1. Give the definition of Statistics due to Croxton and Cowden, and note down the importance of Statistics in the field of Economics. 2. Find Mean, Variance and Coefficient of vari... 1. Give the definition of Statistics due to Croxton and Cowden, and note down the importance of Statistics in the field of Economics. 2. Find Mean, Variance and Coefficient of variation for first n natural numbers. 3. The first three moments of a distribution about 2 are 1, 22 and 10. Find its mean, standard deviation and the third central moment. Read more

Steps to Solve

1. Definition of Statistics Statistics, as defined by Croxton and Cowden, is "the science of collection, classification, analysis and interpretation of numerical data". In Economics, statistics is vital for making informed decisions, understanding trends and patterns, and conducting quantitative analyses to inform policy.

2. Finding Mean, Variance, and Coefficient of Variation for First n Natural Numbers

   • Mean Calculation: The mean of the first $n$ natural numbers can be calculated using the formula: $$ \text{Mean} = \frac{1 + 2 + 3 + \ldots + n}{n} = \frac{n(n+1)/2}{n} = \frac{n+1}{2}$$

   • Variance Calculation: The variance formula for the first $n$ natural numbers: $$ \text{Variance} = \frac{(1^2 + 2^2 + 3^2 + \ldots + n^2)}{n} - \left( \frac{n+1}{2} \right)^2 $$

   The sum of squares for the first $n$ numbers is: $$ 1^2 + 2^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6} $$ Thus, variance becomes: $$ \text{Variance} = \frac{n(n+1)(2n+1)/6}{n} - \left( \frac{n+1}{2} \right)^2 = \frac{(n+1)(2n+1)}{6} - \frac{(n+1)^2}{4} $$

   • Coefficient of Variation: The coefficient of variation is calculated as: $$ \text{Coefficient of Variation} = \frac{\sigma}{\mu} = \frac{\sqrt{\text{Variance}}}{\text{Mean}} $$

3. Mean, Standard Deviation, and Third Central Moment

   • The first moment about $2$ is $1$, which implies: $$ \text{Mean} = 2 + 1 = 3 $$

   • The second moment about $2$ is $22$. Variance can be derived as: $$ \text{Variance} = E(X^2) - (E(X))^2 = 22 - 3^2 = 22 - 9 = 13 $$

   • The standard deviation is: $$ \text{Standard Deviation} = \sqrt{13} $$

   • The third moment about $2$ is $10$. The third central moment can be found as: $$ \text{Third Central Moment} = E((X - \mu)^3) = 10 $$ Thus, the third central moment is simply $10$.