Multinomial Distribution
World Premier League Soccer Game
Outcomes
Multinomial Distribution
• Used to model a series of n independent trials, where each
trial has k possible outcomes (categories)
• The probability of ith category occurring on any given trial is
pi subject to p1 + p2 + … + pk = 1
• The random variable Yi denotes the number of trials in
which the ith category occurred: Y1 + Y2 + … + Yk = n
• Note that once we have observed Y1, Y2, …, Yk-1, we have
Yk = n - Y1 - … - Yk-1
similarly, pk = 1 - p1 - … - pk-1
Multinomial Distribution – Mathematical Form - I
n
n!
 k 
n
Multinomial Expansion:   ai    a1  a2  ...  ak  
a1m1 a2m2 ...akmk

m1  m2 ... mk  n m1 ! m2 !...mk !
 i 1 


In terms of the RVs: Y1 ,..., Yk 1 : P Y1  y1 , Y2  y2 ,..., Yk 1  yk 1   p  y1 , y2 ,... yk 1  

n!
n  y  y ... yk 1
p1y1 p2y2 ... pkyk 11 1  p1  p2  ...  pk 1  1 2
y1 ! y2 !... yk 1 ! n  y1  y2  ...  yk 1  !
Note that these probabilities sum to 1 over all possible outcomes where y1  y2  ...  yk  n


Moment Generating Function: m  t1 , t2 ..., tk 1   E et1Y1 ...tk 1Yk 1 
  et1 y1 ...tk 1 yk 1

n!
n  y  y ... yk 1
p1y1 p2y2 ... pkyk 11 1  p1  p2  ...  pk 1  1 2

y1 ! y2 !... yk 1 ! n  y1  y2  ...  yk 1 !

y1
n!
p1et1  ... pk 1etk 1

y1 ! y2 !... yk 1 ! yk !

yk 1

pkyk  p1et1  ...  pk 1etk 1  pk

n
Marginal Distribution of Y1 obtained by evaluating
m  t1 , 0,....0    p1et1  p2  ...  pk 1  pk    p1et1  1  p1  
n
n
 Yi ~ Bin  n, pi  i  1,..., k
Multinomial Distribution – Mathematical Form - II
Marginal Distribution of Y1 obtained by evaluating
m  t1 , 0,....0    p1et1  p2  ...  pk 1  pk    p1et1  1  p1  
n
n
 Yi ~ Bin  n, pi  i  1,..., k
 E Yi   npi V Yi   npi 1  pi 
Alternative Approach (also used to obtain Covariances) :
1 if Trial i is in Category j
Ui  
otherwise
0
E U i   1 p j   0 1  p j   p j
E U i2   12  p j   02 1  p j   p j
 V U j   p j   p j   p j 1  p j 
2
1 if Trial i is in Category j '
Wi  
otherwise
0
E Wi   p j ' V Wi   p j ' 1  p j ' 
Trials are independent  COV U i , U i '   COV U i ,Wi '   COV Wi ,Wi '   0 i  i '
n
Y j  U i
i 1
n
Y j '   Wi
i 1
 E Y j   np j V Y j   np j 1  p j 
 E Y j '   np j ' V Y j '   np j ' 1  p j ' 
Multinomial Distribution – Mathematical Form - III
Obtaining COV Y j , Y j ' 
j  j ':
1 if Trial i is in Category j
Ui  
otherwise
0
1 if Trial i is in Category j '
Wi  
otherwise
0
Note: U iWi  0 since only 1 category can occur on each trial 
E U iWi   0
 COV U i ,Wi   E U iWi   E U i  E Wi   0  p j p j '   p j p j '
Trials are independent 
COV U i ,U i '   COV U i ,Wi '   COV Wi ,Wi '   0 i  i '
n
n
 n
 n
COV Y j , Y j '   COV  U i ,  Wi    COV U i ,Wi    COV U i , Wi '  
i 1
i 1 i ' i
 i 1
 i 1
 n   p j p j '   np j p j '
Examples – World Premier Soccer Leagues
• Nations: England, France, Germany, Italy, Spain (2013)
• League Play: Each team plays all remaining teams twice
(once Home, once Away)
• Games can end in one of 3 possible ways with respect to
Home Team: Win, Draw (Tie), Lose
Win (1)
League
p1
England
0.4711
France
0.4395
Germany 0.4739
Italy
0.4763
Spain
0.4711
Draw (2) Lose (3)
p2
p3
0.2053
0.3237
0.2868
0.2737
0.2092
0.3170
0.2368
0.2868
0.2263
0.3026
Probability Calculations
All calculations based on assumption of their being an infinite population of games
Sample 8 Games from each league, probability of: 4 Wins, 2 Draws, 2 Losses:
P Y1  4, Y2  2, Y3  2   p  4, 2, 2 
8!
4
2
2
.4711 .2053 .3237   .0914
4!2!2!
8!
4
2
2
France: p  4, 2, 2  
.4395  .2868  .2737   .0966
4!2!2!
8!
4
2
2
Germany: p  4, 2, 2  
.4739  .2092  .3170   .0932
4!2!2!
8!
4
2
2
Italy: p  4, 2, 2  
.4763 .2368  .2868   .0997
4!2!2!
8!
4
2
2
Spain: p  4, 2, 2  
.4711 .2263 .3026   .0970
4!2!2!
England: p  4, 2, 2  
Distribution of Number of Points in Sample
Points are assigned such that a Win gets 3 Points, Draw gets 1, Loss gets 0
P  3Y1  1Y2  0Y3  3Y1  1Y2
E  P  3E Y1  E Y2   3np1  np2
V  P  32 V Y1  12 V Y2   2  31 COV Y1 , Y2   9np1 1  p1   np2 1  p2   6  np1 p2 
n  8  E  P  24 p1  8 p2 V P  72 p1 1  p1   8 p2 1  p2   48 p1 p2
England: E  P  24 .4711  8 .2053  12.95 V  P  14.60  P  3.82
France:
E  P  24 .4395   8 .2868   12.84 V  P  13.32  P  3.65
Germany: E  P  24 .4739   8 .2092   13.05 V  P  14.52  P  3.81
Italy:
E  P  24 .4763  8 .2368   13.33 V  P  13.99  P  3.74
Spain:
E  P  24 .4711  8 .2263  13.12 V  P  14.22  P  3.77