# 🎲 Mathematics for Machine Learning 3: Probability and Statistics for Data-Driven Models

By Better Wise · Generated by Everwise

Gain the probability and statistics fluency models rely on: reason about distributions, apply Bayes and maximum likelihood, and quantify uncertainty in real data.

- **Level:** Intermediate
- **Goal:** Fundamentals
- **Language:** English
- **Focus:** [probability distributions](/courses?q=probability%20distributions), [Bayes' rule](/courses?q=Bayes'%20rule), [maximum likelihood](/courses?q=maximum%20likelihood), [statistical inference](/courses?q=statistical%20inference), [sampling and uncertainty](/courses?q=sampling%20and%20uncertainty)
- **Prior knowledge:** Basic calculus (derivatives, integrals), Vectors and matrices, Some Python for small simulations
- **Practice:** In-depth examples
- **Time:** ~5.3h
- **Voice:** a data scientist who grounds every distribution in a real dataset question before showing the formula
- **Not covered:** Measure-theoretic probability, Bayesian deep learning, Causal inference

## What you will achieve

By the end of this course you will be able to look at a real dataset question and choose the right probability model for it, rather than reaching for a formula by habit. Concretely, you will be able to:

- State the axioms of probability and use conditional probability and independence to reason about events in data you actually have.
- Compute and interpret expectation and variance on a real dataset, and explain what each number tells you about the underlying process.
- Distinguish covariance from correlation, and recognize when two variables are dependent even though their correlation is near zero.
- Match a data question to a distribution: Bernoulli and binomial for yes/no and counts, Poisson for rare events, Gaussian for continuous measurements, exponential for waiting times.
- Apply Bayes' rule with base rates, and explain why a highly accurate test can still produce mostly false positives when the condition is rare.
- Derive maximum likelihood and MAP estimates and explain in plain terms what the prior is doing.
- Reason about the bias-variance tradeoff when fitting a model, and connect it to overfitting you can observe.
- Build a confidence interval and run a hypothesis test, then state the result in a way that is statistically honest.

Each of these is tied to a concrete dataset question, so you finish with working intuition, not just definitions.

## Who this course is for

This is an intermediate course. It assumes you have seen the basics before and want to tighten your understanding and connect the math to real analysis work.

You should be comfortable with:

- Basic calculus: derivatives and integrals, at the level of knowing what a derivative measures and being able to read an integral sign.
- Vectors and matrices: enough to recognize a matrix and follow notation that uses them.
- Some Python: you have written small simulations or data scripts before. The course references Python for quick simulations but does not teach programming.

You do not need prior coursework in statistics or machine learning. If you have used a library like scikit-learn or scipy without fully understanding what it was doing, this course is designed for exactly that gap.

## Course roadmap

**Module 1: Probability Foundations for Data**
Start with the axioms, conditional probability, and independence, then move to expectation and variance computed on a real dataset. The module closes with covariance, correlation, and the difference between them, plus a checkpoint quiz.

**Module 2: Distributions and Bayes' Rule**
Cover the discrete workhorses first: Bernoulli, binomial, and Poisson for counts. Then the continuous ones: Gaussian and exponential models. The module ends with Bayes' rule and base rates, including the classic base-rate trap, followed by a checkpoint.

**Module 3: Estimation and Inference**
Learn maximum likelihood and MAP estimation, then see the bias-variance tradeoff play out on a real fitting problem. Finish with sampling, confidence intervals, and hypothesis testing, and a final checkpoint that ties the whole course together.

## How the course works

The course runs about 315 minutes across 12 lessons, so roughly five hours of focused study. Most learners spread it over one to two weeks.

Lessons come in two main shapes:

- **Theory lessons** built around a real dataset question. Each one opens with the question, shows why the naive answer fails, then introduces the distribution or method that resolves it. Examples are worked in depth inside the lesson, so there are no separate labs.
- **Checkpoint quizzes** at the end of each module. These are short and designed to catch the specific misconceptions that trip people up, such as confusing correlation with dependence or misreading a p-value.

Practice is embedded, not separate. Expect short exercises inside theory lessons where you compute something by hand or predict a result before the explanation.

**How to use the course:** if you are new to the material, go linearly. Module 1 sets up notation and intuition that Modules 2 and 3 lean on. If you already know the axioms and basic expectation cold, you can skim 1.1 and 1.2 and start at 1.3, where covariance and dependence usually need the most attention. Do not skip the checkpoints even if you skip lessons; they are the fastest way to find out whether a shortcut was safe.

One suggestion before you begin: keep a dataset you care about open in another window. Every concept in this course is easier to hold onto when you have a real number to attach it to.

## Contents

- **🎲 Probability Foundations for Data**
  - [🧩 Axioms, Conditional Probability and Independence](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/probability-axioms-conditional-independence.md)
  - [📊 Expectation and Variance on a Real Dataset](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/expectation-variance-worked-dataset.md) (sign-in required)
  - [🔗 Covariance, Correlation and Dependence](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/covariance-correlation-dependence.md)
  - [✅ Checkpoint: Probability Foundations](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/checkpoint-probability-foundations.md)
- **📈 Distributions and Bayes' Rule**
  - [🔢 Bernoulli, Binomial and Poisson Counts](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/bernoulli-binomial-poisson-count-data.md)
  - [🔔 Gaussian and Exponential Models](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/gaussian-exponential-continuous-data.md)
  - [🧠 Bayes' Rule and Base Rates](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/bayes-rule-base-rates.md)
  - [🧪 Checkpoint: Distributions and Bayes](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/checkpoint-distributions-and-bayes.md)
- **🎯 Estimation and Inference**
  - [⚙️ Maximum Likelihood and MAP Estimation](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/maximum-likelihood-map-estimation.md)
  - [⚖️ Bias-Variance Tradeoff in Practice](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/bias-variance-tradeoff-practice.md)
  - [📐 Sampling, Confidence Intervals and Hypothesis Tests](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/sampling-confidence-intervals-hypothesis-tests.md) (premium content)
  - [🏁 Checkpoint: Estimation and Inference](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/checkpoint-estimation-and-inference.md) (premium content)
- [Course cheatsheet](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/cheatsheet.md) (premium content)