Statistics and Mathematics for Economics

1
Statistics and Mathematics for Economics

• Statistics Component Lecture Five

2
Objectives of the Lecture

• To provide an interpretation and to give an
  example of the calculation of the value of the
  third central moment of a probability
  distribution
• To provide an interpretation and to give an
  example of the calculation of the value of the
  fourth central moment of a probability
  distribution
• To inform you of a measure that can be used to
  summarise a joint probability distribution

3
The Third Central Moment

• The first moment of a probability distribution
  represents a measure of central tendency
• The second central moment constitutes a measure
  of dispersion
• The value of the third central moment provides
  information on whether or not a probability
  distribution is symmetrical
• If a probability distribution is symmetrical
  then, in terms of a graph, the right-hand side of
  the distribution is the mirror image of the
  left-hand side

4
Mathematical Definition of the Third Central
Moment
X is a random variable The third central moment
of the probability distribution of X is E(X
EX)3.
5
Implications of Values of the Third Central Moment

• When the value of the third central moment is
  equal to zero, the implication is that the
  probability distribution is symmetrical
• When the value of the third central moment is
  greater than zero, the implication is that the
  probability distribution is positively skewed. In
  terms of a graph, the right-hand tail is longer
  than the left-hand tail
• When the value of the third central moment is
  less than zero, the implication is that the
  probability distribution is negatively skewed. In
  terms of a diagram, the left-hand tail is longer
  than the right-hand tail.

6
An Example of the Calculation of the Value of the
Third Central Moment
X is the number of wickets which are taken by a
bowler during the first innings of a cricket
match Probability distribution of
X x 0 1 2 3 4 5 P(X x) 1/10 2/10 3/10 2/10 1/
10 1/10 Third central moment E(X EX)3
?(x EX)3.P(X x).
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First Step
The first step is to calculate the expected value
of X. EX ?x.P(X x) x P(X x) x.P(X
x) 0 1/10 0 1 2/10 2/10 2 3/10 6/10 3 2/10 6/1
0 4 1/10 4/10 5 1/10 5/10 EX 2.3 wickets
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Second Step Construct a Table
Third Central Moment E(X EX)3 ?(x
EX)3.P(X x)
9
Implication of the Result

• The value of the third central moment is greater
  than zero (0.864 wickets cubed)
• The implication is that the probability
  distribution is positively skewed
• In terms of a graph, the right-hand tail is
  longer than the left-hand tail of the probability
  distribution

10
Diagrammatic Presentation of the Probability
Distribution
P(X x)
3/10
2/10
1/10
x
0 1 2 3 4 5
11
Second Example
Y is the number of goals which are scored by a
football team during the course of an individual
match. Probability distribution of
Y y 0 1 2 3 4 5 P(Y y) 2/10 3/10 2/10 1/10 1/
10 1/10 Third central moment E(Y EY)3
?(y EY)3.P(Y y).
12
Calculation of the Expected Value of Y
y P(Y y) y.P(Y y) 0 2/10 0 1 3/10 3/10 2 2/
10 4/10 3 1/10 3/10 4 1/10 4/10 5 1/10 5/10
EY ?y.P(Y y) 19/10 1.9 goals
13
Calculation of the Value of the Third Central
Moment of the Probability Density Function of Y
Third central moment E(Y EY)3 ?(y
EY)3.P(Y y)
14
Implication of the Result

• The value of the third central moment is greater
  than zero (2.448 goals cubed)
• The implication of this value is that the
  probability distribution of Y is positively
  skewed
• In terms of a diagram, the right-hand tail is
  longer than the left-hand tail of the probability
  distribution

15
Diagrammatic Presentation of the Probability
Distribution of Y
P(Y y)
3/10
2/10
1/10
y
0 1 2 3 4 5
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A Limitation of the Third Central Moment

