ST 4900 – MATHEMATICAL STATISTICS – II

(Also equivalent to ST 4953)

Date & Time : 03-05-2006/9.00-12.00 Dept. No. Max. : 100 Marks

SECTION-A (10 ´ 2 = 20)

Answer ALL questions. Each question carries 2 marks.

Let T have a t-distribution with 10 degrees of freedom. Find P(|T| > 2.228).
Find the variance of S² = (1/n) ∑ ( x_i – x )² , when X₁, X₂,…., X_n is a random sample from N(µ , σ² ).
How do you obtain the joint p.d.f. of any two order statistics Y_r and Y_s when Y_r < Y_s ?
What do you understand by a sufficient statistic for a parameter?
Define: UMVUE.
State Rao-Cramer Inequality.
Distinguish between randomized and non-randomized tests.
Illustrate graphically, the meaning of UMPT of level α test.
Define a renewal process.
When do you say that a stochastic matrix is regular?

Answer any 5 questions. Each question carries 8 marks.

Let and S² be the mean and the variance of a random sample of size 25 from a distribution N (3, 100). Evaluate P (0 < < 6, 55.2 < S² < 145.6).
Derive the central F-distribution with (r₁, r₂) degrees of freedom.
Let Y₁ < Y₂ < Y₃ be the order statistics of a random sample of size 3 from the uniform distribution having p.d.f.
f(x; θ ) = 1/θ, 0 < x < θ, 0 < θ < ∞, zero elsewhere. Show that 4Y₁, 2Y₂ and (4/3)Y₃ are all unbiased estimators of θ. Find the variance of (4/3)Y₃.
If az² + bz + c = 0 for more than two values of z, then show that a = b = c = 0. Use this result to show that the family{ B(2, p): 0 < p < 1} is complete.
State and prove Lehmann-Scheffe’s theorem.
Let X have a p.d.f. of the form f(x; θ) = θ x^θ-1 , 0 <x < 1, θ =1,2, zero elsewhere. To test H₀ : θ =1 against H₁: θ =2, use a random sample X₁, X₂ of size n = 2 and define the critical region to be C = { (x₁, x₂) : ¾ ≤ x₁x₂}. Find the power function of the test.
Prove or disprove: “UMPT of level α always exists for all types of testing problems”. Justify your answer.
A certain genetic model suggests that the probabilities of a particular trinomial distribution are, respectively, p_{1 =}p², p₂ = 2p (1-p), and p₃ = (1-p)² , where 0 < p < 1. If X_1,X₂, X₃ represent the respective frequencies in ‘n’ independent trials, explain how we could check on the adequacy of the genetic model.

Answer any 2 questions. Each question carries 20 marks.

a) State and prove Factorization theorem. (12)
b) Given the p.d. f. f(x; θ) = 1 / ( π [1 + ( x – θ)² ) , -∞ < x < ∞ , -∞ < θ < ∞. Show that the Rao-Cramer lower bound is 2/n, where n is the size of a random sample from this Cauchy distribution. (8)

a) State and prove the sufficiency part of Neyman-Pearson theorem. (12)
b) Let X₁, X₂,…, X_n denote a random sample from a distribution having the p.d.f.
f(x; p) = p^x (1-p)^1-x , x = 0,1, zero, elsewhere. Show that C = { (x₁, …,x_n) : Σ x_i ≤ k }is a best critical region for testing H₀: p = ½ against H₁: p = 1/3. Use the central limit theorem to find n and k so that approximately the level of the test is 0.05 and the power of the test is 0.9. (8)

a) Derive the likelihood ratio test for testing H₀: θ₁=0, θ₂ > 0 against
H₁: θ₁ ≠ 0, θ₂ >0 when a random sample of size n is drawn from N(θ₁ , θ₂ ). (12)
b) By giving suitable examples, distinguish between unpaired and paired t-tests. (8)

a) Show that the Markov chain is completely determined by the transition matrix and the initial distribution. (8)
b) Give an example of a random walk with an absorbing barrier. (4)
c) Explain in detail the properties of a Poisson process. (8)