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Statistics and Mathematics for Economics

Statistics and Mathematics for Economics. Statistics Component: Lecture Five. Objectives of the Lecture. To define and to provide an example of the calculation of the value of the third central moment

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Statistics and Mathematics for Economics

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  1. Statistics and Mathematics for Economics Statistics Component: Lecture Five

  2. Objectives of the Lecture • To define and to provide an example of the calculation of the value of the third central moment • To define and to provide an example of the calculation of the value of the fourth central moment • To inform you of a measure which can be used to summarise a joint probability distribution

  3. The First Three Moments of a Probability Distribution The first moment of a probability distribution is a measure of central tendency, e.g., E[X]. The second central moment of a probability distribution is a measure of dispersion, e.g., E[(X – E[X])2]. The third central moment of the probability distribution of X can be defined as E[(X – E[X])3].

  4. The Third Central Moment • The value of the third central moment indicates whether the probability distribution is symmetrical or skewed • The implication of the value of the third central moment being zero is that the probability distribution is symmetrical • The implication of the value of the third central moment being greater than zero is that the probability distribution is positively skewed • The implication of the value of the third central moment being less than zero is that the probability distribution is negatively skewed

  5. 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 – E[X])3] = (x – E[X])3.P(X = x).

  6. First Priority The first priority is to calculate the expected value of X. E[X] = 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 6/10 4 1/10 4/10 5 1/10 5/10 E[X] = 2.3 wickets

  7. Second Step: Construct a Table Third Central Moment = E[(X – E[X])3] = (x – E[X])3.P(X = x)

  8. 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 not symmetrical • More specifically, the probability distribution is positively skewed • In terms of the graph of the probability distribution, the right-hand tail is longer than the left-hand tail

  9. Diagrammatic Presentation of the Probability Distribution P(X) 3/10 2/10 1/10 X 0 1 2 3 4 5

  10. 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 – E[Y])3] = (y – E[Y])3.P(Y = y).

  11. 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 E[Y] = y.P(Y = y) = 19/10 = 1.9 goals

  12. Calculation of the Value of the Third Central Moment of the Probability Distribution of Y Third central moment = E[(Y – E[Y])3] = (y – E[Y])3.P(Y = y)

  13. 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 not symmetrical • More specifically, the probability distribution of Y is positively skewed • In terms of the graph of the probability distribution, the right-hand tail is longer than the left-hand tail

  14. Diagrammatic Presentation of the Probability Distribution of Y P(Y) 3/10 2/10 1/10 Y 0 1 2 3 4 5

  15. A Comparison of Results • It has been established that both the probability distribution of X and the probability distribution of Y are positively skewed • 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 • 2.448 > 0.864, so, on the surface, it would seem that the probability distribution of Y is skewed to a greater extent than the probability distribution of X • However, the results are not comparable because the values are expressed in terms of different units

  16. A Standardised Measure of Skewness For a fair comparison of the extent to which two probability distributions are skewed, a contrast should be performed of the values of a standardised form of the third central moment. 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 – E[X])3] ------------------ (E[(X – E[X])2])3/2

  17. Calculation of the Value of the Variance of X • var.(X) = E[X2] - (E[X])2 • E[X] = 2.3 wickets, E[X2] = 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 • E[X2] = 73/10 wickets2 var.(X) = 73/10 – (23/10)2 • = 2.01 wickets2

  18. Calculation of the Value of the Variance of Y • var.(Y) = E[Y2] - (E[Y])2 • E[Y] = 1.9 goals, E[Y2] = 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 • E[Y2] = 61/10 wickets2 var.(Y) = 61/10 – (19/10)2 • = 2.49 goals2

  19. Calculation of the Values of the Standardised Measure of Skewness Probability Distribution of X S = E[(X – E[X])3] = 0.864 = 0.3032 ------------------ -------- (E[(X – E[X])2])3/2 (2.01)3/2 Probability Distribution of Y S = E[(Y – E[Y])3] = 2.448 = 0.6230 ------------------ -------- (E[(Y – E[Y])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.

  20. The Fourth Central Moment of a Probability Distribution The fourth central moment of the probability distribution of the random variable, X, can be defined mathematically as: E[(X – E[X])4]. The value of the fourth central moment provides information on the kurtosis or the peakedness of the probability distribution.

  21. 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 E[X] = 7/2, var.(X) = 35/12 units squared, E[(X – E[X])3] = 0

  22. Construct a Table • x x – E[X] (x – E[X])4 P(X = x) (x – E[X])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 – E[X])4] = (x – E[X])4.P(X = x) • = 1414/96 units4

  23. 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. It would be useful to have a measure of kurtosis, the value of which independent of the units in which the associated random variable is expressed. Thus, a standardised version of the fourth central moment is: K = E[(X – E[X])4] ------------------ (E[(X – E[X])2])2

  24. Implications of Different Values of K • In connection with the standardised measure of kurtosis, a critical value is 3 • When K is > 3, the implication is that, in terms of a diagram, the probability distribution is tall and thin (leptokurtic) • When K is < 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)

  25. Calculation of the Value of K In connection with the probability distribution of X: K = E[(X – E[X])4] = 1414/96 ------------------ ---------- (E[(X – E[X])2])2 (35/12)2 Thus, K = 1.7314, and so the probability distribution is platykurtic, i.e., short and wide.

  26. A Summary Measure of a Joint Probability Distribution • Consideration will now be given to a measure that can be used for the purpose of summarising a joint probability distribution • In fact, such a measure has already been encountered, and is the covariance of the respective random variables • Recall that Cov.(X, Y) = E[(X – E[X])(Y – E[Y])]

  27. 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

  28. 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 E[X] = 2, E[Y] = 2 Cov.(X, Y) = E[(X – E[X])(Y – E[Y])] = E[X.Y] – E[X].E[Y]

  29. Calculation of E[X.Y] Given that X and Y are discrete random variables, then: E[X.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

  30. Covariance of X and Y Cov.(X, Y) = E[X.Y] - E[X].E[Y] = 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 gain an appreciation of 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.

  31. Creation of a Standardised Measure It is possible to eliminate the dependence upon the units in terms of which the two random variables are expressed 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.

  32. 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

  33. Calculation of the Value of the Variance of X var.(X) = E[X2] – (E[X])2, where E[X] = 2 and E[X2] = 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 E[X2] = 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.

  34. 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.

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