Loyola College Mathematical Statistics – II Question Papers

Loyola College M.Sc. Statistics April 2003 Mathematical Statistics II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – STATISTICS

FourTH SEMESTER – APRIL 2003

St 4953 / s 1072  –  mathematical statistics -ii

 

26.04.2003

1.00 – 4.00                                                                                                      Max : 100 Marks

 

                                                                section – A                                (10´ 2=20 marks)

      Answer ALL the questions. Each carries two marks.

 

  1. Define empirical distribution function.
  2. For the sample central moment compute E (b2).
  3. Let X1, X2, …, Xn be a sample from a Compute the .
  4. Define t and F statistics.
  5. Let X1, X2, X3, X4 be iid random variables with pdf f. Find the marginal pdf of the order statistic X(2) .
  6. Let X1, X2, … be iid N random variables. Show that is an unbiased and consistent estimator of  .
  7. Let X ~ N (0,q). Check if the family of pdf’s is complete.
  8. State Neyman-Pearson fundamental lemma.
  9. Define a likelihood ratio test.
  10. Define a process with independent increments.

 

 

 

                                                         section – B                                         (5´ 8=40 marks)

      Answer any FIVE questions. Each carries EIGHT marks.

 

  1. Let X(1), X(2), X(3) be the order statistics of iid random variables X1, X2, X3 with common pdf

Let Y1 = X(3) – X(2) and Y2 = X(2) . Show that Y1 and Y2 are independent.

  1. State and prove factorization criterion for determining sufficient statistics.
  2. State and prove Rao-Blackwell theorem.
  3. Let X1, X2, …,Xn be iid random variables, where both n and p are unknown. Find the estimates of p and n by the method of moments.
  4. Let X1, X2, …,Xn be a sample from U [q -, q + ]. Show that the statistic T(X1,X2,…,Xn) such that max Xi – T (X1, X2, …, Xn) min Xi +

 

is an MLE of q .

 

  1. Let X1, X2,…,Xn ~ U[0, q], q>0. Show that the family of uniform densities on [0,q] has
    an MLR in max

 

  1. (a)  Explain Chi-square test of goodness of fit.
  • Explain normal test for single proportion.
  1. (a) Classify the stochastic processes with respect to time and state space.
  • State the characteristics of the Brownian motion process.

 

 

 

                                                        section – C                                         (2´20=40 marks)

Answer any TWO questions. Each carries TWENTY marks.

 

  1. (a) Derive the distribution of   in sampling from a normal population.
  • Derive the pdf of Chi-square distribution.
  1. (a) State and prove Cramer-Rao inequality.
  • Let X ~ P (). Find the UMVUE of based on a sample of size one.
  1. (a)  Let X ~ N . Obtain a 100 confidence interval for based on a
    random sample of size n.
  • Let X ~ N (0, 1) under H0 and X ~ C (1, 0) under H1. Find an MP size  test of H0 against H1, and obtain the power of the test.
  • Explain F – test for the equality of population variances.
  1. (a) Establish Chapman – Kolmogorov equation and hence show that the   is
    the power of 1-step tpm.
  • State the postulates for Poisson process.
  • Show that if {N (t)} is a Poisson process, then for , the conditional distribution of N (s) given N (t) = n is binomial .

 

 

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Loyola College M.Sc. Statistics April 2004 Mathematical Statistics – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

  FOURTH SEMESTER – APRIL 2004

ST 4953 – MATHEMATICAL STATISTICS – II

12.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

  1. Define a consistent estimator and give an example.
  2. Show that unbiased estimators do not always exist.
  3. Let X1, X2, …, Xn be iid b (1,p) random variables.

Show that T =  is sufficient for p.

  1. State Lehmann – Scheffe theorem.
  2. State Bhattacharya Inequality.
  3. Write the test function associated with i) a non-randomized test ii) a randomized test.
  4. Define UMP test for testing a simple hypothesis against a composite hypothesis.
  5. Write any four applications of chi-square distribution.
  6. State the postulates for Poisson process.
  7. Define Markov chain and give an example.

