Shark attacks and the Poisson approximationByron Schmuland.docx

Shark attacks and the Poisson approximation
Byron Schmuland
A story with the dramatic title “Shark attacks attributed to
random
Poisson burst” appeared on September 7, 2001 in the National
Post news-
paper. In this story, Professor David Kelton of Penn State
University used
a statistical model to try to explain away the surprising number
of shark at-
tacks that occurred in Florida last summer. According to this
article, Kelton
thinks the explanation for the spate of attacks may have nothing
to do with
changing currents, dwindling food supplies, the recent rise in
shark-feeding
tourist operations, or any other external cause.
“Just because you see events happening in a rash like this does
not imply
that there’s some physical driver causing them to happen. It is
characteristic
of random processes that they exhibit this bursty behaviour,” he
said.
What is this professor trying to say? Can mathematics really
explain the
increase in shark attacks? And what are these mysterious
Poisson bursts?

The main point of the Professor Kelton’s comments is that
unpredictable
events, like shark attacks, do not occur at regular intervals as in
the first
diagram below, but tend to occur in clusters, as in the second
diagram below.
The unpredictable nature of these events means that there are
bound to be
periods with an above average number of events, as well as
periods with a
below average number events, or even no events at all.
Regular Events
* * * * * * * * * *
time −→
Random Events
* * * **** * **
time −→
The statistical model used to study such sequences of random
events gets
its name from the French mathematician Siméon Denis Poisson
(1781-1840),
who first wrote about the Poisson distribution in a book on law.
The Pois-
son distribution can be used to calculate the chance that a
particular time
period will exhibit an abnormally large number of events

(Poisson burst), or
that it will exhibit no events at all. Since Poisson’s time, this
distribution
has been applied to many different kinds of problems such as
decay of ra-
dioactive particles, ecological studies on wildlife populations,
traffic flow on
the internet, etc. Here is the marvellous formula that gives us
information
on random events:
The chance of exactly k events occurring is
λk
k!
× e−λ, for k = 0, 1, 2, . . .
The funny looking symbol λ is the Greek letter lambda, and it
stands for
“the average number of events”. The symbol k! = k × (k − 1) × ·
· · × 2 × 1
means the factorial of k, and e−λ is the exponential function ex
with the
value x = −λ plugged in. Let’s take this new formula out for a
spin.
Shark attack!
If, for example, we average two shark attacks per summer, then
the chance
of having six shark attacks next summer is obtained by plugging
λ = 2 and
k = 6 into the formula above; and this gives
probability of six attacks ≈ (26/6!) × e−2 = 0.01203,

which is a little over a 1% chance. This means that six shark
attacks are
quite unlikely in one year, though this would happen about once
every fifty
years. The chance that we go the whole summer without any
shark attacks
can also be calculated by plugging λ = 2 and k = 0 into the
formula. This
gives
probability of no attacks ≈ (20/0!) × e−2 = 0.13533,
which is a 13% chance. We expect a “sharkless summer” every
seven or eight
years.
In this hypothetical shark problem the number of attacks
followed the
Poisson distribution exactly. The Poisson distribution is most
often used
to find approximate probabilities in problems with n repeated
trials and
probability p of success. Let me show you what I mean.
Lotto 6-49
One of my favorite games to study is Lotto 6-49. Six numbers
are randomly
chosen from 1 to 49 and if you match all six numbers you win
the jackpot.
Since the number of possible ticket combinations is
(
49
6

)
= 13,983,816, your
chance of winning the jackpot with one ticket is one out of
13,983,816, which
is p = 7.15 × 10−8. Let’s say you are a regular Lotto 6-49 player
and that
you buy one ticket, twice a week, for 100 years. The total
number of tickets
you buy is n = 100 × 52 × 2 = 10,400. What is the chance that
you win a
jackpot sometime during this 100 year run?
Now this is a pretty complex problem, but the Poisson formula
makes it
simple. First of all, the average number of jackpots during this
time period
is λ = np = 10,400/13,983,816 = 0.0007437. Plugging this in
with k = 0
shows that the chance of a “jackpotless 100 years” is
probability of no jackpots ≈ e−0.0007437 = 0.99926.
Wow! Even if you play Lotto 6-49 religiously for 100 years,
there is a better
than 99.9% chance that you never, ever win the jackpot.
Coincidences
Take two decks of cards and shuffle both of them thoroughly.
Give one deck
to a friend and place both your decks face down. Now, at the
same time,
you and your friend turn over your top card. Are they the same
card? No?

