Poisson Modeling and Predicting English Premier League Goal

The New England Journal of Statistics in Data Science Volume 1, 95–101 (2023)

DOI: https://doi.org/10.51387/21-NEJSDS1

Scoring

Quang NGUYEN

Abstract

The English Premier League is well-known for being not only one of the most popular professional sports leagues in the

world, but also one of the toughest competitions to predict. The ﬁrst purpose of this research was to verify the consistency

between goal scoring in the English Premier League and the Poisson process; speciﬁcally, the relationships between the

number of goals scored in a match and the Poisson distribution, the time between goals throughout the course of a

season and the exponential distribution, and the time location of goals during football games and the continuous uniform

distribution. We found that the Poisson process and the three probability distributions accurately describe Premier League

goal scoring. In addition, Poisson regression was utilized to predict outcomes for a Premier League season, using diﬀerent

sets of season data and with a large number of simulations being involved. We examined and compared various soccer

metrics from our simulation results, including an English club’s chances of being the champions, ﬁnishing in the top four

and bottom three, and relegation points.

keywords and phrases: English Premier League, sports statistics, Poisson process, Poisson regression.

1. INTRODUCTION

Association football, also commonly known as soccer in

America, is undoubtedly the most widely-played sport in

the world. Often referred to as the “king of sport,” football

can be played almost anywhere, from grass ﬁelds to indoor

gyms, streets, parks, or beaches, due to the simplicity in its

principal rules and essential equipment. Europe is known

to be the birthplace of modern football [6], and the Euro-

pean soccer culture is unlike any other. The Old Continent

is home to numerous top-level professional football leagues,

and the English Premier League (EPL) distinguishes itself

because of its competition quality, overall balance, and pop-

ularity. Some of the best football coaches and players in the

world come together to compete for the prestigious Premier

League trophy.

The EPL was founded in 1992, and over the last three

decades, we have witnessed numerous memorable matches

and countless outstanding performances by clubs and their

players. The EPL is currently a competition of twenty En-

glish football clubs. At the end of each season, the bottom

three teams get relegated to the second-highest division of

English football, in exchange for three promoted teams. A

Premier League season usually takes place from mid-August

to mid-May. Each team gets to play every other team twice,

once at home and once on the road, hence there are a total

of thirty-eight ﬁxtures in a season for each team [8].

The most important aspect of the game of football is

indisputably scoring goals. Despite the signiﬁcance of other

factors like ball possessing or disciplined defending, we have

to admit that the main reason we pay to watch soccer is

to see the ball being put in the back of the net. The rule

is very simple: in order to win, you must score more than

your opponent. In the Premier League, each match happens

within the span of ninety minutes (plus stoppage time), and

the match consists of two 45-minutes halves. Each team can

get one of these three results after each match: a win, a draw,

or a loss. If there is a draw, the two clubs receive a point

apiece, and for non-drawing matches, the winner is rewarded

with three points and the losing team gets punished with

zero points. Thus the club with the most points at the end of

the year will have their hands on the exquisite EPL trophy,

and the total points also determines the fates of teams in

the relegation zone [8]. This makes every single match so

critical, as losing one single point could end up costing a

team’s chance of winning a title or remaining in the top tier

football league in England.

In this paper, we attempt to use statistical methods to

model and predict goal scoring and match results in the

Premier League. We will ﬁrst determine whether notable

aspects of goal scoring, namely, the number of goals scored,

the time between goals, and time location of goals in a

match, ﬁt the characteristics of a Poisson process. We will

then use Poisson regression to predict what would happen in

the 2018–19 EPL season, for instance, which clubs are more

likely to win the title or get relegated, using diﬀerent subsets

of data from prior seasons. The paper is outlined as follows:

We ﬁrst introduce the data that we used for our analyses

in Section 2. Next, our methodologies are described in Sec-

tion 3. We then spend section 4 on our two main topics of

96 Q. Nguyen

this research – using the Poisson process to model goal scor-

ing, and utilizing Poisson regression to predict the 2018–19

season outcomes. Lastly, in Section 5, we give a quick sum-

mary of our results as well as discuss possible future work

related to this research.

