LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 51

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2007

ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

 

 

 

Date & Time: 23/04/2007 / 9:00 – 12:00      Dept. No.                                          Max. : 100 Marks

 

 

SECTION A

Answer ALL the Questions                                                                  (40 X 1 = 40 Marks)

 

1. The events A = {1, 2}, B = {2, 3} and C= {2, 4} are exhaustive and A and B are independent .If P (A) = ½ and P (B) = ⅓, what must be P(C)?

(A) 1/6            (B) ⅔           (C) ½                (D) 5/6

2. If P AUB) =5/6 and P (A) = ½, then P(B/A^C) is

(A) 2/3            (B) 3/5         (C) 1/3              (D) ½

3. For what value of λ, the random variable, whose distribution function is

F(x) =      0                 if x < -1

1-λe ^–x/2         if x ≥ -1

is continuous?

(A) 1               (B) 1 /√e      (C) ½                (D) √e

4. A random variable X takes the values -1, 0, 2, and 4 with respective probabilities 1/6, ⅓, ⅓, 1/6. What is the expected value of X/(X+2)?

(A) 1/18          (B) 1/9         (C) 1/36            (D) -1/36

5. If X₁ and X₂ are independent and identically distributed Geometric random variables with parameter 1/3, then the distribution of Y= min (X₁, X₂) is Geometric with the parameter

(A) 5/9            (B) 1/3         (C) 1/9              (D) 4/9

6. A box contains 7 marbles of which 3 are red and the rest are green. If 4 marbles are drawn from the box at random without replacement, what is the probability that 3 marbles are green?

(A) 1/35          (B) 18/35    (C) 4/35             (D) 12/35

7. A random variable X distributed uniformly is such that P(X < 9) =1/8 and

P(X > 22) = 1/3. What is P (11 < X < 19)?

(A) 1/3            (B) 1/8         (C) ½                (D) 4/15

8. If (X, Y) has standard bivariate normal distribution with correlation coefficient ρ, what should be the value of λ in order that (X + λY) and Y are independently distributed?

(A) -1              (B) 1            (C) -ρ               (D) ρ

9. If X₁ and X₂ are independent random variables each having the distribution function G, then the distribution function of min(X₁, X₂) is

(A) G              (B) G ²           (C) G (2-G ² )     (D) G (2-G)

10. If X is an exponential random variable with E (e^X) = 2, then E(X) is

(A) 1/2            (B) 1             (C) 2                 (D) 6

 

11. A random variable X has the probability density function

f (x) =  (e^-x x ^m )/m!         if x > 0,m>0

=    0     otherwise

The lower bound for Pr (0 < X < 2(m+1)) is

(A)  m/m+1    (B) 1/m+1    (C) 1/2              (D) 1/m

12. If R_1.23 =1, then the value of R_2.13 is

(A) 0              (B)-1            (C) ½                (D) 1

13. If (0.1, 0.2) is the strength of a SPRT, its approximate stopping bounds are

(A) (2/9, 8)     (B) (1/8, 9/2)  (C) (1/8,8)         (D) (2/9, 9/2)

 

14. Choose the correct statement in connection with a standard LP problem.

(A)  Variables can be unrestricted

(B)  All constraints can be less than or equal to type (or) greater than or equal to type

(C) An LP involving only equal to type constraints may not require application of  big-M  method

(D) All variables must be nonnegative

 

15. When a LPP has feasible solution, at then end of Phase-I in two-phase method,

the objective function’s value will be

(A) >0       (B) <0              (C) 0                (D) infinity

 

16. The number of basic vells in any IBS solution for a TP is

 

(A) m+n+1            (B) m+n-1       (C) m-n+1       (D) n-m+1

 

1. Given the following simplex table (associated with a maximization problem)

 

Basic   z          x1        x2        x3        x4        Solution

 

z          1          -4         -2         0          0          8

 

x3        0          0          2          1          0          1

 

x4        0          -1         1          0          1          2

 