Then try again with the second card, the third card, etc. If you
go through
the whole deck what is the chance that, at some point, you and
your friend
turn over the same card?
In this problem there are n = 52 trials and the chance of a
success
(coincidence) on each trial is p = 1/52. The average number of
coincidences
is λ = np = 52/52 = 1, and so putting k = 0 in the Poisson
formula gives
probability of no coincidences ≈ e−1 = 0.36788.
The chance that you do see a coincidence is 1 − 0.36788 =
0.63212. You will
get a coincidence about 63% of the times you play this game.
Try it and see!
Birthday problem
Suppose there are N people in your class. What are the odds that
at least
two people share a birthday? Imagine moving around the class
checking
every pair of people to see if they share a birthday. The number
of trials is
equal to the number of pairs of people, i.e., n =
(
N
2

)
= N(N − 1)/2. The
probability of success on a given trial is the chance that two
randomly chosen
people share a birthday, i.e., p = 1/365. This gives the average
number of
shared birthdays to be λ = N(N −1)/2(365), so the probability of
“no shared
birthdays” is
probability of no shared birthdays ≈ e−N(N−1)/2(365),
and so the probability of at least one shared birthdays is
approximately
1 − e−N(N−1)/2(365). Here’s what happens when you try
different values of N
in this expression.
Probability of a shared birthday
N Prob N Prob
10 0.115991 60 0.992166
20 0.405805 70 0.998662
30 0.696320 80 0.999826
40 0.881990 90 0.999983
50 0.965131 100 0.999999
With N = 10 people there is only about an 11.5% chance of a
shared
birthday, but with N = 30 people there is a 69.6% chance. In a
large class
(like at university!) with N = 100 students, a shared birthday is
99.9999%
certain.

For large class sizes maybe it is possible to have a triple
birthday. Follow-
ing the same pattern as before, we work out the chances that
there is at least
one triple shared birthday in a class of N people. This time you
go through
the class checking each triple of people, there are
(
N
3
)
= N(N − 1)(N − 2)/6
trials, and the chance of success on each trial is p = 1/(365)2.
This gives λ = N(N − 1)(N − 2)/6(365)2, so the probability of
“no triple
shared birthdays” is
probability of no triple shared birthdays ≈
e−N(N−1)(N−2)/6(365)2,
and so the probability of at least one triple is 1 −
e−N(N−1)(N−2)/6(365)2. Let’s
look at different values of N for this formula.
Probability of a triple shared birthday
N Prob N Prob
10 0.000900 60 0.226522

20 0.008520 70 0.336936
30 0.030015 80 0.460278
40 0.071477 90 0.585970
50 0.136809 100 0.702915
My large first year statistics courses usually have about 100
students in
them, and I always check their birthdays. According to the
table, there
should be a triple birthday over 70% of the time. It really is
true; there is
usually a triple birthday in those classes.
The Great One
During Wayne Gretzky’s days as an Edmonton Oiler, he scored
a remark-
able 1669 points in 696 games, for a rate of λ = 1669/696 = 2.39
points per
game. From the Poisson formula with k = 0 we estimate that the
probability
of Gretzky having a “pointless game” is
probability of no points ≈ (2.39)0/0! e−2.39 = 0.0909.
Over 696 games, this ought to translate to about 696 × 0.0909 =
63.27
pointless games. In fact, during that period he had exactly 69
games with
no points.
Now for one point games we find an approximate probability of
probability of one point ≈ (2.39)1/1! e−2.39 = 0.2180,
for a predicted value of 696 × 0.2180 = 151.71 one point games.

Let’s try
the same calculation for other values of k, and compare what
the Poisson
formula predicts to what actually happened.
Points Actual # Games # Predicted by Poisson
0 69 63.27
1 155 151.71
2 171 181.90
3 143 145.40
4 79 87.17
5 57 41.81
6 14 16.71
7 6 5.72
8 2 1.72
9 0 0.46
As you see there is a remarkable agreement between the
predictions based
on the Poisson formula, and the actual number of games with
different point
totals. This shows that Gretzky was not only a high scoring
player, but a
consistent one as well. The occasional pointless game, or
occasional “Poisson
burst” in seven or eight point games were not due to
inconsistent play, but
are exactly what is expected in any random sequence of events.
Another
reason why he really was the great one!

2
Job Satisfaction-What is it and Why is it Important in
Organizational Behavior
University of Arkansas at Little Rock (UALR)
December 04, 2017
Introduction
Organizational behavior is an important factor in management
that is successful can increase the production of the business.
Several goals for organizational behavior includes; organization
of resources, analyzing of behavior, and motivating of
productivity. Company culture involves studying the micro and
macro organization. “Good management was based on good
discipline, specific and detailed job descriptions” (Wren, 2005
p.84). Some principles of management include; a proper
division of responsibilities, authority to carry out
responsibilities, promptness in reporting if responsibilities are
carried out so if necessary corrections can be made, information
gathered daily on reports and checks and a system that will
allow the prompt reporting of errors and the areas of
delinquency (Wren, 2005). To understand job satisfaction it is
important to understand job context. Hygienic factors include
positive attitude, motivation, include challenging work and an
increase in job responsibilities. The traditional assumption of
motivation did not increase productivity and job satisfaction,
only the motivators led to positive work environment results”
(Wren, 2005, p.439).Expectancy theory is complex and asks the
question-what is on the workers mind. Many other theories
attempt to explain job satisfaction. They are; goal-setting theory
and social learning theory.
World Services for the Blind in Little Rock Arkansas offers a
variety of services and programs to help the visually impaired
(legally blind) and blind. Their approach is to offer many types

