2.  Inequalities in the distribution of income have created fat tail
distributions, where 20% of the population controls 80% of the
wealth. A tail index that is greater than 0 results in fat tailed
distributions, which produces a more modest value for unlikely
situations. Knowing that the high income distribution for every
country is modeled by a Pareto, the research was aimed at proving
that the world’s distribution was also modeled by a Pareto, which
would be proven by the world’s Pareto index. The Pareto function
for each nation was discerned by finding its Pareto index and
coefficient, and threshold. With these known, the Pareto function
was calculated and added together, yielding the probability
associated with an income, x. Five income values were chosen and
their corresponding probabilities were plotted on a log-log
graph, revealing the world’s Pareto index. The acquired results were
that the model for the world’s and each country’s high incomes was
a Pareto since their tail indexes were greater than 0. It was
concluded that the sum of a sample of Pareto distributions yields a
Pareto distribution.

3.  The Pareto Distribution is a power law function whose
tail fall slower than that of a normal distribution.
 The tail index is responsible for the thickness of the
tails, such that as its value increases, the tail becomes
thicker, yielding a distribution where there are not a lot
of inequalities in the dispersion of wealth.
 A tail index less than 0 is bounded, one equal to 0 falls
at an exponential rate and one greater than 0 is fat or
long tailed.
 There is less inequality in the spread of wealth as the
Pareto index increases.

4.  The Pareto Principle says that 20% of the
population controls 80% of the wealth.
 The relationship between the spread of the
wealth of investors and their return from their
stocks was seen to be proportional such that
the Pareto coefficient is equal to that of the
Levy distribution’s exponent.
 The objective is to discern the distribution that
would result from a sample of high incomes in
the world.

5.  Galton experimented with peas to show how the distribution of weight among
offspring is influenced by inheritance. He then used these distributions to
show that the sum of a sample of normal distributions yields a normal
distribution. Galton separated the 490 peas he used into seven groups
according to their weights and distributed 10 peas from each of those seven
groups to each of his friends. His friends then grew these peas which created
seeds whose weights were measured and models were combined together to
see which type of model he would yield. From his experiments, Galton found
out that the group of seeds that he supplied to each of his friends to
grow, yielded seeds with weights that had a normal distribution. In
addition, the variances for each group were synonymous, which correlates to
the line that Galton made to be AB on the Galton board, where the drops are
separated into different sections and would rest on a line set to be AB'. If the
line AB is not near the top of the Galton board, these drops would have a
normal distribution, and if one section was opened and the drops fell to the
bottom, a normal distribution would composed. If the other sections were
opened one by one, the outcome would be a normal distribution as well.

6.  The research would allow people to see the
distribution of the population of high incomes
in the world based on the sample that
comprises it.
 The conjecture is that “If the Pareto
distribution for the incomes of each country
were added together, the new distribution
would also be a Pareto if the wealth is
dispersed in the same fashion and the Pareto
index is greater than 0.”

7.  Materials needed were the total income and
population, as well as the gini indexes for each
country studied, and the equations for the
Pareto function, ,Pareto mean,
, and gini coefficient, .
 Since the gini coefficient was known, its
equation was used to solve for the Pareto
index.
 The mean is the total income over the total
population, the Pareto index was plugged into
the equation and the threshold was solved for.

8.  With the Pareto index and threshold
known, the Pareto function was extracted
for each country.
 To determine the relative probability of a
certain income, the Pareto functions for
each country were multiplied by the ratio of
its population to the sum of the populations
of the countries used.

9.  With the product determined, five random
high income values were
chosen, 1,000,000, 1,500,000, 2,000,000, 2,
500,000 and 9,000,000, calculating their
corresponding probabilities.
 These values were plotted on a log-log
graph, and the slope of the line was
taken, finding the Pareto index for the
model of income in the world.

10.  Calculation of the Income Probability for
the United States:
Table 1: Pareto and Threshold Parameter
A list of the Pareto indexes for each country studied, Japan has the highest index
at 2.5, yielding fewer inequalities in the spread of wealth. With the mean income
for each country shown, the threshold parameter for each can be seen, with
China having the smallest threshold at $3,424.20 and Switzerland having the
greatest at $20,919.62.

11. Table 2: Calculation of the Pareto probability for a certain
income value, x. The Pareto functions for each country
allowed the probabilities for an income value to be
calculated, predicting that the United States and China
would yield the greatest probabilities since their Pareto
indexes are the smallest.

13. The Probabilities for World Income Values
1
0.1
0.01
0.001
0.0001 Probabilities for World
Probability
0.00001 Income Values
0.000001
0.000000
1E-08
1E-09
1E-10
1E-11
1E-12
1E-13
1,000,000 10,000,000
Income

14.  The Pareto model of world incomes
involves the collaboration of individual
Pareto functions. Canada has the
greatest equality in the distribution of
wealth since it’s gini index is the smallest
and Pareto index is the greatest.
 With
$1,000,000, $1,500,000, $2,000,000, $2,500,
000 and $9,000,000 and their
corresponding probabilities, the slope of
the log-log graph ended up with an
index of 1.74.

15.  Recall the hypothesis: If the Pareto distribution
for the incomes of each country were added
together, the new distribution would also be a Pareto
if the wealth is dispersed in the same fashion and the
Pareto index is greater than 0.”
 The proposed hypothesis was valid because once the
Pareto probabilities were added together, the model
for the world’s incomes was concluded to be a
Pareto.

16.  Galton showed that the distribution of the
weight’s of peas is affected by inheritance such
that when he split them into seven groups, they
yielded seeds with weights that had a normal
distribution and when those groups were added
together, a normal distribution came about.
 Galton showed that the drops in each section of
a quincunx resting on a line above the bottom is
a normal distribution such that when he opened
each section one by one, they generate a
normal distribution.

17.  Although the research was able to show
the weighted sum of a sample of Paretos
yielded a Pareto, there were limitations to it.
 The Pareto optimal values cannot be
compared to each other with regards to
the Pareto principle and the median
income cannot be determined since the
Pareto is not symmetrical.

18.  Further research would be utilizing another
methodology to tackle this problem. Rather
than pinpointing the probability for each
income value, the moments for each
country would be calculated and the
gamma function would be used to find the
moment generating function. From there
the world’s Pareto index could be found.

19.  The study demonstrated that the sum of a
sample of Pareto distributions does in fact
yield a Pareto.
 The Pareto model for the incomes in the
entire world is a fat-tailed distribution with a
thick tail since its tail index is 1.74, which
shows that there is not a lot of inequality in
the spread of wealth.
 By finding the total income and population
of all the countries used, the mean income
in the world was calculated to be
$12,449.53, with a threshold of $5,294.63.

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