The above table indicates

(A) several optima       (B) degeneracy

(C) unbounded solution  (D) all of them

 

18. An LPP has 6 variables and 3 constraints. How many sets of basic variables are possible ?

 

(A) 10        (B) 6                (C) 3                (D) 20

 

• The power function associated with the UMPT for testing against the alternative in is always

(A) Strictly increasing in      (B) Strictly decreasing

(C) Periodic in                                 (D) can’t say

 

1. Which of the following is the form of UMPT for testing  against the alternative in

(A)                        (B)

(C)                        (D)

1. A UMPUT can be found for a testing problem on finding the UMPT in the class

of all

 

(A) Unbiased tests                  (B) Similar tests

(C) Invariant tests       (D) All the three mentioned in (A), (B) and (C)

 

22. Two phase sampling is resorted when

(A) variance of an estimator can not be estimated

(B) we can not use systematic sampling schemes

(C) auxiliary information is not fully known

(D) sensitive information is to be gathered

 

23. Among the following which can not use Yates-Grundy estimated variance

(A) Simple random sampling              (B) PPSWR

(C) Linear systematic sampling           (D) Midzuno sampling

 

24. When N=20 and n=4 which of the following represents ideal group size under

random group method ?

(A) 4 each       (B) 5  each       (C) 4,6,5,5       (D) 6,6,2,2

 

25. Choose the correct statement

(A) Ratio estimator is always unbiased

(B) Regression estimator is always unbiased

(C) RR methods are not associated with sensitive attributes

(D) Yates Grundy estimator is non-negative under Midzuno scheme

 

26. Choose the correct statement

(A) Systematic sampling is a particular case of cluster sampling

(B) Cluster sampling is a particular case of systematic sampling

(C) While forming strata we should ensure that within-stratum variability is more

(D) Proportional allocation is better then optimum allocation

 

 

27. Which of the following is a tree wrt a network consisting of 5 nodes ?

 

 

 

 

(A)                                                          (B)

 

 

 

 

 

 

 

 

 

(C)                                                                   (D)

 

 

 

 

 

 

 

 

• The function f(x) = | x + 1| is NOT differentiable at

(A) 0                (B) 1                (C) –1              (D) f is differentiable everywhere

 

29.A tosses a fair coin thrice and  B throws a fair die twice. Let

a = Probability of getting an odd number of heads

b = Probability that the sum of the numbers that show up is at least 7.

Then

(A) a > b        (B) a < b         (C) a = b         (D) a + b < 1

 

• The system of equations

x – y +3z = 3

4x – 3y +2z = 7

(3m – 1)x – y – 4 = 2 m² – 1

has infinitely many solutions if m equals

(A) 0                (B)1                 (C)2                 (D)3

 

• The mean and variance of 6 items are 10 and 5 respectively. If an observation 10 is deleted from this data set, the variance of the remaining 5 items is

(A)5                 (B)6                 (C)7                 (D)8

 

• Which of the following forms a basis for R³ along with (1, 2, – 1) and (2, – 2, 4)?

(A) ( 0, 0, 0)     (B) (2, 1, 1)      (C) (3, 0, 3)      (D) (1, 4, – 2)

 

• In a bivariate dataset {(X_i, Y_i), i =1, 2, …,n}, X assumed only two values namely 0 and – 1 and the correlation coefficient was found to be –0.6. Then , the correlation coefficient for the transformed data {(U_i, V_i), i =1, 2, …,n}, where U_i = 4 – 2 X_i³ and V_i = 3Y_i + 5, is

(A) 0,6             (B) – 0.6          (C)0                 (D) cannot be determined

 

• Which one of the following cannot be the 1^st and 2^nd raw moments for a Poisson distribution?

(A) 2, 6                        (B) 4, 12          (C) 5, 30          (D) 6, 42

 

• Let X and Y be random variables with identical means and variances. Then

(A)  X + Y and X – Y are uncorrelated

(B)  X + Y and X – Y are independent

(C)  X + Y and X – Y are independent if X and Y are uncorrelated

(D)  X + Y and X – Y are identically distributed if X and Y are uncorrelated

 

• If X₁, X₂, …, X_n is a random sample form N (q ,1), – µ < q < µ, which of the following is a sufficient statistic?