of different services with the ultimate goal of strong, healthy
independent living. A team of professionals provides an
evaluation plan that will meet the specific needs of their clients.
Some programs include; training and counseling, independent
living skills, counseling individually and family, activates of
daily living programs, communication, orientation and mobility
with learning was to access transportation, adaptative aids and
leisure activities like community group activities (World
Services for the Blind, n.d.).
Job Satisfaction refers to the level of how well a job is done.
When an employee experiences job satisfaction it refers to the
degree in which an employee feels positively or negatively
about their job (Chinn, Dane, n.d.). Some theories of job
satisfaction include content theories, process theories, and
situational theories. Content theories are concerned with human
needs, growth and self actualization. Maslow’s hierarchy of
needs explains this theory on the five tier model of human
needs. In Maslow’s hierarchy of needs must be met before
moving up to higher needs. Maslow believed that needs exist in
a logical order for the individual. Process theories attempt to
explain job satisfaction by examining expectations and valves.
This theory believes workers select behaviors according to their
needs. Situational theories believe job satisfaction is achieved
by situational characteristics and situational occurrences.
Situational theories include situational characteristics like
working conditions and situational occurrences like extra
vacation time. Situational charactertic happen before accepting
the job and situational occurrences seem to happen after taking
a job (Chinn, n.d.).
Job satisfaction is measured by surveys and direct observation.
Popular questionnaires included the Minnesota Satisfaction
Questionnaire (MSQ) and the jobs description index (JDI). The
MSQ measures working conditions, chances for advancement,
freedom to use one’s own judgment and praises for doing a
good job (Chenn, n.d.). A five point scale from very satisfied to
very dissatisfied is used. The JDI looks at five areas of

measurement. These include the work itself, quality of
supervision, relationships with coworkers, promotion
opportunities and pay scale (Chenn, n.d.).
Employee morale is critical for success of company. This is an
important factor in job satisfaction. There seems to be a
connection between job satisfaction and organizational
performance and with worker behavior and job satisfaction. The
relationships of employees with the company affect
productivity, the quality of products and services of employee’s
turnover and eventually to the success of the organization
(Dunn, n.d.).
Organizational behavior is known as organization citizenship
behavior (OCB) or organizational commitment. OCB is the
employee behavior that benefits the organization. Parts of OCB
include; altruism, courtesy, conscientiousness, civic virtue,
sportsmanship, peace keying and cheerleading behaviors (Job
Satisfaction Can Affect a Person’s Level of Commitment to the
Organization, Absenteeism and Job Turnover, n.d.).There seems
to be three forms of employee commitment. They are; affective
or emotional, normative or obligation and continuance (Job
Satisfaction Can Affect a Person’s Level of Commitment to the
Organization, Absenteeism and Job Turnover, n.d.). When
people are satisfied with the work they are doing what
influences a more positive mood and higher levels of passions,
Job satisfaction can be improved by management strategies. The
managers are responsible for discovering what motivates
employees and creating a positive work environment. Job
turnover will be less with job satisfaction. When employees
have clear job descriptions they understand their employees
roles better which can lead to becoming more productive in
their work environment.
Three main factors are important to an employee experiencing
job satisfaction. These include the ability to complete required
task, the level of communication in an organization and the way
management treats employees measuring job performance. One
influence on job satisfaction appears to be the communication

and relationship between supervisors and their direct
subordinates. Job satisfaction seems to be correlated with life
satisfaction. With positive job satisfaction there is a decrease in
absenteeism and turnover rates (Defining Job Satisfaction, n.d.).
Communication demands on the employee in their work
environment affect job satisfaction. The employee needs to have
adequate training on their jobs. If an employee receives too
many messages at the same time, the employee is more likely to
become dissatisfied with the work environment. This can cause
the employee to have lower levels of job satisfaction (Defying
Job Satisfaction, n.d.).
Literature Review
Does seeing “Eye to Eye” Affect Work Engagement and
Organizational Citizenship Behavior? A role Theory Perspective
on LMX agreement discusses the effects of leader member
exchange (LMX) quality has meaningful effects on employee
motivation and behavior. This article discusses how LMX
theory leadership develops relationships with subordinates from
high-quality socio-emotional relationships with some
subordinates to low-quality transaction relationship with others.
The quality of these relationships is associated with critical
outcomes including work attitudes job performance and
retention. This article stresses the importance of developing, to
the extent possible, high-quality relationships with multiple
subordinates (Matta, Scott, Koopman, Conlon, 2015). There is
not really a gap in this research, just need to further research
the topic.
Good Citizen Interrupted: Calibrating a Temporal Theory of
Citizenship Behavior discusses the effects of organizational
citizenship behavior (OCB). The four main effects are lag, rate
of change, magnitude and permanence. One concept of
discussion is the meaning of being a good citizen. These are
employee who has high levels of OCB. These researchers
believe employees perform OCB because of factors like
personality traits, including agreeableness, prosaically

