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1. Discuss convergence in probability, convergence in distribution and convergence in Mean and their relationships. (6)
2. Explain and compare the fundamental concepts and properties of probability generating functions, moment generating functions and characteristics functions in the context of probability theory and statistical analysis. (6)
3. Define sufficiency and minimal sufficiency, and explain their significance in the context of estimation. Provide examples to illustrate how sufficiency is determined and how it influences the selection of estimators. (6)
4. Define Fisher information, discuss its role in quantifying the amount of information in a sample, and explain how it relates to the Cramer- Rao inequality. (6)
5. Describe Mann- Whitney U test. (4)
6. Explain briefly the Latin Square- Model in design of experiment. (4)
7. Find the best critical region to test the null H₀: θ = 0 against the alternate H₁: θ = 1 for a population with pdf f(x, θ) = (1/√(2π))e^(-(x − θ)²/2), −∞ < x < ∞. (6)
8. Derive the test statistic for testing the significance of the difference of proportions of a certain attribute of two populations, when the sample size is large. (6)
9. In a simple random sample of 200 from a population of 2400 colleges, 120 colleges were in favour of a proposal. Estimate the total number of colleges in the population that favoured the proposal and find the variance of this estimate. (4)
10. If X₁ and X₂ are independent and Xᵢ ~ Binomial(nᵢ, 1/2), i = 1, 2, find the distribution of X₂ − X₁ + n₁. (4)
11. Consider a systematic sampling of selecting n units from a population of N units, where N = nk. Let S²_wsy be the variance among units within the same systematic sample and S² be the variance as a whole. Obtain the condition for which mean of the systematic sample ȳ_sy is more efficient than the mean of the simple random sample ȳ of same size. (6)
12. Let X and Y be independent N(0, 1) random variables. Obtain the distribution of W = X/|Y|. (6)
13. Let X be distributed as Nₚ(μ, Σ). Write down the characteristic function of X and hence find the distribution of CX, where C is non-singular. Explain the procedure of estimating the parameter μ. (4)
14. Distinguish between Hotelling T² in two sample problem and Mahalanobis D² Statistics and obtain the interrelationship between them. What are they primarily used for? (4)
15. Define recurrent state of a Markov chain. Prove or disprove: A state ‘i’ is recurrent if and only if ∑ₙ₌₁^∞ pᵢᵢⁿ is divergent. (4)
16. Carry out the distinction between Poisson, pure birth and birth- death processes. (6)
17. The least square estimator β̂₀ is an unbiased estimator of the model parameter β₀ in fitted regression model. Prove. (4)
18. A = [[1, 0, 2], [0, −1, 1], [0, 1, 0]]. Find A³ by using Cayley-Hamilton theorem. (4)
19. If f ∈ R(α) and g ∈ R(α) on [a, b], then prove fg ∈ R(α). (4)
20. In a logistic Regression model the odds ratio can be interpreted as the estimated increase in the probability of success associated with a one unit change in the value of the predictor variable. Prove. (6)
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