• The value of the third central moment of the
  probability distribution of X is 0.864 wickets
  cubed
• The value of the third central moment of the
  probability distribution of Y is 2.448 goals
  cubed
• Both of the distributions are positively skewed
• It would seem as though the probability
  distribution of Y is skewed to a greater extent
  than the probability distribution of X
• However, the two figures are not comparable for
  the reason that they are expressed in terms of
  different units

17
A Standardised Measure of Skewness
If we are seeking to compare the extent to which
two different probability distributions are
skewed then, for each, we should calculate the
value of a standardised measure of skewness. The
third central moment should be transformed so as
to eliminate its reliance upon the units of
measurement of the associated random
variable. Standardised measure of
skewness S E(X EX)3
------------------ (E(X EX)2)3/2
18
Calculation of the Value of the Variance of X

• var.(X) EX2 - (EX)2
• EX 2.3 wickets, EX2 ?x2.P(X x)
• x x2 P(X x) x2.P(X x)
• 0 0 1/10 0
• 1 1 2/10 2/10
• 2 4 3/10 12/10
• 3 9 2/10 18/10
• 4 16 1/10 16/10
• 25 1/10 25/10
• EX2 73/10 wickets2 var.(X) 73/10
  (23/10)2
• 2.01 wickets2

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Calculation of the Value of the Variance of Y

• var.(Y) EY2 - (EY)2
• EY 1.9 goals, EY2 ?y2.P(Y y)
• y y2 P(Y y) y2.P(Y y)
• 0 0 2/10 0
• 1 1 3/10 3/10
• 2 4 2/10 8/10
• 3 9 1/10 9/10
• 4 16 1/10 16/10
• 25 1/10 25/10
• EY2 61/10 goals2 var.(Y) 61/10 (19/10)2
• 2.49 goals2

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Calculation of the Values of the Standardised
Measures of Skewness
Probability Distribution of X S E(X
EX)3 0.864 0.3032
------------------ --------
(E(X EX)2)3/2
(2.01)3/2 Probability Distribution of Y S
E(Y EY)3 2.448 0.6230
------------------
-------- (E(Y EY)2)3/2
(2.49)3/2 Hence, the probability
distribution of Y is skewed to a greater extent
than the probability distribution of X.
21
The Fourth Central Moment of a Probability
Distribution
The value of the fourth central moment provides
information on the peakedness or the kurtosis of
the probability distribution. It gives an
indication of whether the distribution is tall
and thin or short and wide. Mathematical
definition if X is a random variable then the
fourth central moment of the probability
distribution of X is E(X EX)4.
22
An Example of the Calculation of the Value of the
Fourth Central Moment
X is the discrete random variable, the number
which is obtained following a single throw of a
dice. Probability distribution of
X x 1 2 3 4 5 6 P(X x) 1/6 1/6 1/6 1/6 1/6 1/
6 EX 7/2, var.(X) 35/12 units
squared, E(X EX)3 0
23
Construct a Table

• x x EX (x EX)4 P(X x) (x EX)4.P(X
  x)
• 1 -5/2 625/16 1/6 625/96
• -3/2 81/16 1/6 81/96
• 3 -1/2 1/16 1/6 1/96
• 4 ½ 1/16 1/6 1/96
• 5 3/2 81/16 1/6 81/96
• 6 5/2 625/16 1/6 625/96
• ---------
• 1414/96 units4
• E(X EX)4 ?(x EX)4.P(X x)
• 1414/96 units4

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A Limitation of the Fourth Central Moment
A limitation of the fourth central moment is that
its value is sensitive to the units in which the
associated random variable is expressed. In
order to acquire information on the kurtosis of a
probability distribution, then, it is more
helpful to calculate the value of a standardised
measure such as K E(X EX)4
------------------
(E(X EX)2)2
25
Implications of Different Values of K

• In connection with the standardised measure of
  kurtosis, a critical value is 3
• When K is gt 3, the implication is that, in terms
  of a diagram, the probability distribution is
  tall and thin (leptokurtic)
• When K is lt 3, the implication is that, in terms
  of a diagram, the probability distribution is
  short and wide (platykurtic)
• A normal distribution corresponds to a value of K
  which is equal to 3 (mesokurtic)

