Poisson Distribution

Goals in English Premier Football League – 2006/2007 Regular

Season

Poisson Distribution• Distribution often used to model the number of

incidences of some characteristic in time or space:– Arrivals of customers in a queue– Numbers of flaws in a roll of fabric– Number of typos per page of text.

• Distribution obtained as follows:– Break down the “area” into many small “pieces” (n pieces)– Each “piece” can have only 0 or 1 occurrences (p=P(1))– Let =np ≡ Average number of occurrences over “area”– Y ≡ # occurrences in “area” is sum of 0s & 1s over “pieces”– Y ~ Bin(n,p) with p = /n– Take limit of Binomial Distribution as n with p = /n

Poisson Distribution - Derivation

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Poisson Distribution - Expectations

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Example – English Premier League

• Total Goals Per Game (Both Teams)– Mean=2.47 Variance=2.49

• Goals by Team by Half– Home Team, 1st Half: Mean=0.68 Variance=0.73– Road Team, 1st Half: Mean=0.44 Variance=0.39– Home Team, 2nd Half: Mean=0.77 Variance=0.75– Road Team, 2nd Half: Mean=0.58 Variance=0.83*

*Does not reject based on Goodness-of-Fit test

Goals by Team by Half

Goals All Home1 Road1 Home2 Road20 828 199 236 175 2181 492 121 122 134 1152 157 46 21 56 343 31 11 0 12 84 9 3 1 3 25+ 0 0 0 0 0

Observed Counts

Goals All Home1 Road1 Home2 Road20 818.97 192.72 244.22 175.30 212.991 506.47 130.84 107.97 135.63 123.312 156.60 44.42 23.87 52.47 35.693 32.28 10.05 3.52 13.53 6.894 4.99 1.71 0.39 2.62 1.005+ 0.69 0.26 0.04 0.46 0.13

Expected Counts Under Poisson Model

Goodness of Fit Tests (Lumping 3 and More Together for Team Halves)

Goals Home1 Road1 Home2 Road20 0.2048 0.2766 0.0005 0.11811 0.7407 1.8229 0.0195 0.55972 0.0563 0.3444 0.2381 0.08043+ 0.3263 2.1967 0.1563 0.4928

Chi-Square 1.3282 4.6407 0.4144 1.2509P-value 0.7225 0.2001 0.9373 0.7408

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Correlations Among Goals Scored

Correlations Home1 Road1 Home2 Road2Home1 1.0000 0.0491 0.0262 -0.0587Road1 0.0491 1.0000 -0.0388 -0.0475Home2 0.0262 -0.0388 1.0000 -0.0771Road2 -0.0587 -0.0475 -0.0771 1.0000

t-test (r=0) Home1 Road1 Home2 Road2Home1 #N/A 1.0047 0.5239 -1.0774Road1 1.0047 #N/A -0.7259 -0.8808Home2 0.5239 -0.7259 #N/A -1.3910Road2 -1.0774 -0.8808 -1.3910 #N/A

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