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1. A point is chosen at random inside a circle. Find the probability that the point is more close to the centre of the circle than the circumference of the circle. (3)
2. Show that Binomial distribution belongs to the exponential family of distributions. (3)
3. In order to test whether a coin is perfect, it is tossed 5 times. The null hypothesis of perfectness is rejected if and only if more than 4 heads are obtained. Find the probability of type I error. (3)
4. Give the layouts of a 4 × 4 Latin square design with the treatments A, B, C and D. (3)
5. Define Binomial distribution. Find its mean and variance. (3)
6. Examine whether f(x) = x for 0 < x < 1, f(x) = 2 − x for 1 ≤ x ≤ 2, and f(x) = 0 otherwise is a probability density function. (3)
7. Define Hotelling’s T²-statistics. (3)
8. What is the difference between random sampling and non-random sampling ? (3)
9. S. T. sample variance is an unbiased estimate of the population variance. (3)
10. Explain the problem of auto-correlation in general regression model. (3)
11. Define quadratic form. (3)
12. Define the following:
(i) Matrix
(ii) Idempotent matrix
(iii) Rank of a matrix. (3)
13. Define transition probability matrix. (3)
14. State Ergodic theorem. (3)
15. If X is a random variable with mean α, show that E(X − t)² is minimum when t = α. (4)
16. Let X₁, X₂, …, Xₙ be a random sample from a population having uniform distribution in the interval (μ − √3σ, μ + √3σ). Find the estimators of μ and σ by the method of moments. (4)
17. Given H₀: Median = 5. Compute T⁺, T⁻ and T for the following observations: 6, 10, 3, 5, 2, 12. (4)
18. Define Weibull distribution. Find its mean and variance. (4)
19. Define χ², t and F distributions. (4)
20. Describe spectral decomposition of a symmetric matrix. (4)
21. State Cayley–Hamilton theorem. Also using Cayley–Hamilton theorem, compute A⁻¹ in terms of powers of A. (4)
22. Let X̄ₙ be the mean of a random sample of size n from a population with mean μ and variance σ². Show that X̄ₙ converges stochastically to μ, if σ² is finite. (5)
23. Derive the confidence interval for the difference of means of two normal populations, based on small samples of sizes n₁ and n₂ taken from the populations by assuming the variances of the populations are same. (5)
24. Let X have the distribution f(x, λ) = λˣ(1 − λ)¹⁻ˣ, x = 0, 1; 0 < λ < 1. Construct the SPRT for testing H₀: λ = λ₀ against H₁: λ = λ₁. (5)
25. For a resolvable BIBD with parameters v, b, r, k and λ, show that b ≥ v + r − 1. (5)
26. a) Define joint, marginal and conditional p.d.f. of multivariate distributions.
b) What is lack of memory property ? Give discrete and continuous distributions having lack of
memory property.
c) Define multivariate normal distribution. (5)
27. In the general linear model Y = Xβ + U, with the usual notation, and if U follows Nₙ(0, σ²Iₙ), derive an unbiased estimator for σ². (5)
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