2. DATA

The ﬁrst dataset for our investigation simply consists of

match ﬁnal scores of all Premier League games from its inau-

gural competition, the 1992–93 season, to the last ﬁxture of

2018–19 season. The main attributes of this dataset are the

season, the home and away teams, and the number of goals

scored by each team. We rely on Football-Data.co.uk’s data

[4], which contains all Premier League match ﬁnal scores

from 1993 to 2019. Each season has its own data ﬁle, and

we read in and then join the individual datasets together

to get our desired data table. We utilize this data to model

the number of goals scored and then to make predictions of

the 2018–19 season, using three diﬀerent subset of seasons:

1) data from all seasons prior to 2018–19, 2) data from only

the 2010s, and 3) data from all seasons, but assigning more

weight to more recent competitions.

To obtain the data for the ﬁrst two season subsets, we

simply ﬁlter out the seasons that don’t belong to the year

ranges from the initial table. For the assigning weight sim-

ulation method, our weight allocation approach is very sim-

ple, as we let the weight number be equivalent to the number

of times the data for a particular season is duplicated. We

have decided that the previous ﬁve years before 2018–19 are

almost all that matter. Thus, every season from 1992–93 to

2012–13 are given weight 1, then the weight increases by 1

for each one of 2013–14, 2014–15, and 2015–16. After that,

we have the 2 most recent years left and we multiply the

weight by 2. Our weight values are depicted in Figure 1.

Finally, since we are also interested in examining goal scor-

ing time and the time between goals in the EPL, data on

these two topics for Manchester United, a well-known Pre-

mier League club, are collected and stored in a spreadsheet

by ourselves. This data set contains ﬁve columns: minutes,

which is the point of time during a match at which a goal

is scored; matchweek, which is the ﬁxture number of each

game; the stoppage time in minutes for both halves of each

game, and ﬁnally, the time between goals, which is the dif-

ference in minutes between the scoring events, taking into

account stoppage time. We collected these ﬁve variables for

all Manchester United goals during their 2018–19 Premier

League campaign [9]. In addition to the initial attributes,

we created a new variable for normalizing the goal scoring

minutes by dividing each one of them by the total minutes of

their respective game. The reason for re-scaling the minutes

is because the match total time varies, since we also take

into account stoppage time of football games, which means

some matches take place for a longer time than others.

Figure 1: Weight values across seasons from 1992–93 to

2017–18 for predicting 2018–19 season outcomes.

3. METHODOLOGY

3.1 Goal Scoring and the Poisson Process

The Poisson process [3] is a stochastic process used to

model the occurrence of phenomena over a continuous in-

terval, which in most cases represents time. There are several

characteristics of the Poisson process that can be observed,

including, the number of events happening in a given time

period; the time between those events; and when (at what

point of time) the events occur. Playing a huge role in the

Poisson process is the Poisson distribution [1], which deals

with the number of occurrences of an event in a ﬁxed period

of time, with a rate of occurrence parameter λ. Named for

French mathematician Simèon Denis Poisson, the Poisson

distribution is a discrete probability distribution that ex-

presses the number of occurrences of an event over a given

period of time. The probability density function of a Poisson

random variable X with parameter λ is given by

(x)=

−λ

; x =0, 1, 2,... and λ>0, (3.1)

where X represents the number of occurrences of an event

in a given unit time period, and λ is the constant rate of

occurrence per time period. The mean and variance of our

Poisson random variable X, denoted by μ

and σ

respec-

tively, are

= λ and σ

= λ. (3.2)

Another key distribution in this process is the exponential

distribution [1], which has a strong connection with the Pois-

son distribution, in that if the number of occurrences per

interval of time are illustrated by Poisson, then the descrip-

tion of the length of time between occurrences are provided

Poisson Modeling and Predicting English Premier League Goal Scoring 97

by the exponential distribution. If we have a non-negative

random variable X that is the time until the next occur-

rence in a Poisson process, then X follows an exponential

distribution with probability density function

(x)=λe

−λx

−

; x ≥ 0, (3.3)

where λ represents the average rate of occurrence and β is

the average time between occurrences. The mean and vari-

ance of an exponentially distributed random variable X are

= β and σ

= β

. (3.4)

Furthermore, there is a connection between Poisson and an-

other famous probability distribution – the continuous uni-

form distribution [1]. If a Poisson process contains a ﬁnite

number of events in a given time interval, then the unordered

times, or locations, or positions, or points of time at which

those events happen are uniformly distributed on that con-

tinuous interval. The continuous uniform distribution is a

probability distribution with equally likely outcomes, mean-

ing that its probability density is the same at each point in

an interval [A, B]. A continuous random variable X is uni-

formly distributed on [A, B] if its probability density func-

tion is deﬁned by

(x)=

B − A

; A ≤ x ≤ B. (3.5)

In addition, X has mean and variance

A + B

and σ

(B − A)

. (3.6)

We postulate that goal scoring in football can be modeled

by a Poisson process. According to the characteristics de-

scribed above, if goal scoring for a club happens at a certain

rate in a given time period, then a Poisson distribution can

be used to model the number of goals scored. Additionally,

the waiting time (in minutes) between successive goals can

be described using an exponential distribution. Moreover,

the time positions (or “minute marks”) in a game at which

scoring events transpire may be uniformly distributed. We

will explore these relationships in more detail in Section 4.

