Skellam Goal Difference & Dixon-Coles Tau Formula

Correct independent Poisson under-prediction of 0-0 and 1-1 draws while preserving unit probability!

When Home Goals X ~ Poisson(lambda_H) and Away Goals Y ~ Poisson(lambda_A) are modeled independently, Goal Difference D = X - Y follows John Gordon Skellam’s (1946) distribution with Mean E[D] = lambda_H - lambda_A and Variance Var(D) = lambda_H + lambda_A (a frequent rookie mistake is subtracting variances instead of adding them!). Yet Mark Dixon and Stuart Coles (1997, Royal Statistical Society) proved empirically that Premier League matches exhibit tactical low-score dependence (rho < 0): 0-0 and 1-1 draws occur significantly MORE often than independent Poisson predicts, while 1-0 and 0-1 occur slightly less often! Their exact probability-preserving multiplier tau_{x,y}(rho) scales (0,0) by (1 - lambda_H * lambda_A * rho), (1,0) by (1 + lambda_A * rho), (0,1) by (1 + lambda_H * rho), and (1,1) by (1 - rho)—so the four probability shifts (-lambda_H*lambda_A*rho + lambda_H*lambda_A*rho + lambda_H*lambda_A*rho - lambda_H*lambda_A*rho)*e^{-(lambda_H+lambda_A)} cancel to ZERO!

Verify on Home_xG lambda_H = 1.50 (B2), Away_xG lambda_A = 1.00 (B3), and Dependence_Parameter rho = -0.12 (B4): Skellam Mean Goal Difference is 1.50 - 1.00 = +0.50 goals, Skellam Variance is 1.50 + 1.00 = 2.50 goals^2 (SD = 1.5811), Independent 0-0 Probability EXP(-2.50) = 0.0821 (8.21%) rises to Dixon-Coles P_DC(0,0) = 0.082085 * (1 - 1.5 * 1.0 * (-0.12)) = 0.082085 * 1.18 = 0.0969 (9.69%), and P_DC(1,1) rises by 1 - (-0.12) = 1.12x to 0.1379 (13.79%)! Enforce positivity bounds MAX(-1/lambda_H, -1/lambda_A) <= rho <= MIN(1/(lambda_H*lambda_A), 1) so no scoreline probability ever turns negative!

Feasibility-Guarded Dixon-Coles P(0-0) & P(1-1) Formula

Check all four tau positivity conditions at once: 1 - lambda_H*lambda_A*rho >= 0, 1 + lambda_H*rho >= 0, 1 + lambda_A*rho >= 0, and 1 - rho >= 0!

=LET(lm_h, B2, lm_a, B3, rh_d, B4, IF(OR(lm_h <= 0, lm_a <= 0, 1 - lm_h * lm_a * rh_d < 0, 1 + lm_h * rh_d < 0, 1 + lm_a * rh_d < 0, 1 - rh_d < 0), "Invalid Dixon-Coles Bound", ROUND(EXP(-(lm_h + lm_a)) * (1 - lm_h * lm_a * rh_d), 4)))

Dixon-Coles P(1-1) & Zero-Dependence Independence Check

Set rho = 0 in cell B4 to verify exact collapse back to independent Skellam/Poisson probabilities!

=IF(ISTEXT(E4), E4, ROUND(B2 * B3 * EXP(-(B2 + B3)) * (1 - B4), 4))

Further reading

• Arte Sheets Lesson — Colley Bias-Free Laplace Sports Rating Formula
• Arte Sheets Lesson — Elo Chess Rating Expected Score & K-Factor Update Formula