 

SECTION – B

 

Answer any FIVE questions                                                                        (5 ´ 8 = 40 marks)

 

  1. Let X1, X2,…, Xn be a random sample from N (m, s2). Show that  and S2 are independent.
  2. Derive the pdf of F-distribution.
  3. Let X1, X2, …., Xn be a random sample from a distribution of continuous type with pdf f(x; q). Derive the joint pdf of jth and kth order statistics, 1 £ j < k £
  4. State and prove Factorization Criterion for determining sufficient statistics.
  5. State and prove Rao-Blackwell theorem.
  6. Let X1, X2, …, Xn be a random sample from Poisson distribution that has the mean q > 0.  Show that  is an efficient estimator of q.
  7. Let X1, X2,.., Xn be iid N (m, s2) random variables where both m and s2 are unknown. Obtain a MP test for testing Ho­: m = mo­ ; s2 =  against H1: m = m1; s2 = .
  8. Show that if {N(t)} is a Poisson process, then for s< t, the conditional distribution of N(s) given N(t) = n is binomial b (n, ).

 

 

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

  1. a) State and Prove Cramer-Rao Inequality.            (10)

 

  1. b) Let X1, X2, …, Xn be iid N(m, s2) random variables. Obtain a confidence interval for m

when (i) s2 is known (ii) s2 is unknown.                                                                    (10)

 

  1. a) State and prove Neyman-Pearson lemma.            (10)

 

  1. b) Let X1, X2, …, Xn be a random sample for N(m, s2) where both m and s2 are unknown.

Derive the likelihood ratio test for testing Ho: m = mo against H1 : m ¹ mo.                  (10)

 

  1. a) Let X1, X2,…, XN be iid b (n, p) random variables, where n and p are unknown. Find

the method of moments estimator for (n,p).                                                                 (7)

 

  1. b) Let X1, X2, …,Xn be a sample for U [q – , q + ]. Show that the maximum

likelihood estimator of q is not unique.                                                                        (7)

 

  1. c) Explain normal test of significance for single mean and give an example. (6)

 

  1. a) Classify the stochastic processes with respect to time and state space. (2)

 

  1. b) State the characteristics of the Brownian motion process (4)

 

  1. c) Establish Chapman – Kolmogorov equaion and hence show that the m – step tpm is the

mth power of 1 – step tpm.                                                                                           (8)

 

  1. d) Explain chi-square test for goodness of fit and give an example. (6)

 

 

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Loyola College M.Sc. Mathematics April 2006 Mathematical Statistics – II Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

AC 50

FOURTH SEMESTER – APRIL 2006

                                              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.

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

SECTION-B   (8 x 5 = 40)

Answer any 5 questions.  Each question carries 8 marks.

 

  1. 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).
  2. Derive the central F-distribution with (r1, r2) degrees of freedom.
  3. 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.
  4. 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.
  5. State and prove Lehmann-Scheffe’s theorem.
  6. Let X have a p.d.f. of the form f(x; θ) = θ xθ-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 ≤ x1­ x2 }.  Find the power function of the test.
  7. Prove or disprove: “UMPT of level α always exists for all types of testing problems”. Justify your answer.
  8. A certain genetic model suggests that the probabilities of a particular trinomial distribution are, respectively, p1 =p2, p2 = 2p (1-p), and p3 = (1-p)2 , where  0 < p < 1.   If X1, X2, X3 represent the respective frequencies in ‘n’ independent trials, explain how we could check on the adequacy of the genetic model.

 

SECTION-C       ( 20 ´ 2 = 40 )

Answer any 2 questions. Each question carries 20 marks.

  1. a) State and prove Factorization theorem. (12)
  2. 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)

 

  1. a) State and prove the sufficiency part of Neyman-Pearson theorem. (12)
  2. b) Let X1, X2,…, Xn denote a random sample from a distribution having the p.d.f.
    f(x; p) = px (1-p)1-x , x = 0,1, zero, elsewhere.  Show that C =  { (x1, …,xn) :  Σ xi ≤ 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)

 

  1. 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)
  2. b) By giving suitable examples, distinguish between unpaired and paired t-tests. (8)

 

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

 

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Loyola College M.Sc. Mathematics April 2007 Mathematical Statistics – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 56

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2007

ST 4900 – MATHEMATICAL STATISTICS – II

 

 

 

Date & Time: 25/04/2007 / 9:00 – 12:00      Dept. No.                                       Max. : 100 Marks

 

 

 

SECTION-A (10 x 2 = 20)

Answer ALL the questions.   Each carries 2 marks.