personality (Methot, Lepak, Shipp, & Bowell, 2017). It is
believed employees who are good citizens show pride in their
work and commitment to the organization. It seems employees
who perform OCB are responsible for goodwill and social
capital and this is a form of competitive advantage. There are
unexpected cognitive processes that promote OCB. The
unexplained changes cause changes to employee’s behavior
(Methot, Lepak, Shipp, & Bowell, 2017).
From Good Soldiers to Psychologically Entitled: Examine When
and Why Citizenship Behavior Leads to Deviance discusses that
some employees: engage in OCB’s not because they want to, but
because they feel they have to, and it is not clear whether
OCB’s performed for external motives have the same positive
effects on individuals and organizational functioning as do
traditional OCB’s” (Yam, Kluz, He, & Reynolds, 2017, p.373).
This article discusses self-determined theory (SDT) as people
engaging in motivational behaviors like OCB’s due to either
autonomous or controlled motives. Moral Licensing Theory and
entitlement is produced when “individuals engage in good
deeds, moral licensing produces a sense of entitlement to some
more laxity” (Yan, Klotz, He, & Reynolds, 2017, p.376). It is
believed that moral licensing theory can lead to externally
motivated OCB’s of interpersonal and organizational deviance.
The social exchange theory believes “if employees do not feel
fully compensated they may retaliate against their organization”
(Yam, Klotz, He, & Reynolds, 2017, p. 377). This article
focused on using moral licensing and self-determination
theories to explain the attitudinal and behavioral consequences
of externally motivated OCB’s.
Linking Passion to Organizational Citizenship Behavior and
Employees, Performance: the Mediating Role of Work
Engagement discusses how employees that are passionate about
their work are “more effective in both task and non-task related
performance and in creating a conductive environment for
efficient and affective functioning of the organizations”
(Ahmad, Hameed, Mahmood, 2016, p.316). The article discusses

two kinds of passion. They are obsessive passion and
harmonious passion. Harmonious passion is an independent
internalization of the activity into the person’s identity.
Obsessive passion “is a controlled internalization of the activity
into one’s identity” (Ahmad, Hameed, Mahmood, 2016, p.318).
Harmonious passion positively relates to work engagement and
employee’s OCB. It is believed harmonious passion is of great
importance in generating employees, work engagement and
OBC behaviors (Ahmad, Hameed, Mahmood, 2016).
The Effect of Employer Branding on Employee’s Organizational
Citizenship Behaviors discuses the positive relationship
between overall employer brand and organizational citizenships
employer brand discusses the unique parts of the company’s
employment offerings or environments, There are;
psychological, functional, and economic valves. Employee
brand can affect what new employees expect from an
organization. There are five factors to the citizenship. They are;
altruism, sportsmanship, conscientiousness, civic virtue and
courtesy, Favorable employer brand can lead to decreased
employment expenses due to better recruitment process.
Employer brand provides valve to employees who work for
certain companies “It is possible that employer attractiveness
and employer brand drive employees to engage in organizational
citizenship behaviors” (Gozukara, & Hatipglu, 2016, p. 249).
The Role of Work –Family Culture Personality Traits in
Organizational Citizenship Behavior (OCB) at First level
Managerial Personnel discusses the role of work-family culture
and personality traits in OCB among first-level managers. There
seems to be a positive relationship between “a supportive work-
family culture and engaging in OCB’s” (Singh, Gupta, Karma,
Dubey, & Singh, 2017, p.60). This article found “individual
differences play an important role in predicting whether an
employee would exhibit OCB” (Singh, Gupta, Karma, & Dubey,
& Singh, 2017, p 60.).
Serving First For The Benefit of Others; Preliminary Evidence
for a Hierarchical Conceptualization of Servant Leadership

discusses the important of leaders becoming servants to others
first. It is understood that leadership is an important factor in
producing desired organizational outcomes like OCB with
servant leaderships. A leader whose first priority is to serve
others is seen. The leader puts others interest ahead of self
interest. “A servant leader’s dominant focus is on others is
likely to cultivate various extra-role behaviors” (Grisaffe,
Douglas, Van Meter, & Chonke, 2016, p.58).
Shared Values and Organizational Citizenship Behavior of
Generational Cohorts: A Review and Future Directions
discusses how important for “management to effectively handle
different generations who possess various valve systems”
(Yogamalar & Samuel, 2016, p.249). It is important to
understand generational diversity. “Values influence the work
behavior and direct their efforts toward organizational
citizenship behavior” (Yogamalar & Samuel, p. 250). This
article discusses how culture in India is different from culture
in the Western world. It is important to understand the values
and attitudes of different generations in employees help
managers increase their learning styles and potentials.
Differences in generational values can lead to conflict in the
workplace. They are at the center of culture in an organization.
Values are what is important to us they are very personal and
individualistic. Organizational values are the set of common
beliefs of the workplace concerning the ways of reaching goals.
“OCB is considered positive behavior and it moves toward
organizational well-being without external motivations or
formal reward system: (Yogamalar, & Samuel, 2016, p262).
This article discusses how culture in India is different from
work culture in the Western world. Understanding the values
and attitudes of different generations in employees helps
managers increase their learning styles and potentials.
Differences in generational values can lead to conflict in the
workplace. They are at the center of culture in an organization.
Valves are what is important to us they are very personal and
individualistic. Organizational valves are the set of common