3.2 Simulating and Predicting Season

Outcomes Using Poisson Regression

Our second goal of this research is to use the method of

Poisson regression to predict the outcomes for EPL matches.

Poisson regression is a member of a broad class of models

known as the Generalized Linear Models (GLM) [5]. A gen-

eralized linear model has the general form

E(Y

)=μ

= g

−1

(β

+ β

+ ···+β

). (3.7)

There are three main components to a generalized linear

model:

1. A random component, indicating the conditional dis-

tribution of the response variable Y

(for the ith of n in-

dependently sampled observations), given the values of the

explanatory variables. Y

’s distribution must be a member of

an exponential family, such as Gaussian, Binomial, Poisson,

or Gamma.

2. A linear predictor (β

+ β

+ ···+ β

which is a linear combination of the predictors (the X’s),

with the β’s as the regression coeﬃcients to be estimated.

3. A canonical link function g(·), which transforms the

expected value of the response variable, E(Y

)=μ

,tothe

linear predictor.

Poisson regression models are generalized linear models

with the natural logarithm as the link function. It is used

when our response’s data type is a count, which is appro-

priate for our case since our count variable is the number

of goals scored. The model assumes that the observed out-

come variable follows a Poisson distribution and attempts

to ﬁt the mean parameter to a linear model of explanatory

variables. The general form of a Poisson regression model is

ln(μ

)=β

+ β

+ ···+ β

. (3.8)

To make predictions for Premier League matches and to de-

termine what would happen in the 2018–19 season using

Poisson regression, we ﬁtted two models to get the scoring

rates for every EPL team, 1) at home, and 2) away from

home. Here we are interested in evaluating the model equa-

tion at diﬀerent values of the explanatory variables. Since

the link function for Poisson regression is the natural log

function, we would back-transform the equation with the

corresponding exponential function. This will then give us

the home and away mean (expected) scoring rates for every

EPL club, aggregated across all opponents.

After that, we executed a large number of simulations, to

get the hypothetical 2018–19 season results and then ana-

lyzed and compared what we got for each of the three subsets

of season mentioned in the previous section. For each sub-

set of data, we performed 10000 simulations, and this was

accomplished by randomly generating the match ﬁnal score

for every team matchup, using the clubs’ average scoring

rates that we got from ﬁtting the Poisson regression mod-

els, which returns a random integer for each team’s number

of goals scored. In addition, the number of points for every

match outcome based on the teams’ number of goals scored

were also calculated (see Table 1), as a side gets 3 points

if they score more than their opponent, 1 point if the ﬁ-

nal score is a tie, and 0 points if the opposing roster has

more goals. For each simulated season (out of 10000 total

for each method), we tallied up the points, calculated the

goal diﬀerentials, and obtained the ﬁnal standings for EPL

clubs (see Table 2). From this information, we kept track of

various metrics for EPL clubs and utilized them to evaluate

and compare the models and their predictions, which will

be discussed in the next section.

98 Q. Nguyen

Table 1. Simulation table of 2018–19 EPL matches. The two columns HomeScore and AwayScore indicate the number of

goals scored by the home and away clubs, randomly generated from their average scoring rates. The points each team receives

versus their opponent (3 for a win, 1 for a draw, 0 for a loss) are determined from the match ﬁnal score.

HomeTeam HomeRate AwayTeam AwayRate HomeScore AwayScore HomePoints AwayPoints

Newcastle 1.667 Arsenal 1.510 1 3 0 3

Bournemouth 1.474 Southampton 0.989 0 0 1 1

West Ham 1.429 Brighton 0.526 1 0 3 0

Fulham 1.413 Tottenham 1.256 2 3 0 3

Cardiﬀ 1.053 Leicester 1.148 0 2 0 3

Table 2. Final standings for a simulated 2018–19 season. The

team ranks are arranged by total points, followed by goal

diﬀerential (goal scored minus goal conceded).

