LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 49

FOURTH SEMESTER – APRIL 2006

                               ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

 

 

Date & Time : 25-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

SECTION A

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

 

1. If the difference between two numbers is 1.2, then the variance of them is

(A) 0.18          (B) 1.44        (C) 0.72       (D) 0.36

2. To test the hypothesis that the variance of a normal distribution is 2, the test procedure used is

(A) Normal test        (B) Chi-square test    (C) F-test   (D) t-test

3. To test which one of the following hypothesis, F-test is used?

(A) Goodness of fit (B) equality of means of two normal populations

(C)  Significance of correlation coefficient (D) Equality of Variances of two normal populations

4. If X has Poisson distribution with 3P[X = 2] = 2P[X ≤ 1], then the expected value of X is

(A) 3               (B) -2/3           (C) 2           (D) 3/2

5. X₁, X₂ and X₃ are independent observations on a normal random variable with

mean μ and variance σ².What is the efficiency of   (3X₁+2X₂+X₃ ) / 6 as an estimator of  μ ?

(A) 6/7             (B) 1           (C) 1/3           (D) 1/12

1. If E(Y/X) = α X + β and X has standard normal distribution, then E(Y) is

(A) 0               (B) 1                (C) β          (D) α

∞

1. If P ( A_n) = 1, n=1, 2, 3… then the value of P (A_n) is

n=1

(A) 0               (B) 1                (C) 1/2       (D) ¼

8. A random variable X has characteristic function

Φ (t) = (sin t)/t,    t ≠0

1          otherwise

Then, Var(X) is equal to

(A) 1        (B) 0       (C) 1/6        (D) 1/3

9. If X₁ and X₂ are independent and identically distributed random variables with p(x) = q^x.p, x = 0, 1, 2, 3… and (p+q) = 1, then the distribution of (X₁+X₂) is

(A) Geometric        (B) degenerate          (C) Negative Binomial    (D) Hyper- geometric

 

10. In a random sample of size n from the distribution

dF (x) = e^-x.dx,   0<x<∞,

the mean of the smallest sample value is

(A) 1/n      (B) 1/n²      (C) 0       (D) 1

11. If the degrees of freedom for error in the analysis of Variance for a Latin square design is 30, the number of treatments is

(A) 5               (B) 6           (C) 7           (D) not possible to determine

12. If a population consists of 10 units and the population Variance is 20, the Variance of the sample mean of a simple random sample pf size 4 without replacement is

(A) 5               (B) 2           (C) 20           (D) 3

13. The number of simple random samples of size 4 that can be drawn without replacement from a population with 12 units is

(A) 12⁴           (B) 495        (C) 11,880        (D) 48

14. The standard deviation of a symmetric distribution is 4. The value of the fourth moment about the mean in order that the distribution be leptokurtic is

(A) greater than 768              (B) equal to 768           (C) equal to 256       (D) less than 48

15. Given Maximize subject to

 

 

 

For what values of the above problem will have several optimum solutions?

(A) 2    (B) 3    (C) 6    (D) 1

 

1. The objective function in the phase-I (when we use two phase simplex method) is formed by

• summing all the variables
• summing all the artificial variables

(C)   taking the product of artificial variables

• subtracting the sum of artificial variables from the sum of other variables

 

17. The following set of constraints require x artificial variables

 

 

where x is

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

 

18. 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          4          3          1          0          1

 

x4        0          -1         1          0          1          2

 

The leaving and entering variables are

(A) x1, x3        (B) x1, x4        (C) x2,x3         (D) x2,x4

 

1. An LPP has 4 variables and 2 constraints. How many sets of basic variables are possible?

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

1. 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)

22. Choose the correct statement

(A) Power functions of UMPTs are always monotone

(B) A UMPT is always UMPUT

(C) MPT’s are not unique

(É)All similar tests will have Neyman structure

23. Choose the correct statement

(A) RR methods are not associated with sensitive attributes

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

(C) HTE can not be used under PPSWOR

(D)Balanced systematic sampling is not recommended for populations with linear

trend.

24. Lahiri’s method

(A) is a PPS selection method involving a given number of attempts

(B) is a PPS selection method involving unknown number of attempts

(C) is an equal probability selection method involving a given number of attempts

(D) is an equal probability selection method involving unknown number of

attempts

 

25. Ratio estimator is

(A) a particular case of regression estimator

(B) an unbiased estimator

(C)more suitable when y and x have high negative correlation

(D) more suitable when y and x have no correlation

 

26. Random group method is due to

(A) Desraj       (B) Murthy      (C) Hartley-Ross         (D) Rao-Hartley-Cochran

 

27. Randomised response methods are meant for

(A) homogeneous data

(B) heterogeneous data

(C) sensitive data

(D) stratified populations

 

28. Which name is associated with shortest route problems

(A) Kuhn-Tucker

(B) Floyd

(C) Charnes

(D) Karmakar

 

29. Which of the following functions is NOT continuous at 0?

(A) |x|              (B) e^x               (C) x – [x]                   (D)Sin x

 

