Expert Distribution Theory: The Complete Toolkit

## The Distribution Toolkit An expert quantitative bettor maintains fluency with the full distribution toolkit: | Distribution | When to Use | Key Parameters | |---|---|---| | Poisson | Goals, points scored, events per time period | λ (rate) | | Negative Binomial | Over-dispersed count data | μ (mean), r (dispersion) | | Binomial | Binary outcomes over n trials | n (trials), p (probability) | | Normal | Large-sample aggregates, P&L distributions | μ, σ | | Beta | Uncertainty about a probability | α, β | | Elo/Bradley-Terry | Pairwise comparison and ranking | Team ratings | ## Combining Distributions The most powerful models combine multiple distributions: - Poisson for each team's goals → scoreline matrix - Dixon-Coles correction for low-score adjustment → improved scoreline matrix - Beta distribution for team rating uncertainty → probability intervals, not points - Normal approximation for portfolio P&L → risk management ## The Simulation Approach When analytical formulae become complex, simulation is often more practical: 1. Draw each team's expected goals from its beta distribution (incorporating uncertainty) 2. Draw actual goals from a Poisson distribution with the drawn expected goals 3. Generate the match result 4. Repeat 100,000 times The output is a full probability distribution over match results that properly accounts for both model uncertainty and outcome variance. ## Building Distributional Intuition The expert bettor develops distributional intuition: the ability to immediately recognise what distribution governs a specific situation and roughly what that distribution predicts. This intuition is built through years of working with these distributions in real betting contexts — seeing predictions vs outcomes thousands of times. The combination of formal distribution knowledge and calibrated intuition is the foundation of expert quantitative analysis.