Rank Team Played Points GD

1 Man United 38 77 30

2 Chelsea 38 73 27

3 Liverpool 38 69 20

4 Arsenal 38 68 26

5 Leicester 38 62 9

6 Man City 38 61 10

7 Newcastle 38 56 5

8 Everton 38 56 2

9 Bournemouth 38 55 −1

10 Southampton 38 53 0

11 West Ham 38 53 −3

12 Tottenham 38 51 −2

13 Wolves 38 48 −5

14 Burnley 38 48 −6

15 Cardiﬀ 38 47 −10

16 Fulham 38 40 −9

17 Huddersﬁeld 38 39 −33

18 Watford 38 38 −14

19 Crystal Palace 38 36 −19

20 Brighton 38 24 −27

4. RESULTS

4.1 Goal Scoring and the Poisson Process

For our ﬁrst analysis of the relationship between the num-

ber of goals scored and the Poisson distribution, we used

Manchester United (MU) as our case of inspection. Our

question here was “Does MU’s distribution of number of

goals scored follow a Poisson distribution?” Table 3 and

Figure 2 are numerical and visual summaries of Manchester

United number of goals scored in every EPL season until the

ﬁnal ﬁxture of their 2018–19 campaign. To examine MU’s

number of goals scored closer, we divided the goals vari-

able into four levels: 0, 1, 2, 3, and 4 or more goals; and

also tally up the number of matches in real life from 1992

to 2018 having those goal values. From there, we obtained

the probability of Manchester United scoring each individ-

ual goal value, using the probability density function of a

Poisson distribution with mean λ =1.916 (see Table 3);

Table 3. Descriptive statistics of Manchester United’s number

of goals scored.

min Q1 median Q3 max mean sd n

0 1 2 3 9 1.916 1.405 1038

Figure 2: Histogram of Manchester United’s goals scored.

and then multiplied the probabilities with the total number

of matches of n = 1038 (see Table 3), to get the expected

number of matches for each number of goals category. The

counts for the observed and expected number of matches

are very close to one another, as illustrated by Table 4 and

Figure 3, which is a signiﬁcant validation for us. The re-

sults from a Chi-square goodness-of-ﬁt test (χ

=0.381,

p-value =0.984) further conﬁrm that there is no signiﬁcant

diﬀerence between the data’s and the empirical distribution,

hence the number of goals scored by Manchester United is

consistent with a Poisson distribution.

We were also interested in verifying the connections be-

tween the time between goals in a season and the exponen-

tial distribution, and the re-scaled goal scoring minutes in a

match and the standard uniform distribution. We continued

to use Manchester United to investigate these topics and ex-

plore their goal scoring time data described in Section 2.We

ﬁrst plotted the cumulative distribution curve of our data

Poisson Modeling and Predicting English Premier League Goal Scoring 99

Table 4. Observed and expected frequencies of the number of

matches for each goal value, alongside their Poisson

probabilities.

Goals Probability Observed Expected

0 0.147 158 153

1 0.282 286 293

2 0.270 282 280

3 0.173 178 180

4 or more 0.128 134 133

Figure 3: Side-by-side bar graph comparing the observed

and expected matches.

and compare them with the reference distributions (expo-

nential and uniform), as shown in Figures 4 and 5.Aswe

can see, the diﬀerences between the curves in each ﬁgure are

marginal, which is a validation and gives us more reason to

believe in our claim that our goal scoring data are consistent

with the speciﬁed distributions. After that, we conducted a

Kolmogorov-Smirnov goodness-of-ﬁt test to conﬁrm the ﬁt

between our data and the hypothetical distributions – expo-

nential and uniform. Based on the test statistics and p-values

(D =0.089, p-value =0.679 for Time between goals vs. Ex-

ponential and D =0.085, p-value =0.731 for Time location

of goals vs. Uniform), there is insuﬃcient evidence to sup-

port a conclusion that our data for the time between goals

and minutes of scoring are not consistent with the exponen-

tial and uniform distributions. Therefore, the time between

goals and re-scaled scoring time for Manchester United in

the 2018–19 EPL season are exponentially and uniformly

distributed, respectively.

4.2 Simulating and Predicting Season

Outcomes Using Poisson Regression

In this section, we discuss the results of our Poisson

regression models and simulations described in Section 3.

After ﬁtting the models and conducting the simulations,

Figure 4: Cumulative distribution curves of time between

goals and the exponential distribution.

