PD101

Understand the Poisson Distribution with Diagrams and Hand Calculations

An introductory course for building intuition about count data over a fixed interval, through λ, the probability mass function, mean = variance, and the sum of intervals. Short exercises for computing P(X = k), cumulative probabilities, and interval scaling by hand are placed in every chapter, and an in-browser simulator lets you watch how the shape changes.

Diagram → hand calculation → simulator → implementation 39 questions in total Graded in-browser Saved to localStorage

Time

3–4 hours

Question count

39 questions

Format

7 chapters + practice + simulator

Cost

Free

Line up 5-minute "boxes" and look only at how many events fall into each box. How many land in each box is the main subject of the Poisson distribution.

What makes this course different

Numeric practice problems placed in the middle of every chapter

You compute P(X = k), cumulative probabilities, and interval conversions on the spot, so the structure makes it hard to skim through passively.

Progresses in the order diagram → hand calculation → simulator → code

In Chapter 5 you move the shape with an in-browser simulator, and in Chapter 6 you read a minimal implementation in vanilla JavaScript.

Everything graded and stored in-browser

Practice problems and the comprehensive review are graded without sending anything to a server, and answers and progress are stored only in localStorage.

The five calculations that come up again and again

λ = rate × interval length

First, transform the rate into the value for the target interval.

P(X = k) = e^(−λ) λ^k / k!

The probability of exactly k events.

P(X ≤ k) = Σ_{i=0}^k P(X = i)

The cumulative probability of at most k events.

E[X] = λ, Var(X) = λ

Both the mean and the variance are λ.

Poisson(λ₁) + Poisson(λ₂) = Poisson(λ₁ + λ₂)

The sum of independent Poissons adds the λ's.

Chapter overview

1 Introduction — what kind of counts does the distribution describe? 5 questions. Develop a feel for "how many events occurred" within a fixed time window or a fixed area. 2 λ and the probability mass function — compute P(X = k) by hand 6 questions. Compute P(X = k), the core of the Poisson distribution, starting from 0, 1, and 2 events. 3 Mean = variance — reading the shape and spread 5 questions. Read how λ determines both the center and the spread, through mean, variance, and standard deviation. 4 Interval scaling and sums — transforming λ across intervals 5 questions. Get a feel for how λ moves when you change the time window or add independent streams. 5 Simulator — explore PMFs interactively 5 questions. Drag λ and watch the bar-chart peak, spread, and cumulative probability change before your eyes. 6 Implementation — reading a minimal Poisson sampler 4 questions. Read a minimal implementation of pmf / cdf / sampler built without external libraries. 7 Comprehension check — comprehensive review and full-question audit 9 questions. A mixed-review case plus simulator readouts and minimal-implementation reading exercises tie all chapters together.

Tips for studying

Read the body text and then solve the exercise right below it.
Three decimal places is plenty for probabilities. Write at least one intermediate step on paper before typing the answer.
Do both the simulator in Chapter 5 and the mixed review in Chapter 7.

Prerequisites

A high-school feel for exponentials, square roots, and factorials is enough. We do not go into calculus or the rigorous derivations of stochastic processes. The goal, first of all, is to be able to read counts over a fixed interval.