beliefs of the workforce concerning the ways of reaching goals.
“OCB is considered positive behavior and it moves toward
organizational well-being without external motivators or formal
reward systems” (Yogamalar & Samuel, 2016, p.262).
Discussion
I chose this organization to research, because of the apparent
job satisfaction of the employees with a relaxed but professional
work culture. Some of my friends are employed here and really
enjoy their work. I ask them what the main reason for job
satisfaction was. They told me they liked their co-workers, felt
their bosses listened to their opinions; they had real chances of
affecting policy changes and their jobs a clear path to job
advancement. A workers level of job satisfaction impacts their
job performance.
Organizational behavior is an important factor in deciding if a
business is successful or not. Job satisfaction is an important
consideration in organizational behavior. Job satisfaction can be
influenced by several components. They are; a person’s ability
to complete job required task, level of communication in an
organization, and the way management treats employees.
Superior subordinate communication is an important element of
job satisfaction in the workplace. It seems job satisfaction falls
into two levels; affective job satisfaction factor and cognitive
job satisfaction. Emotions are involved in affective job
satisfaction. Cognitive job satisfaction is concerned with how
satisfied employees feel concerning some aspect of their job or
benefits (Defining Job Satisfaction, n.d.). When Human
Resources (HR) departments start encouraging positive
practices the results usually lead to financial gain for the
organization. It seems positive work environments and
increased shareholder valve are directly related (Defining Job
Satisfaction, n.d.). There are five factors used to measure and
influence job satisfaction. They are; pay or total compensation,
work responsibilities, promotion opportunities, relationships

with responsibilities, promotion opportunities relationship with
supervisor and interaction and working relationships with co-
workers (Defining Job Satisfaction, n.d.).
Communication concerns are demands felt by the employee on
the job. Verbal and non-verbal communication is very important
to an organizations success. These demands are called
communication load. Demands can be defined as “the rate and
complexity of communication inputs an individual must process
in a particular time frame (Refining Job Satisfaction, p.3). How
subordinates perceive supervisors behavior can positively or
negatively influence the job satisfaction employee feels. Non-
verbal communication like facial expressions, voice tone etc. is
more important than actual words said (Defining job
Satisfaction, n.d.).
Management may implement new job strategies to increase job
satisfaction. They are; create new challenges, mentor a
colleague, expand skills, learn from mistakes and stay positive
(Defining Job Satisfaction, n.d.). Management uses
questionnaires and surveys. Some are; The Job Satisfaction
Description Index (JDI), the Minnesota Satisfaction
Questionnaire (MSQ). The JDI, MSQ, and Faces Scale are
considered closed questionnaires because they provide various
answers. The NSQ is considered an open-ended questionnaire
(Defining Job Satisfaction, n.d.). The negative indicators of
organizational commitment include absenteeism, sabotage, and
violence (Defining Job Satisfaction, n.d.).
The development of job satisfaction researches has discovered
three common approaches. They are; job charactertic, social
information processing (organizational charactertic) and
dispositional (worker charactertic). Job characteristics are part
of a job that generates ideal conditions for higher levels of
motivation, satisfaction and performance. Five core job
charactertic include skill variety, task identity, task
significance, autonomy and feedback. Four personal and work
outcomes include “internal work motivation, growth
satisfaction, general satisfaction, and work effectiveness”

(Defining Job Satisfaction, n.d.). Job satisfaction differs
according to the perception of individual workers (Defining Job
Factors that lead to dissatisfaction include; poor pay, poor
consequences, poor work conditions, lack of promotions poor
benefits offered and lack of job security (Defining Job
Satisfaction, n.d.). The factors leading to job satisfaction
includes, good leadership practices good manager relationships,
recognition, advancement personal growth, feedback, and
support and clear direction and objectives (Defining Job
Factors that lead to dissatisfaction include; poor pay, poor
consequences, poor work conditions, lack of promotions, poor
benefits offered, and lack of job security (Defining Job
Satisfaction, n.d.). The factors leading to job satisfaction
includes good leadership practices, good manager relationships,
recognition, advancement, personal growth, feedback and
support and clear direction and objectives (Defining Job
Strategies, n.d.).
Ambition and desire seem to be some of the strongest predictors
for advancement. High levels of ambition resulting from high
standards can cause employees to be unhappy as a result of their
inability to be promoted in the company. General life
satisfaction can positively influence job satisfaction and vice
versa. Engaged employees seem to do more of the work.
Engaged employees are enthusiastic about the company they
work for. There are three possible factors that are responsible
for employee engagement. They are; vigor, dedication, and
absorption. Vigor refers to the amount of energy and effort an
individual will put forth to complete a task. Dedication refers to
the amount of overall significance a task carries. Finally,
absorption is the depth of work emersion employees feel
(Defining Job Strategies, n.d.).
Job satisfaction can be associated with performance,
absenteeism, and turnover. Happier workers are more