Figure 5: Cumulative distribution curves of the re-scaled

goal scoring minutes and the standard uniform distribution.

there are various analyses we could do, from comparing how

teams’ chances of winning diﬀer across the three subsets of

data used for simulation, to analyzing which method gives

us the most consistent predictions compared to the actual

outcomes. In this paper, we focus on the title race and the

relegation battle in the Premier League, namely, the chances

of winning the league title and getting relegated for EPL

teams.

We ﬁrst look at the chances of ﬁnishing ﬁrst in the EPL

table at the end of the 2018–19 season for the “Big 6” in En-

glish football, which includes Manchester United, Liverpool,

Arsenal, Chelsea, Manchester City, and Tottenham (see Ta-

ble 5). Overall, it is clear that MU’s chance of having higher

ranks decreases drastically if we prioritize recent data over

just using data from all seasons. This is consistent with the

club’s situation in real life, as MU has not been doing well

lately, but was winning a lot prior to the 2010s. Their in-

town rivals, Manchester City, on the other hand, have much

higher percentages of winning the league in 2018–19 if we

focus on data from recent years. In fact, they were the cham-

pions of the 2018–19 EPL season. Similar to Manchester

100 Q. Nguyen

Table 5. Chances of winning the 2018–19 Premier League

title for the Big 6.

Team All Seasons 2010s Assign Weight

Arsenal 19.68 15.05 14.07

Chelsea 14.28 11.70 9.03

Liverpool 12.50 10.96 17.61

Man City 7.00 41.71 38.09

Man United 36.99 10.47 10.53

Tottenham 3.91 7.38 8.34

Table 6. Chances of getting relegated after the 2018–19

season for Premier League teams.

Team All Seasons 2010s Assign Weight

Huddersﬁeld 69.15 72.91 71.03

Cardiﬀ 49.99 54.38 53.47

Brighton 41.65 45.03 44.03

Burnley 31.60 43.38 33.34

Watford 26.28 17.24 15.82

Wolves 22.04 14.22 22.32

Crystal Palace 17.54 14.42 10.65

Fulham 9.87 7.91 11.74

West Ham 7.46 6.73 6.39

Southampton 6.49 5.23 11.65

Leicester 5.53 1.75 2.39

Bournemouth 4.76 5.46 5.80

Everton 3.48 2.17 3.63

Newcastle 2.40 8.78 7.41

Tottenham 1.02 0.16 0.08

Man City 0.38 0 0

Chelsea 0.15 0.09 0.13

Arsenal 0.10 0.04 0.06

Liverpool 0.10 0.05 0.02

Man United 0.01 0.05 0.04

United, our results for Manchester City also make sense,

due to the fact that City has emerged into an EPL title

contender over the last decade, but they were just an aver-

age team dating back to the old 1990s and 2000s days. It is

also notable that both assigning weight to more recent years

and using only data from the 2010s lower Chelsea and Arse-

nal’s likelihood of ﬁnishing ﬁrst, whereas these two methods

increase Tottenham’s chance of winning the league, though

their winning chance is still the smallest in each category.

Next, we investigate the likelihood of getting relegated

after the 2018–19 season for EPL clubs. The relegation zone,

or the last three places in the ﬁnal rankings, is where no

teams in the Premier League want to end up at the end

of the season, because after each season, the bottom three

clubs get relegated to the second highest division of English

football. The results from our three subsets of season data

(see Table 6) are pretty consistent with each other, with

Huddersﬁeld and Cardiﬀ essentially being the two “locks”

to play in the English football division below the EPL in

the following season. Brighton and Burnley also have high

Table 7. 40-point safety rule comparison between the three

subsets of season data. We tallied up the total number of

teams as well as distinct simulated seasons with teams being

relegated while having at least 40 points for each simulation

method.

Subset Seasons Teams

All Seasons 3434 4424

2010s 2066 2470

Assign Weight 2346 2889

chances of being in the bottom three, although in reality,

both of these teams successfully remained in the league for

another year. The third team that got dismissed in 2019,

Fulham, does not have high chances of relegation in any of

the three simulation methods. Unsurprisingly, Big 6 teams

have the smallest chances of getting related from the EPL

at the completion point of the 2018–19 season.

