Further Distributions — AHL 4.11 Further distributions and...

SECTION 1

Historical Context & Motivation

Probability theory didn't spring up overnight. For centuries, mathematicians wrestled with games of chance, gambling odds, and eventually the question of how to model events that happen randomly over time or space. The Poisson distribution emerged from this rich history as a way to count rare events — things like the number of typos on a page, the number of phone calls received in an hour, or even the number of soldiers kicked by horses in the Prussian army. Meanwhile, mathematicians discovered that under the right conditions, one distribution can serve as a surprisingly good approximation for another, saving enormous amounts of computation.

1733

De Moivre's Normal Curve

Abraham de Moivre discovered the normal curve as an approximation to the binomial distribution, paving the way for the idea that one distribution can stand in for another.

1837

Poisson Publishes His Law

Siméon Denis Poisson published his probability work, introducing what we now call the Poisson distribution as a limiting case of the binomial for rare events.

1898

Bortkiewicz and Horse Kicks

Ladislaus Bortkiewicz famously applied the Poisson distribution to deaths from horse kicks in the Prussian army, demonstrating its real-world power for modeling rare events.

1920s

Approximation Methods Mature

By the early 20th century, statisticians had formalized the conditions under which Poisson, binomial, and normal distributions can approximate each other, forming the core of modern applied statistics.

The central question this topic addresses is: how do we model the count of events that occur randomly, and when can we use a simpler distribution in place of a more complex one? In your IB HL course, you already know the binomial and normal distributions. Now we add the Poisson distribution and the powerful techniques of distributional approximation to your toolkit.

SECTION 2

Core Principles & Definitions

Before diving into formulas, let's establish the foundational ideas that underpin this topic. You need to understand what the Poisson distribution models, when it applies, and why approximations between distributions work.

Poisson Distribution

Models the number of independent events occurring in a fixed interval of time or space, when events happen at a constant average rate. Defined by a single parameter, λ (the mean).

Poisson Conditions

Events must be independent, occur singly (not in clusters), and happen at a constant average rate. The random variable counts how many events occur, with no upper limit.

Poisson as Binomial Approximation

When n is large and p is small (rare events), the binomial B(n, p) can be approximated by a Poisson with λ = np. This avoids huge factorial calculations.

Normal Approximation to Binomial

When both np and n(1 − p) are greater than about 5, the binomial B(n, p) is well approximated by a normal distribution N(np, np(1 − p)). A continuity correction improves accuracy.

Normal Approximation to Poisson

When λ is large (typically λ > 15), the Poisson distribution is approximately normal with mean λ and variance λ: N(λ, λ). Again, a continuity correction is used.

✦ KEY TAKEAWAY

Think of distributions like different lenses for viewing the same scene. A binomial lens is perfect for counting successes in a fixed number of trials. But when those trials become very numerous and successes very rare, the Poisson lens gives you a clearer, simpler picture of the same situation. And when the numbers get large enough, the smooth bell-curve lens of the normal distribution fits over either one. Choosing the right lens — or the right approximation — is about knowing which conditions are met.

SECTION 3

Visual Explanation — The Poisson Distribution

The shape of a Poisson distribution depends entirely on one parameter: λ (lambda), the average number of events per interval. When λ is small, the distribution is strongly right-skewed — most of the probability is packed near zero. As λ increases, the distribution becomes more symmetric and begins to resemble a normal curve. The diagram below shows Poisson probability mass functions for three different values of λ.

The three bar groups show how the Poisson PMF changes shape with λ. At λ = 2 (cyan), the distribution is clearly skewed right. At λ = 5 (violet), it is more spread out and less skewed. At λ = 10 (pink), it closely resembles a symmetric bell curve — hinting at the normal approximation.

Notice how at λ = 2, the most likely outcomes are 1 and 2 events, with a noticeable tail stretching to the right. As λ grows to 5 and then 10, the peak shifts rightward and the spread widens. The key observation is that by the time λ reaches about 15 or more, the Poisson distribution is nearly indistinguishable from a normal distribution with mean λ and variance λ. This visual intuition is essential for understanding when approximations are valid.

SECTION 4

Mathematical Framework

Let's formalize the distributions and the conditions under which approximations apply. Each formula below is one you may need to use in your IB HL exam, so pay close attention to when each is appropriate.

POISSON PROBABILITY MASS FUNCTION

P(X = k) = (e⁻λ × λᵏ) / k!

X ~ Po(λ), where λ > 0 is the mean number of events per interval, k = 0, 1, 2, … is the number of events observed, and e ≈ 2.71828 is Euler's number.

POISSON MEAN AND VARIANCE

E(X) = λ, Var(X) = λ

A unique property of the Poisson distribution: the mean and variance are both equal to λ. This is a useful diagnostic — if sample data has mean ≈ variance, the Poisson model may be appropriate.