productive workers. However in times of high unemployment,
unhappy employees will perform choosing unsatisfied work
over employment. The more satisfied we are in life the more
satisfied we generally are in our jobs and less likely we are to
be absent from our jobs (Job Satisfaction Can Affect a Persons
Level of Commitment to the Organization, Absenteeism, and
Job Turnover, n.d.).
When a person has low job satisfaction they are more likely to
be actively searching for another job. People that are satisfied
with their job are less likely to be actively job hunting. For
employees to perform at their optimum levels they need to be
happy with life in general and their job satisfaction needs to be
high. When employees are satisfied they seem to handle
pressure easier than frustrated ones (Job Satisfaction Can Affect
a Persons Level of Commitment to the Organization,
Absenteeism, and Job Turnover, n.d.).
Job dissatisfaction can be from intrinsic and extrinsic factors.
When an employee is dissatisfied in one area of their life that
does not necessarily mean they are completely dissatisfied with
their job. There could be work-life conflicts the employee is
trying to balance. The work-life balance needs to be addressed
for an employee to maintain job satisfaction (Job Satisfaction
Can Affect a Persons Level of Commitment to the Organization,
There are many factors to ensuring job satisfaction. They
include company policies that are transparent, fair, and applied
equally to all employees. Salary and benefits should be
comparable to other organization salaries. Employees that is
more satisfied when they are paid competitive wages instead of
being underpaid. Developing teamwork helps workers
relationships carry their own weight so they will not let their
co-workers down. Employees with adequate personal work
space and upgraded equipment are more satisfied. Achievement,
recognition, autonomy, advancement, and job security are

important. It is important for them to be a balancing act
between employee’s personal life and work life (Job
Satisfaction Can Affect a Persons Level of Commitment to the
Organization, Absenteeism, and Job Turnover, n.d.).
Some aspects of a job that encourages job satisfaction from
employees is; work/life balance, opportunities to learn and
grow, ability to accomplish goals, positive relationship with co-
workers, and committed to their workers, and a good working
relationship with their boss. When employees are satisfied they
are usually more productive, creative, and committed to their
employers. There seems to be motivating and de-motivating
factors to consider. The motivating and de-motivating factors to
consider are varied work, sense of achievement, and
recognition. De-motivators include poor pay, company policy,
and continual pressure (Job Satisfaction Can Affect a Persons
Level of Commitment to the Organization, Absenteeism, and
Job Turnover, n.d.).
A difference exists between correlation and causation. A
correlation indicates that there is a relationship between
variables. Correlation does not explain which variable is
responsible. Consequences of job dissatisfaction usually involve
four forms. They include active vs passive and constructive
verses destructive. The active response refers to the exit or
leaving the organization. Active-constructive is the voice. This
refers to employee initiative to improve conditions in the
organization. Loyalty refers to the passive constructive form.
Finally, there is neglect or passive destructive (Job Satisfaction
Can Affect a Persons Level of Commitment to the Organization,
Job Satisfaction is an important factor in organizational
behavior. It is important for managers to communicate clearly
and effectively with employees as possible. Organizational
success increases when employees have higher levels of job
satisfaction. Strategies can be implemented from management as

a result of measuring job satisfaction.
References
Chinn, Danne. N.d. Importance of Job Satisfaction and
Organizational Behavior
Retrieved from:
www.chav.com/info_8481669_importance_job_satisfaction-
organizational
Defining Job Satisfaction n.d. Job Satisfaction n.d. Job
Satisfaction is the level of
Contentment Employees Feel About Their Work, Which Can
Affect Performance
N.d. Retrieved from: www.boundless.com
Gozukara,Irlem & Hatipoglu, Zeynep 2016. The Effect of
Employer Branding on
Employees Organizational Citizenship Behaviors. International
Journal of
Business Management and Economics Research Vol 7(1) p.477-
485
Grisaffe, Douglas B., Van Meter, Rebecca, & Chonko,
Lawrence B., 2016 Serving
First for the Benefit of Others: Preliminary Evidence for a

Hierarchical
Conceptualization of Servant Leadership Journal of Personal
Selling &
Sales Management, 2016 Vol.1 40-58.
Absenteeism and Job Turnover N.d. Retrieved from https:
www.boundless.com/management/textbooks/boundless-
management
Matta, Fadel F., Scott, Brenda, Koopman, Joel, & Conlon,
Donald E. Does
Seeing “Eye to Eye” Affect work Engagement and
Organizational Citizenship
Behavior? A Role Theory Perspective on LMX agreement.
Academy of
Management Journal 2015 Vol. 58(6) pp.1686-1708
Methot, Jessica R., Lepak, David, Shipp, Abbie, J., & Bowell,
Wendy R., 2017
Academy of Management Review Vol 42, (1), pp. 10-31.
Qadeer Faisal, Ahmad, Aftab, Hameed, Iman, & Mahmood,
Shahid 2016 Linking
Passion to Organizational Citzenship Behavior and Employer
Performance:
The Mediating Role of Work Engagement

Pakistan Journal of Commerce and Social Sciences Vol 10 (2)
pp. 316-334.,
World Services for the Blind. N.d. Arkansas Community
Foundation. Retrieved from:
www.bing.com/search?q=world
Wren, Daniel A. The History of Management Thought. 5th Ed.
2005 Oklahoma: John
Willey and Sons, Inc
Yam, Kai, Chi, Klotz, Anthony C., He, Wei, Reynolds, Scott, J.,
2017 from Good
Soldiers to Psychologically Entitled: Examine When and Why
Citzenship
Behavior and Employee Performance Academy of Management
Journal Vol 60 (5) pp. 373-396.
Yogomalar, I., Samuel, Anad, A. 2016. Shared Values and
Organizational Citizenship
Behavior of Generational Cohorts: A Review and Future
Directions Management
Vol.21 (2) pp. 249-27
Shark attacks and the Poisson approximation
Byron Schmuland