26
Calculation of the Value of K
In connection with the probability distribution
of X K E(X EX)4 1414/96
------------------ ----------
(E(X EX)2)2 (35/12)2 Thus, K
1.7314, and so the probability distribution is
platykurtic, i.e., short and wide.
27
Summarising a Joint Probability Distribution
A marginal probability distribution can be
summarised using the moments of the probability
distribution. A joint probability distribution
can be summarised using the covariance of the
two random variables. (The value of a covariance
provides information on the nature and the
strength of the linear relationship between two
random variables.) Given the two random
variables, X and Y, then a mathematical definition
of their covariance is Cov.(X, Y) E(X
EX)(Y EY)
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An Example of the Calculation of the Value of the
Covariance

• A black velvet bag contains three balls which are
  of equal size
• In order to be able to distinguish between them,
  the balls have the numbers, 1, 2 and 3, marked on
  them
• Two balls are drawn from the bag, in sequence and
  without replacement
• In the context of this game, X is defined as the
  discrete random variable, the number which is
  marked on the first ball that is drawn from the
  bag
• Y is the discrete random variable, the number
  which is marked on the second ball

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Joint Probability Distribution of X and Y
Value of Y 1 2 3 P(X
x) 1 0 1/6 1/6 1/3 Value of
X 2 1/6 0 1/6 1/3 3 1/6 1/6 0 1/3 P(Y
y) 1/3 1/3 1/3 EX 2, EY 2 Cov.(X, Y)
E(X EX)(Y EY) EX.Y
EX.EY
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Calculation of EX.Y
Given that X and Y are discrete random variables,
then EX.Y ??(x.y)P(X x, Y y)
(1)(1)(0) (1)(2)(1/6) (1)(3)(1/6)
(2)(1)(1/6) (2)(2)(0) (2)(3)(1/6)
(3)(1)(1/6) (3)(2)(1/6)
(3)(3)(0) 5/6 8/6 9/6
22/6
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Covariance of X and Y
Cov.(X, Y) EX.Y - EX.EY
22/6 - (2)(2)
22/6 - 24/6 -2/6 or
-1/3 The implication of this value is that there
is a negative linear relationship between X and
Y. However, it is difficult to interpret the
strength of this linear relationship as the
value of the covariance is dependent upon the
units in which the two random variables are
expressed.
32
Creation of a Standardised Measure
It is possible to eliminate the dependence upon
the units of measurement of the two random
variables by dividing the covariance by the
product of the standard deviations of the two
variables. Cov.(X, Y)
----------------------
?var,(X) ?var.(Y) The measure which has been
formed is the correlation coefficient
corresponding to X and Y.
33
Implications of Different Values of a Correlation
Coefficient

• By construction, the value of a correlation
  coefficient cannot be less than 1 and cannot be
  greater than 1
• The nearer that the value of the correlation
  coefficient is to 1, the stronger is the
  negative linear relationship between the two
  variables
• The nearer is the value of the correlation
  coefficient to 1, the stronger is the positive
  linear relationship between the two variables
• The nearer that the value of the correlation
  coefficient is to zero then the weaker is the
  linear relationship between the two variables

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Calculation of the Value of the Variance of X
var.(X) EX2 (EX)2, where EX 2 and
EX2 ?x2.P(X x) x x2 P(X x) x2.P(X
x) 1 1 1/3 1/3 2 4 1/3 4/3 3 9 1/3 9/3 EX2
14/3 So, var.(X) 14/3 4 2/3 units
squared Probability distribution of Y is
identical to the probability distribution of X,
with the consequence that var.(Y) 2/3 units
squared.
35
Correlation Coefficient corresponding to X and Y
Corr.(X, Y) Cov.(X, Y)
---------------------
?var.(X) ?var.(Y)
-1/3
--------------------- ?(2/3) ?(2/3)
(-1/3)/(2/3)
-½ Hence, the linear relationship is neither
weak nor strong.