1. Statistics And Probability

This section is my go-to reference for essential probability and statistics concepts, that act as the building blocks that underpin everything from risk modeling to machine learning.

Think of this as a personal binacle where we dig into the "why" behind commonly-used equivalences and definitions. Instead of just accepting formulas at face value, we'll explore their origins and derivations. Whether you're developing a new model, validating assumptions, or simply need a quick refresher on the math that matters, you'll find it here.

We're covering the core stuff: distributions, expected values, moments, convergence theorems, and those elegant mathematical relationships that make quantitative work possible. Clear, rigorous, and ready for real-world application.

1.1 Statistics vs Probability - Two Sides of the Same Coin

While often used interchangeably, statistics and probability are fundamentally different disciplines that approach uncertainty from opposite directions. Understanding their distinction is essential for rigorous quantitative work.

Probability is the mathematical framework for modeling uncertainty. It starts with a known model (a distribution, parameters, or generating process) and derives what outcomes we should expect. It's deductive reasoning: given the rules of the game, what will happen?

Statistics, on the other hand, works backward. It starts with observed data and attempts to infer the underlying model, parameters, or process that generated it. It's inductive reasoning: given what happened, what are the rules of the game?

1.1.1 How They Complement Each Other

These two disciplines form a complete cycle in quantitative analysis:

1. Probability provides the foundation: Statistical methods rely on probability theory to quantify uncertainty. When we construct confidence intervals or perform hypothesis tests, we use probabilistic calculations to assess how likely our observations are under various models.

2. Statistics validates probability models: Real-world applications require us to choose appropriate probability models. Statistics gives us tools to test whether our probabilistic assumptions match observed data, estimate unknown parameters, and refine our models.

3. Iteration drives insight: In practice, we move back and forth: propose a probabilistic model, use statistics to estimate its parameters and test its fit, refine the model based on diagnostics, and repeat. This iterative process is central to risk modeling, machine learning, and quantitative finance.

4. Both handle uncertainty rigorously: Together, they provide a complete framework for reasoning under uncertainty. Thereby, probability tells us what to expect from known models, while statistics tells us which models are consistent with reality.

In this section, we'll explore both perspectives: the probabilistic foundations that define the rules, and the statistical techniques that let us learn those rules from data.

1.2 Topics Covered

1.2.1 Probability

Explore the deductive framework for modeling uncertainty with known models and deriving expected outcomes.

• Expected Value —
• Filtration —
• Tower Property —
• Gaussian Integral —
• Jensen's Inequality —
• Moment Generating Functions —

→ View all Probability topics

1.2.2 Statistics

Explore the inductive framework for learning from data and inferring underlying models.

• Maximum Likelihood Estimation —

→ View all Statistics topics

Building confidence through rigorous validation