POISSON APPROXIMATION TO BINOMIAL

B(n, p) ≈ Po(np) when n > 50 and p < 0.1

Use λ = np. This works because when n is large and p is small, the binomial converges to the Poisson. Some textbooks use the condition n ≥ 20 and np < 5 as an alternative threshold.

NORMAL APPROXIMATION TO BINOMIAL (WITH CONTINUITY CORRECTION)

B(n, p) ≈ N(np, np(1 − p)) when np > 5 and n(1 − p) > 5

Apply a continuity correction: to find P(X ≤ k), compute P(Y ≤ k + 0.5) where Y ~ N(np, np(1 − p)). Similarly, P(X ≥ k) becomes P(Y ≥ k − 0.5). This correction accounts for using a continuous distribution to approximate a discrete one.

📐 Continuity Correction

When approximating a discrete distribution (binomial or Poisson) with a continuous one (normal), you must adjust by ±0.5. For P(X ≤ k), use P(Y ≤ k + 0.5). For P(X ≥ k), use P(Y ≥ k − 0.5). For P(X = k), use P(k − 0.5 ≤ Y ≤ k + 0.5). This is called the continuity correction and it significantly improves the accuracy of normal approximations.

SECTION 5

When to Use Which Approximation

One of the most practical skills in this topic is knowing which approximation to apply in a given situation. The flowchart below provides a decision framework. Start with the distribution you have, check the conditions, and follow the arrows to the appropriate approximation.

Start with a binomial distribution at the top. If n is large and p is small, go left to the Poisson approximation. If np and n(1 − p) are both greater than 5, go right to the normal approximation. The Poisson itself can be approximated by the normal when λ is large. Both normal paths require a continuity correction before using Z-tables or a GDC.

Summary of distribution approximations and their conditions

Original Distribution	Approximation	Conditions	Parameters
B(n, p)	Po(λ)	n > 50, p < 0.1	λ = np
B(n, p)	N(μ, σ²)	np > 5, n(1 − p) > 5	μ = np, σ² = np(1 − p)
Po(λ)	N(μ, σ²)	λ > 15	μ = λ, σ² = λ

SECTION 6

Worked Example

Let's work through a problem that uses both the Poisson distribution directly and the normal approximation. This is the kind of multi-step question you can expect in an IB HL exam.

Emails to a Help Desk

Step 1 — Read and Model the SituationA university help desk receives an average of 4.2 emails per hour. Assume the number of emails follows a Poisson distribution. (a) Find the probability of receiving exactly 3 emails in a given hour. (b) Find the probability of receiving more than 50 emails in a 12-hour day using a suitable approximation.

Step 2 — Part (a): Apply the Poisson PMFWe have X ~ Po(4.2), and we want P(X = 3). Using the formula: P(X = 3) = (e⁻⁴·² × 4.2³) / 3! First, compute e⁻⁴·² ≈ 0.01500. Then 4.2³ = 74.088. And 3! = 6. So P(X = 3) = (0.01500 × 74.088) / 6 = 1.11132 / 6.

P(X = 3) ≈ 0.1852

Step 3 — Part (b): Identify the Appropriate Distribution for 12 HoursOver 12 hours, the average number of emails is λ = 4.2 × 12 = 50.4. Since a Poisson distribution is additive (sum of independent Poisson variables is Poisson), the 12-hour count Y ~ Po(50.4). Because λ = 50.4 > 15, we can use the normal approximation: Y ≈ N(50.4, 50.4), which means μ = 50.4 and σ = √50.4 ≈ 7.099.

Step 4 — Apply the Continuity CorrectionWe want P(Y > 50). Since Y is discrete, P(Y > 50) = P(Y ≥ 51). With the continuity correction: P(Y ≥ 51) ≈ P(Z ≥ (50.5 − 50.4) / 7.099) where Z is a standard normal variable.

Step 5 — Compute the Z-Score and Find the ProbabilityZ = (50.5 − 50.4) / 7.099 = 0.1 / 7.099 ≈ 0.01409. From the Z-table (or GDC), P(Z ≥ 0.0141) = 1 − Φ(0.0141) ≈ 1 − 0.5056 = 0.4944.

P(Y > 50) ≈ 0.494 — there is roughly a 49.4% chance of receiving more than 50 emails in a 12-hour day.

🖩 GDC Tip

On a TI-84, use poissonpdf(4.2, 3) for part (a). For part (b), use normalcdf(50.5, 1E99, 50.4, √50.4). On a TI-Nspire, use the Statistics > Distributions menu. Always state the approximation and continuity correction in your working to earn full method marks.

SECTION 7

Strengths & Limitations of Approximations

Approximations are powerful shortcuts, but they come with trade-offs. Understanding when an approximation is good enough — and when it breaks down — is just as important as knowing the formulas. The table below compares the key approximation methods.

