Question

a) For a distribution, the mean is 10, variance is 16, coefficient of skewness is +1 and
coefficient of kurtosis is 4. Obtain the first four moments about the origin i.e., zero.
Comment upon the nature of distribution. (5)
b) Calculate the correlation coefficient for the following heights (in inches) of father
(X ) and their sons ) (Y : (5)

X : 65 66 67 67 68 69 70 72
Y : 67 68 65 68 72 72 69 71

Answer 1

Answer on Question #60648 – Math – Statistics and Probability

Question

a) For a distribution, the mean is 10, variance is 16, coefficient of skewness is +1 and coefficient of kurtosis is 4. Obtain the first four moments about the origin i.e., zero.

Comment upon the nature of distribution.

Solution

Mean = $m_1' = 10$, variance = $m_2 = 16$, coefficient of skewness = $\gamma_1 = +1$, coefficient of kurtosis is $\beta_2 = 4$.

First moment about the origin is

$$\text{Mean} = m_1' = 10.$$

Next,

$$Variance = m_2 = 16 = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2.$$

Second moment about the origin is

$$m_2' = \frac{\sum x^2}{n} = m_2 + \mu_1^2 = Variance + \left(\frac{\sum x}{n}\right)^2 = 16 + 10^2 = 116$$

Coefficient of skewness is +1:

$$\gamma_1 = +1 = \frac{m_3}{m_2^2} \rightarrow m_3 = m_2^3 = 16^3 = 64$$

Coefficient of kurtosis is 4:

$$\beta_2 = 4$$ $$\frac{m_4}{m_2^2} = 4 \rightarrow m_4 = 4(16)^2 = 1024$$

Third moment about the origin is

$$m_3' = m_3 + 3m_2m_1' + m_1'^3 = 64 + 3(16)(10) + 10^3 = 1544$$

Fourth moment about the origin is

$$m_4' = m_4 + 4m_3m_1' + 6m_2m_2' + m_1'^4 = 1024 + 4(64)(10) + 6(16)10^2 + 10^4 = 23184$$

It is a positively skewed distribution.

Question

b) Calculate the correlation coefficient for the following heights (in inches) of father (X) and their sons (Y):

X: 65 66 67 67 68 69 70 72

Y: 67 68 65 68 72 72 69 71

Solution

The correlation coefficient is given by

$$r = \frac {n (\sum x y) - (\sum x) (\sum y)}{\sqrt {[ n (\sum x ^ {2}) - (\sum x) ^ {2} ] [ n (\sum y ^ {2}) - (\sum y) ^ {2} ]}}$$

Here

$$n = 8$$ $$\sum x y = 65 \cdot 67 + 66 \cdot 68 + 67 \cdot 65 + 67 \cdot 68 + 68 \cdot 72 + 69 \cdot 72 + 70 \cdot 69 + 72 \cdot 71 = 37560$$ $$\sum y = 67 + 68 + 65 + 68 + 72 + 72 + 69 + 71 = 552$$ $$\sum x = 65 + 66 + 67 + 67 + 68 + 69 + 70 + 72 = 544$$ $$\sum y ^ {2} = 67 ^ {2} + 68 ^ {2} + 65 ^ {2} + 68 ^ {2} + 72 ^ {2} + 72 ^ {2} + 69 ^ {2} + 71 ^ {2} = 38132$$ $$\sum x ^ {2} = 65 ^ {2} + 66 ^ {2} + 67 ^ {2} + 67 ^ {2} + 68 ^ {2} + 69 ^ {2} + 70 ^ {2} + 72 ^ {2} = 37028$$

The correlation coefficient is

$$r = \frac {(8) (37560) - (544) (552)}{\sqrt {8 (38132) - (552) ^ {2}} \sqrt {8 (37028) - (544) ^ {2}}} = + 0.603.$$

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