CSS 2023 Statistics Paper I

Define Moments, Skewness, and Kurtosis. How are they useful in describing a frequency distribution? Calculate the first four moments about the mean for a given set of data.

State and prove Bayes' Theorem. Explain its application in real-world scenarios with a numerical example involving conditional probability.

Define Mathematical Expectation. Prove that E(aX + b) = aE(X) + b and Var(aX + b) = a^2Var(X).

What are the properties of a Poisson Distribution? Derive the mean and variance of the Poisson Distribution and discuss the conditions under which Binomial distribution tends to Poisson distribution.

Describe the Normal Distribution. Explain the importance of the Central Limit Theorem in Statistics. Find the area under the normal curve between specified Z-scores.

What is the Principle of Least Squares? Derive the normal equations for fitting a straight line Y = a + bX and explain the coefficient of determination.

Discuss the various problems encountered in the construction of Index Numbers of prices. Why is Fisher’s Index called the 'Ideal Index'?