www.owlsprep.com/study/ib-math-ai-hl-u1-probability-fundamentals/); [Measures of mean and variance](https://www.owlsprep.com/study/ib-math-ai-hl-u4-descriptive-statistics/)

## Learning objectives

- Identify conditions for binomial, Poisson and normal distributions
- Calculate probabilities for each distribution using GDC and formulae
- Recognize when to use approximations between distributions
- Apply these distributions to solve real-world modelling problems

## Binomial Distribution

**Binomial Distribution** — A discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with constant probability of success.

*Notation:* $X \sim B(n, p)$

*Example:* Number of heads in 10 independent coin flips

The probability of getting exactly $k$ successes is given by the probability mass function:

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

The expected value (mean) and variance of a binomial distribution are:

$$E(X) = np, \quad Var(X) = np(1-p)$$

**Worked example:** 5% of light bulbs produced at a factory are defective. A random sample of 10 bulbs is selected. What is the probability that exactly 2 bulbs are defective?

1. Check binomial conditions: fixed $n=10$ trials, two outcomes (defective/not defective), constant $p=0.05$, independent trials (population large), so binomial is appropriate.
2. Substitute $k=2$ into the PMF:
3. $$P(X=2) = \binom{10}{2} (0.05)^2 (0.95)^8$$
4. Calculate $\binom{10}{2} = 45$, then simplify to get:
5. $$P(X=2) \approx 0.0746$$

> **Exam tip:** Marks are often awarded for justifying why you chose a binomial distribution, always confirm the four conditions in your working.

## Poisson Distribution

**Poisson Distribution** — A discrete probability distribution that models the number of events occurring in a fixed interval of time/space, when events occur at a constant known mean rate and independently of each other.

*Notation:* $X \sim Po(\lambda)$

*Example:* Number of customers arriving at a café in 1 hour

The probability of exactly $k$ events occurring is:

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

A key property of the Poisson distribution is that its mean and variance are equal to $\lambda$:

$$E(X) = \lambda, \quad Var(X) = \lambda$$

**Worked example:** On average, 3 cars pass through a crosswalk every 10 minutes. What is the probability that exactly 5 cars pass through in a 10-minute interval?

1. Check Poisson conditions: fixed interval, constant mean rate, independent events. So $X \sim Po(\lambda=3)$.
2. Substitute $k=5$ into the PMF:
3. $$P(X=5) = \frac{e^{-3} 3^5}{5!}$$
4. Calculate $3^5 = 243$, $5! = 120$, so simplify:
5. $$P(X=5) \approx 0.1008$$

## Normal Distribution

**Normal Distribution** — A continuous symmetric bell-shaped probability distribution, widely used to model naturally occurring continuous variables.

*Notation:* $X \sim N(\mu, \sigma^2)$

*Example:* Heights of adult humans

The probability density function is:

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

In IB AI HL, you will use your GDC to calculate cumulative probabilities and quantiles for normal distributions, rather than using the PDF directly. The empirical rule states ~68% of data lies within 1 standard deviation of the mean, ~95% within 2, ~99.7% within 3.

**Worked example:** Given $X \sim N(20, 5^2)$, find $P(X < 24)$ and the value of $k$ such that $P(X > k) = 0.1$.

1. First, identify parameters: $\mu = 20$, $\sigma = 5$ (note that the second parameter is variance, so $\sigma^2 = 25$, $\sigma = 5$).
2. For $P(X < 24)$, use the cumulative normal function on your GDC:
3. Result: $P(X < 24) \approx 0.7881$
4. For $P(X > k) = 0.1$, we have $P(X < k) = 0.9$, use inverse normal:
5. Result: $k \approx 26.41$

> **Exam tip:** Always check if the question gives variance or standard deviation; mixing these up when entering values into your GDC is a very common mistake.