A story with the dramatic title “Shark attacks attributed to
random
Poisson burst” appeared on September 7, 2001 in the National
Post news-
paper. In this story, Professor David Kelton of Penn State
University used
a statistical model to try to explain away the surprising number
of shark at-
tacks that occurred in Florida last summer. According to this
article, Kelton
thinks the explanation for the spate of attacks may have nothing
to do with
changing currents, dwindling food supplies, the recent rise in
shark-feeding
tourist operations, or any other external cause.
“Just because you see events happening in a rash like this does
not imply
that there’s some physical driver causing them to happen. It is
characteristic
of random processes that they exhibit this bursty behaviour,” he
said.
What is this professor trying to say? Can mathematics really
explain the
increase in shark attacks? And what are these mysterious
Poisson bursts?
The main point of the Professor Kelton’s comments is that
unpredictable
events, like shark attacks, do not occur at regular intervals as in
the first
diagram below, but tend to occur in clusters, as in the second
diagram below.
The unpredictable nature of these events means that there are
bound to be

periods with an above average number of events, as well as
periods with a
below average number events, or even no events at all.
Regular Events
* * * * * * * * * *
time −→
Random Events
* * * **** * **
time −→
The statistical model used to study such sequences of random
events gets
its name from the French mathematician Siméon Denis Poisson
(1781-1840),
who first wrote about the Poisson distribution in a book on law.
The Pois-
son distribution can be used to calculate the chance that a
particular time
period will exhibit an abnormally large number of events
(Poisson burst), or
that it will exhibit no events at all. Since Poisson’s time, this
distribution
has been applied to many different kinds of problems such as
decay of ra-
dioactive particles, ecological studies on wildlife populations,
traffic flow on
the internet, etc. Here is the marvellous formula that gives us

information
on random events:
The chance of exactly k events occurring is
λk
k!
× e−λ, for k = 0, 1, 2, . . .
The funny looking symbol λ is the Greek letter lambda, and it
stands for
“the average number of events”. The symbol k! = k × (k − 1) × ·
· · × 2 × 1
means the factorial of k, and e−λ is the exponential function ex
with the
value x = −λ plugged in. Let’s take this new formula out for a
spin.
Shark attack!
If, for example, we average two shark attacks per summer, then
the chance
of having six shark attacks next summer is obtained by plugging
λ = 2 and
k = 6 into the formula above; and this gives
probability of six attacks ≈ (26/6!) × e−2 = 0.01203,
which is a little over a 1% chance. This means that six shark
attacks are
quite unlikely in one year, though this would happen about once
every fifty
years. The chance that we go the whole summer without any
shark attacks
can also be calculated by plugging λ = 2 and k = 0 into the
formula. This

gives
probability of no attacks ≈ (20/0!) × e−2 = 0.13533,
which is a 13% chance. We expect a “sharkless summer” every
seven or eight
years.
In this hypothetical shark problem the number of attacks
followed the
Poisson distribution exactly. The Poisson distribution is most
often used
to find approximate probabilities in problems with n repeated
trials and
probability p of success. Let me show you what I mean.
Lotto 6-49
One of my favorite games to study is Lotto 6-49. Six numbers
are randomly
chosen from 1 to 49 and if you match all six numbers you win
the jackpot.
Since the number of possible ticket combinations is
(
49
6
)
= 13,983,816, your
chance of winning the jackpot with one ticket is one out of
13,983,816, which
is p = 7.15 × 10−8. Let’s say you are a regular Lotto 6-49 player
and that

you buy one ticket, twice a week, for 100 years. The total
number of tickets
you buy is n = 100 × 52 × 2 = 10,400. What is the chance that
you win a
jackpot sometime during this 100 year run?
Now this is a pretty complex problem, but the Poisson formula
makes it
simple. First of all, the average number of jackpots during this
time period
is λ = np = 10,400/13,983,816 = 0.0007437. Plugging this in
with k = 0
shows that the chance of a “jackpotless 100 years” is
probability of no jackpots ≈ e−0.0007437 = 0.99926.
Wow! Even if you play Lotto 6-49 religiously for 100 years,
there is a better
than 99.9% chance that you never, ever win the jackpot.
Coincidences
Take two decks of cards and shuffle both of them thoroughly.
Give one deck
to a friend and place both your decks face down. Now, at the
same time,
you and your friend turn over your top card. Are they the same
card? No?
Then try again with the second card, the third card, etc. If you
go through
the whole deck what is the chance that, at some point, you and
your friend
turn over the same card?
In this problem there are n = 52 trials and the chance of a
success