(A) S X_i²         (B)S (X_i – )²           (C) (SX_i , SX_i²)         (d) None of these

 

• Let be an unbiased estimator of a parameter. The Rao-Blackwell Theorem is used to

(A) get an improved estimator of  by conditioning upon any sufficient statistic

(B) get an equally good estimator of  by conditioning upon any sufficient

statistic

(C) get the UMVUE of   by conditioning upon any sufficient statistic

(D) get the UMVUE of  by conditioning upon a complete sufficient statistic

 

• The upper control limit of a c- chart is 40. The lower control limit is

(A) 0                (B) 10              (C) 20              (D) Cannot be determined

 

• The linear model appropriate for two-way classification is

(A) Y_ij = a_i + b_j + e_ij                                        (B) Y_ij = m + a_i + e_ij

(C) Y_ij = m + b_j + e_ij                                                   (D) Y_ij = m + a_i + b_j + e_ij

 

• In a 2⁴ factorial experiment with 5 blocks, the degrees of freedom for Error Sum of Squares is

(A)20               (B)40               (C)60               (D)70

 

SECTION B

Answer any SIX Questions                                                                   (6 X 10 = 60 Marks)

 

41. Let the joint probability mass function of (X,Y) be

e ^{– (a+b)} a^x b^y-x / [x! (y – x)!]         ,      x = 0,1, 2,…,y ;  y = 0,1 ,2,….

f(x, y) =

0 otherwise, where (a, b) > 0.

Find the conditional probability mass functions of X and Y.

 

 

 

 

 

42. Using central limit theorem , prove that

∞    e^-t t ^n-1

Lim   ∫      ———— dt = ½.

n→∞    0      (n-1)!

 

43. Consider a Poisson process with the rate λ (>0). Let T₁ be the time of occurrence of the first event and let N (T₁) denote the number of events in the next T₁ units of time.

Show that E [N (T₁).T₁] = 2/λ and find the variance of N(T₁).T₁.

 

44. Explain how will you solve the following game theory problem using linear
    programming  technique (Complete solution needed)

 

B1       B2       B3

 

A1       3          -1         -3

 

A2       -2         4          -1

 

A3       -5         -6         2

45. Show that family of binomial densities has MLR in
46. Develop Hartley-Ross ratio type unbiased estimator under simple random

sampling.

 

47(a) Let X₁, X₂, X₃, X₄ have the multinomial distribution with parameters q₁,q₂,q₃,

q₄ and q₅ where q₅ = 1 – (q₁ + q₂ + q₃ + q₄) and n = 30. If the observed values of

the random variables are X₁ = 7, X₂ = 4, X₃ =6, X₄ = 9, find the MLEs of the

parameters.

(b) Obtain the MLE of q based on a random sample of size 7 from the double

exponential distribution with p.d.f  f(x,q_­) = exp (– |x – q | )/2, – µ < x , q < µ .

(7 + 3)

 

48 (a) If X₁,….,X_n is a random sample from U(0, 1), show that the nth order statistic  converges in

probability to 1.

(b)  Let T₁ and T₂ be stochastically independent unbiased estimators of q and let V(T₁) be four

times V(T₂). Find constants c₁ and c₂ so that c₁T₁ + c₂T₂ is an unbiased estimator of q with

the smallest possible variance for such a linear combination.                                          (5 + 5)

 

 

49.(a)Let ‘p’ be the probability that the mean of a sample of size ‘n’ falls outside

the control limits of a control chart. Derive an expression for the following:

P{Atmost ‘x’ samples are to be taken for ‘r’ points to go out of the control

limits}

(b) For samples of size n =2, give the theoretical justification for the value of  A_2­

used to determine the control limits for the Chart for means.                                                (4 + 6)

 

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