 

  1. Distinguish between point and interval estimation.
  2. Examine whether the sample variance is a biased estimator of s2 when a random sample of size ‘n’ is drawn from N(m, s2 ).
  3. When do you say that a statistic is a consistent estimator of a parameter?
  4. Distinguish between power function and power of the test.
  5. What is degree of freedom?
  6. Illustrate graphically the meaning of UMPT.
  7. What do you understand by likelihood ratio tests?
  8. Define F statistic.
  1. Define: Markov Chain.
  1. Classify the stochastic processes with respect to time and state space.

SECTION-B  (5 x 8 = 40)

Answer any FIVE questions.  Each carries 8 marks.

 

  1. Let Y1 and Y­2 be two independent unbiased estimators of θ. Let the variance of Y1 be twice the variance of Y2. Find the constants k1 and k2 so that k1 Y1 + k2 Y2 is an unbiased estimator with smallest possible variance for such a linear combination.
  2. State and prove Rao-Cramer Inequality.

 

  1. Let X1, …, Xn be i.i.d., each with distribution having p.d.f.

f(x; θ1, θ2 ) =     (1 / θ2) exp{ – ( x- θ1 ) /θ2},  -θ1 ≤ x< ∞, -∞ <  θ1 <  ∞,  0 <θ2< ∞,

0;  elsewhere.

Find the maximum likelihood estimators of θ1 and θ2.

  1. Let X ≥ 1 be the critical region for testing H0: θ = 1 against H1: θ = 2 on the basis of a single observation from the population with pdf

f(x ,θ) =  θ exp{ – θ x },  0 < x <∞;  0 otherwise.

Obtain  the size and power of the test.

 

  1. Let X1, X2…Xn be iid U(0,q), q>0.  Show that the family of distributions has MLR in X(n).

 

  1. A random sample of size 14 drawn from a normal population provides a sample mean of 3.22mm with an unbiased standard deviation of 0.34mm. Can you conclude at 5% level of significance that it meets the  company’s specification of 2.7mm against more than 2.7mm?  Construct 95% confidence limits for the  population mean.
  2. Let X1, X2,…,Xn be iid N(m,s2). Derive LRT for testing H0: m = m 0 against

H1: m ¹ m0, s2  is unknown.

 

  1. Describe Poisson process.

 

SECTION-C (2 x 20 =40)

Answer any TWO questions.   Each carries 20 marks.

 

  1. a) State and prove factorization criterion for determining sufficient statistics. (12)
  2. b) Show that the first order statistic Y1 of a random sample of size n from the

distribution having p.d.f.

f(x: θ) = e–(x – θ), -¥ < x < ¥, – ¥ < θ < ¥, zero elsewhere,

is a complete sufficient statistic for θ.

Find the unique function of this statistic, which is the unbiased minimum variance

estimator of θ.                                                                                                   (8)

 

  1. State and prove the necessary and sufficient conditions of Neyman-Pearson

Fundamental Lemma.                                                                                    (10+10)

 

  1. a) Let X1,X2 and X3 be a sample of size 3 from Poisson distribution with mean θ.

Consider the problem of testing H0: θ = 2 against H1: θ = 3.  Find the randomized

MPT of level α =0.05.                                                                                      (12)

 

  1. b) Prove or disprove:

“UMPT of level α always exists for all types of testing problems”.

Justify your answer.                                                                                         (8)

 

  1. a) A number is to be selected from the interval (x : 0 < x < 2) by a random process.

Let Ai = {(x : (i-1)/2 < x < i/2}, i = 1,2,3, and let A4 = {x : 3/2 < x < 2}.  A certain

hypothesis assigns probabilities pio to these sets in accordance with

pio = òAi (1/2) (2-x) dx, i = 1,2,3,4.  If the observed frequencies of the

sets Ai, i = 1,2,3,4, are respectively, 30, 30, 10, 10, would Ho be accepted at the

(approximate) 5 percent level of significance ?                                                 (10)

 

  1. b) Consider a Markov chain having state space S = {0,1,2} and transition probability

matrix

 

1/3      1/3       1/3

 

¼        ½          ¼

 

1/6       1/3        ½

 

 

Show that this chain has a unique stationary distribution p and find p.          (10)

 

 

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