Poisson Distribution in Football Predictions

Football is one of the few sports where goal-scoring closely follows a Poisson distribution — a probability formula for rare, independent events. That’s why almost every serious football model has a Poisson core.

The formula Where λ (lambda) is the team’s expected goals (xG) for the match, and k is the number of goals we want the probability for.

Calculating expected goals We need four ingredients per team: - Attack strength = goals scored per game ÷ league average - Defence strength = goals conceded per game ÷ league average - League average goals - Home advantage multiplier ( 1.3)

From xG to match probabilities Once you have Home xG and Away xG, calculate P(home = i) and P(away = j) for every realistic scoreline (0–5 each), then sum: - Home win = all scorelines where i j - Draw = all scorelines where i = j - Away win = all scorelines where j i

Dixon-Coles refinement Plain Poisson slightly underestimates low-scoring draws (0-0, 1-1). The Dixon-Coles adjustment fixes this. We use a bivariate Dixon-Coles variant in our prediction engine.

Limits Poisson assumes goals are independent and identically distributed — which isn’t perfectly true (red cards, momentum shifts). That’s why we layer Elo, form, and market calibration on top.