## Approximations Between Distributions

When calculations for one distribution become tedious (usually for large $n$), we can approximate it with another distribution if conditions are met. Two key approximations are taught in IB AI HL:

- Poisson approximation to binomial: when $n$ is large ($n>10$) and $p$ is small ($p<0.1$), approximate $B(n,p)$ with $Po(\lambda = np)$
- Normal approximation: for $B(n,p)$, approximate with $N(np, np(1-p))$ when $np>5$ and $n(1-p)>5$; for $Po(\lambda)$, approximate with $N(\lambda, \lambda)$ when $\lambda>10$

> **tip**
>
> Always use continuity correction when approximating a discrete distribution with the continuous normal distribution. For $P(X \leq k)$ discrete, use $P(X \leq k+0.5)$ continuous; for $P(X \geq k)$ discrete, use $P(X \geq k-0.5)$ continuous.

**Worked example:** 1000 seeds are planted, each germinates with probability 0.002. Use Poisson approximation to find the probability that fewer than 3 seeds germinate.

1. Check conditions: $n=1000$ large, $p=0.002$ small, so Poisson approximation is appropriate. $\lambda = np = 1000 \times 0.002 = 2$.
2. We want $P(X < 3) = P(X \leq 2) = P(X=0) + P(X=1) + P(X=2)$
3. $$P(X \leq 2) = \frac{e^{-2} 2^0}{0!} + \frac{e^{-2} 2^1}{1!} + \frac{e^{-2} 2^2}{2!} = 5e^{-2}$$
4. Simplify to get:
5. $$P(X < 3) \approx 0.677$$

## Common pitfalls

- **Wrong:** Forgetting continuity correction when approximating a discrete distribution with normal
  - Why it fails: Discrete distributions only take integer values, so omitting the correction gives inaccurate probability
  - Correct: Always adjust the bound by ±0.5 when doing a normal approximation to a discrete distribution
- **Wrong:** Mixing up variance and standard deviation when entering normal distribution parameters into GDC
  - Why it fails: Notation $X \sim N(\mu, \sigma^2)$ uses variance as the second parameter, and students often enter the wrong value
  - Correct: Always confirm if the question gives variance or standard deviation, and double-check your input before calculating
- **Wrong:** Using binomial distribution when sampling without replacement from a small population
  - Why it fails: Trials are not independent, so the probability of success changes between draws
  - Correct: Only use binomial if the population is at least 10 times larger than the sample size
- **Wrong:** Using Poisson approximation to binomial when p is large
  - Why it fails: Poisson approximation is only accurate when p is very small, it will give large errors for large p
  - Correct: Use normal approximation to binomial when $np>5$ and $n(1-p)>5$, regardless of p size

## Cheatsheet

| Distribution | Type | Notation | Mean | Variance | PMF/PDF |
| --- | --- | --- | --- | --- | --- |
| Binomial | Discrete | $B(n,p)$ | $np$ | $np(1-p)$ | $\binom{n}{k}p^k(1-p)^{n-k}$ |
| Poisson | Discrete | $Po(\lambda)$ | $\lambda$ | $\lambda$ | $\frac{e^{-\lambda}\lambda^k}{k!}$ |
| Normal | Continuous | $N(\mu, \sigma^2)$ | $\mu$ | $\sigma^2$ | $\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ |

## What's next

Understanding these three core distributions is foundational for almost all further statistical inference topics you will encounter in IB AI HL. These distributions are used to model a huge range of real-world processes, from quality control in manufacturing to predicting customer foot traffic, and form the backbone of hypothesis testing and confidence interval calculation. Once you are comfortable identifying when to use each distribution, justifying your choice and calculating probabilities, you are ready to move on to more advanced statistical topics that build directly on this knowledge.

- [Non-linear regression models](https://www.owlsprep.com/study/ib-math-ai-hl-u5-non-linear-regression-models/)
- [Chi-squared tests: goodness of fit and independence](https://www.owlsprep.com/study/ib-math-ai-hl-u5-chi-squared-tests-goodness-of/)
- [t-tests and confidence intervals for means](https://www.owlsprep.com/study/ib-math-ai-hl-u5-t-tests-and-confidence-intervals/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-hl-u5-binomial-poisson-and-normal-distributions/