Comparison of approximation methods

Approximation	Strengths	Limitations
Poisson → Binomial	Avoids computing large factorials. Only one parameter (λ) needed. Excellent for rare events.	Breaks down when p is not small or n is not large. Not suitable when np > 10.
Normal → Binomial	Converts discrete probabilities to continuous Z-scores. Works well for symmetric binomials. Leverages widely available Z-tables.	Requires continuity correction. Poor for skewed distributions (small np or n(1−p)). Gives negative values for small n.
Normal → Poisson	Makes cumulative Poisson calculations straightforward. Uses standard normal tables.	Only valid for large λ (typically > 15). Requires continuity correction. Ignores discrete nature of data.

✦ KEY TAKEAWAY

Think of approximations like rounding in everyday life. Saying "about 30 minutes" instead of "29 minutes and 42 seconds" is fine for planning your schedule, but not fine for timing a chemical reaction. Similarly, the normal approximation to the binomial is a convenient shortcut that works brilliantly when conditions are met, but produces misleading results when they're not. Always check the conditions first.

SECTION 8

Connections to Broader Theory

The approximation relationships you've learned in this topic are actually special cases of deeper results in probability theory. Understanding where they fit in the larger landscape will help you see the logic behind them, rather than just memorizing conditions.

How AHL 4.11 concepts connect to broader theory

Concept in This Topic	Broader Mathematical Idea
Poisson approximation to binomial	Law of Rare Events (Poisson limit theorem): as n → ∞ and p → 0 with np = λ constant, B(n, p) → Po(λ).
Normal approximation to binomial	Central Limit Theorem (CLT): the sum of many independent random variables tends toward a normal distribution, regardless of the original distribution.
Normal approximation to Poisson	Also a consequence of the CLT: Po(λ) can be viewed as a sum of λ independent Po(1) variables. As λ grows, the CLT kicks in.
Continuity correction	Bridges the gap between discrete and continuous probability models; becomes unnecessary with very large sample sizes.

If you continue studying statistics at university, the Central Limit Theorem will become one of the most important results you encounter. It explains why the normal distribution appears so frequently in nature and science — and it's the theoretical backbone of every normal approximation you use in this topic. For now, it's enough to recognize that these approximations aren't just convenient tricks; they are consequences of deep, beautiful mathematics.

📝 IB Exam Note

In your IB HL exam, you must explicitly state which approximation you are using and verify that the conditions are met. For example, write: "Since n = 200 and p = 0.02, we have n > 50 and p < 0.1, so we approximate B(200, 0.02) by Po(4)." This earns method marks even if you make an arithmetic error later.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A student says, "The Poisson distribution can always be used instead of the binomial distribution because it's simpler." Explain why this statement is incorrect, and state the specific conditions under which the Poisson approximation to the binomial is valid.

PROBLEM 2 — BASIC CALCULATION

The number of accidents at a junction follows a Poisson distribution with a mean of 3 accidents per week. Find: (a) the probability of exactly 5 accidents in a given week, and (b) the probability of no more than 2 accidents in a given week.

PROBLEM 3 — INTERMEDIATE

A factory produces 500 items per day, and the probability of any item being defective is 0.008. (a) Explain why a Poisson approximation is appropriate. (b) Using this approximation, find the probability that on a given day, more than 6 items are defective.

PROBLEM 4 — APPLIED

A call centre receives an average of 48 calls per hour. Assuming calls arrive according to a Poisson distribution, use a suitable approximation with a continuity correction to find the probability that in a given hour, fewer than 40 calls are received.

PROBLEM 5 — CRITICAL THINKING

A researcher models the number of typos per page in a 300-page manuscript as X ~ Po(1.5). She wants to estimate the probability that the total number of typos in the entire manuscript exceeds 470. (a) Determine the distribution of the total number of typos T across all 300 pages. (b) Choose an appropriate approximation, justify your choice, and compute the required probability.

SUMMARY

Lesson Summary

The Poisson distribution models the count of independent events occurring at a constant average rate λ in a fixed interval, with the distinctive property that its mean equals its variance. Its PMF is P(X = k) = (e⁻λ × λᵏ) / k!. The Poisson distribution can serve as a powerful approximation to the binomial when n is large (n > 50) and p is small (p < 0.1), using λ = np. Meanwhile, the normal distribution can approximate both the binomial (when np > 5 and n(1 − p) > 5) and the Poisson (when λ > 15).

Whenever you use a normal approximation for a discrete distribution, apply a continuity correction of ±0.5 to improve accuracy. Always state the approximation you are using and verify that the conditions are satisfied — this earns full method marks on your IB exam. These approximation techniques connect to the deeper Central Limit Theorem, which explains why the normal distribution appears throughout statistics whenever many independent contributions are summed together.