(coincidence) on each trial is p = 1/52. The average number of
coincidences
is λ = np = 52/52 = 1, and so putting k = 0 in the Poisson
formula gives
probability of no coincidences ≈ e−1 = 0.36788.
The chance that you do see a coincidence is 1 − 0.36788 =
0.63212. You will
get a coincidence about 63% of the times you play this game.
Try it and see!
Birthday problem
Suppose there are N people in your class. What are the odds that
at least
two people share a birthday? Imagine moving around the class
checking
every pair of people to see if they share a birthday. The number
of trials is
equal to the number of pairs of people, i.e., n =
(
N
2
)
= N(N − 1)/2. The
probability of success on a given trial is the chance that two
randomly chosen
people share a birthday, i.e., p = 1/365. This gives the average
number of
shared birthdays to be λ = N(N −1)/2(365), so the probability of

“no shared
birthdays” is
probability of no shared birthdays ≈ e−N(N−1)/2(365),
and so the probability of at least one shared birthdays is
approximately
1 − e−N(N−1)/2(365). Here’s what happens when you try
different values of N
in this expression.
Probability of a shared birthday
N Prob N Prob
10 0.115991 60 0.992166
20 0.405805 70 0.998662
30 0.696320 80 0.999826
40 0.881990 90 0.999983
50 0.965131 100 0.999999
With N = 10 people there is only about an 11.5% chance of a
shared
birthday, but with N = 30 people there is a 69.6% chance. In a
large class
(like at university!) with N = 100 students, a shared birthday is
99.9999%
certain.
For large class sizes maybe it is possible to have a triple
birthday. Follow-
ing the same pattern as before, we work out the chances that
there is at least
one triple shared birthday in a class of N people. This time you
go through
the class checking each triple of people, there are

(
N
3
)
= N(N − 1)(N − 2)/6
trials, and the chance of success on each trial is p = 1/(365)2.
This gives λ = N(N − 1)(N − 2)/6(365)2, so the probability of
“no triple
shared birthdays” is
probability of no triple shared birthdays ≈
e−N(N−1)(N−2)/6(365)2,
and so the probability of at least one triple is 1 −
e−N(N−1)(N−2)/6(365)2. Let’s
look at different values of N for this formula.
Probability of a triple shared birthday
N Prob N Prob
10 0.000900 60 0.226522
20 0.008520 70 0.336936
30 0.030015 80 0.460278
40 0.071477 90 0.585970
50 0.136809 100 0.702915
My large first year statistics courses usually have about 100
students in
them, and I always check their birthdays. According to the

table, there
should be a triple birthday over 70% of the time. It really is
true; there is
usually a triple birthday in those classes.
The Great One
During Wayne Gretzky’s days as an Edmonton Oiler, he scored
a remark-
able 1669 points in 696 games, for a rate of λ = 1669/696 = 2.39
points per
game. From the Poisson formula with k = 0 we estimate that the
probability
of Gretzky having a “pointless game” is
probability of no points ≈ (2.39)0/0! e−2.39 = 0.0909.
Over 696 games, this ought to translate to about 696 × 0.0909 =
63.27
pointless games. In fact, during that period he had exactly 69
games with
no points.
Now for one point games we find an approximate probability of
probability of one point ≈ (2.39)1/1! e−2.39 = 0.2180,
for a predicted value of 696 × 0.2180 = 151.71 one point games.
Let’s try
the same calculation for other values of k, and compare what
the Poisson
formula predicts to what actually happened.
Points Actual # Games # Predicted by Poisson

0 69 63.27
1 155 151.71
2 171 181.90
3 143 145.40
4 79 87.17
5 57 41.81
6 14 16.71
7 6 5.72
8 2 1.72
9 0 0.46
As you see there is a remarkable agreement between the
predictions based
on the Poisson formula, and the actual number of games with
different point
totals. This shows that Gretzky was not only a high scoring
player, but a
consistent one as well. The occasional pointless game, or
occasional “Poisson
burst” in seven or eight point games were not due to
inconsistent play, but
are exactly what is expected in any random sequence of events.
Another
reason why he really was the great one!
Sheet127602427824017222220203126702324131926222325272
52561718202837418263755720213167496620215243315041212
23964215546343443171822565767201818151732768260626272
24505521152528372343464041431820283334323620208440561
91718535219584772557802726275072512121417212934424825
3343622783355933042373163633946492335
Part one
The data comes from following website, I random select 150

observations, x variable is Age.
https://www.statcrunch.com/app/index.php?dataid=2323968
Mean :37
Variance: 361
Standard deviation: 19
4) Suggest an appropriate probability distribution to
approximate the
histogram.
I suggest exponential distribution
6) Give reasons why your selection of probability distribution is
appropriate.
I would select exponential distribution because it is positively
skewed and probabilities decreases as we move forward
Part two
Use the procedure followed in Gretzky's example to
approximate the distribution by a probability density
curve.（see attachment, shark attacks）
7) Perform Goodness of fit test of the distribution
8) Find the pdf, cdf, empirical cdf, mgf, expected value and
variance.

9) Give conclusions of the results obtained.
10) Discuss any limitations of your study.
11) Discuss your perception about the project before and after
completion.

Shark attacks and the Poisson approximationByron Schmuland.docx

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