On a related note, the 40-point safety rule [2] is an inter-

esting myth associated with the EPL’s relegation zone. Since

the league’s reduction in the number of teams to twenty

clubs before the start of the 1995–96 season, there have been

only three times that a squad got relegated despite hitting

the 40-point mark. They are West Ham in 2002–03 with 42

points, and Bolton and Sunderland both with 40 points at

the end of 1997–98 and 1996–97 seasons respectively. This

mythical 40-point mark has been crucial for the relegation

battle for many years, as subpar teams often view getting

there as their “security blanket” for remaining in the top

division of English football. From Table 7, using data from

2010s and assigning weight to recent years give us signiﬁ-

cantly less number of both teams and seasons that violate

the 40-point rule than using data from every season prior to

2018–19. Therefore, the 40-point safety rule seems to hold

much better for the two simulations that focus on recent

data than the one with data from all previous seasons. This

actually makes sense, since in the past two decades in real-

ity, teams with 40 or more points at the end of EPL seasons

all survived from relegation, as the last time this rule did

not happen was the 1997–98 season.

5. CONCLUSION AND DISCUSSION

Overall, we have found that Premier League goal scoring

ﬁts the characteristics of a Poisson process. Our ﬁrst result

was that a Poisson distribution can be used to predict the

number of matches with each number of goals scored. Addi-

tionally, the time between each individual goal in a season

can be described by an exponential distribution. We also

have evidence that the normalized goal scoring time posi-

tions after are uniformly distributed. We also used diﬀerent

sets of data prior to the 2018–19 Premier League season,

namely, data from all seasons before, data from only the

2010s, and data from all previous years but assigning more

weight to recent competitions, to predict what would hap-

pen in the 2018–19 season. We got each team’s goal scoring

Poisson Modeling and Predicting English Premier League Goal Scoring 101

rate at home and away from home by doing Poisson re-

gression, and then performed simulations using those rate

parameters. Diﬀerent team metrics like how many points

each team got and what place each team ﬁnished were be-

ing kept track of from the simulations, and then we make

use of those variables to analyze and compare our models of

diﬀerent season data subsets.

In the future, there are additional topics we could explore,

including:

1. Besides the number of goals scored, there are many

other factors that can be used to determine outcomes of

football matches. In future research, we could use various

factors to predict goal scoring and ﬁnd out if they will be

as helpful as using just number of goals. We could look into

variables that are likely to contribute to the outcomes of

Premier League football matches such as clean sheets, pos-

session time, pass accuracy, shots on target, and numerous

other soccer statistics. On top of that, we could compare dif-

ferent models with diﬀerent predictors and evaluate them to

ﬁnd out which set of variables best predicts league outcomes,

and then use them to simulate and predict match results.

2. In football and many other sports, team performance

tends to vary throughout a season and across seasons. Some

Premier League teams have the tendency of getting hot in

early months, some clubs reach their peak during the mid-

dle period of the season, and a few others are more likely to

do better at the season’s home stretch. Winning and losing

streaks are also important factors in sports, as some clubs

are streaky, while others tend to be more consistent. Thus,

in future research, we could apply match results of EPL

teams from past games within the season, and maybe ﬁnd a

way to emphasize winning and losing streaks, to predict the

outcome of later matches. As a follow up, we could inves-

tigate a model’s performance throughout the season. Some

models may work better and predict more accurate results

at certain times in the year than others.

3. In addition to predicting match results, another pop-

ular application of statistical modeling in sports analytics

is determining betting odds. We could use the probabili-

ties from our Poisson regression models and simulations to

calculate the odds of possible game outcomes for diﬀerent

team matchups. We could also look into and compare diﬀer-

ent types of bets such as over and under, money line wager,

or point spread; determining if it is a good idea to bet on a

match, and if so, how much proﬁt we could win.

ACKNOWLEDGMENTS

This work was completed as the author’s senior honors

thesis, in partial fulﬁllment of the requirements for earning

Departmental Honors in Mathematics at Wittenberg Uni-

versity in Springﬁeld, Ohio. The author would like to ex-

press his special gratitude and thanks to his advisor, Pro-

fessor Douglas M. Andrews, for his many ideas, suggestions

and guidance throughout the research process. The author

would also like to thank the Department of Mathematics

and Computer Science at Wittenberg University for provid-

ing him the valuable knowledge during his undergraduate

career, and for giving him the opportunity to participate in

the Departmental Honors Program.

SUPPLEMENTARY MATERIAL

All of the materials related to this research are available

on GitHub at https://github.com/qntkhvn/eplgoals.

Accepted 12 July 2021

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Quang Nguyen. Department of Mathematics and Statistics,

Loyola University Chicago, Chicago, Illinois, USA.

E-mail address: [email protected]