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

a = Probability of getting at least two heads

b = Probability that the sum of the numbers that show up is less than 6

Then

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

 

31.The system of equations

2x + 4y – z = 3

x + 2y +2z = 2

x + (m²+1) y + 7z = 4m – 1

has infinitely many solutions if m equals

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

 

32. The mean and variance of 8 items are 10 and 100 respectively. An observation 3

is deleted from the data. The variance of the remaining 7 observations is

(A)100             (B)106             (C)112             (D)120

 

33. Let T: R³ ® R² be defined as T(x, y, z) = (3x + y – z , x + 5z). The matrix

corresponding to this linear transformation is

(A)       (B)               (C)        (D)

 

34. Which of the following is NOT true of a normal variable with mean 0?

(A) E(X²) = 1, E(X³) = 0                        (B) E(X²) = 1, E(X⁴) = 2

(C) E(X²) = 2, E(X⁴) = 12                      (D) E(X²) = 1/2 , E(X⁴) = 3/4

 

35. If X and Y are uncorrelated random variables with equal means and variances,

then

(A) X + Y and X – Y are identically distributed

(B) X + Y and X – Y are independent

(C) X + Y and X – Y are negatively correlated

(D) X + Y and X – Y have equal variance

 

36. 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 –0.3. Then , the correlation coefficient for the transformed data {(U_i, V_i), i =1, 2, …,n}, where U_i = 3 – 5 X_i² and V_i = 2Y_i –3  is

(A) –0.3           (B) 0.3             (C) 0                (D) cannot be determined

 

37. If X₁, X₂,…, X_n is a random sample from U( 0, q), which of the following is a biased estimator of q?

(A) 2           (B) X_(n)                        (C)X₁ + X_n      (D) (n+1)X_(n) / n

 

38. The Cramer-Rao lower bound for estimating the parameter l of a Poisson distribution based on a random sample of size n is

(A) l               (B) nl              (C) l / n           (D) l^1/2

 

39. The lower control limit of a c-chart is 4. The upper control limit is

(A)16               (B)20               (C)24               (D) none of these

 

40. In a 2⁴ factorial experiment with 4 blocks the degree of freedom for Error Sum of Squares is

(A) 25              (B)35               (C)45               (D)55

SECTION  B

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

 

41. Explain the procedure for solving a game theory problem graphically

 

42. Show that family of Uniform densities binomial densities has

MLR in

43. A sample has two strata with relative sizes and . He believes

that . For a given cost , show that (assuming stratum

sizes are large)

 

 

 

44. The exponent of a bivariate normal density is given below:

– ⅔(x²+9y²-13x-3xy+60y+103)

Find μ₁, μ₂, σ₁, σ₂ and ρ.

 

45. The number of accidents in a town follows a Poisson process with the mean of 2

accidents per day and the number of people involved in i^th accident has the

distribution

P[X₁=k] = 1/ 2^k, k≥1.

Find the mean and variance of the number of people involved in accidents per

week.

 

46. If Φ is a characteristic function, show that e ^{λ (Φ -1)} is a characteristic function for

all λ>0.

 

47. (a)Let X­_{1, …,}X_n, X_n+1 be a random sample from N(m, s²). Let M be the average of the first ‘n’ observations and S² be the unbiased estimator of the population variance based on the first ‘n’ observations. Find the constant ‘k’ so that the statistic k( M – X_n+1) /S follows a t- distribution.

 

(b) Let X have a Poisson distribution with parameter q. Assume that the

unknown q is a value of a random variable which follows Gamma distribution

with parameters a = a / ( 1- a) and  p = r, where ‘r’ is a positive integer. Show

that the marginal distribution of X is Negative Binomial.                          (5 + 5)

 

48. (a) Let X₁,…,X_n be a random sample from Poisson distribution with parameter l. Starting with the initial estimator X₁² – X_1­ for l², use Rao-Blackwellization to get an improved estimator by conditioning on the sufficient statistic S X_i. State whether the resulting estimator is UMVUE and justify.

(b) Let X₁, …, X_n be a random sample from N(m, s²). Obtain an unbiased and

consistent estimator of s⁴.                                                                           (6 + 4)

 

49. (a) Derive an expression for E(Mean Treatment Sum of Squares) in LSD.

(b) Consider four quantities T₁, …,T₄ and let T₁ – 2T₂ + T₃ be a contrast. Find

two other contrasts so that all the three are mutually orthogonal.              (7 + 3)

 

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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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NO 57

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2008

ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

 

 

 

Date : 03-05-08                  Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

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

 

1. Let  A, B, and C  be independent and exhaustive events. If P (A) = ½ and P(B) = ⅓, fuid P(C).

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

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 < 0

1-λe ^–x/3         if x ≥ 0

is continuous?

(A) 1               (B) 1 /√e      (C) ⅓                (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-2)?

(A) 1/18          (B) 1/9         (C) -5/6            (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 10 balls of which 4 are red and the rest are black. If 3 marbles are drawn from the box at random without replacement, what is the probability that 2 marbles are red?

(A) 1/10          (B) 1/2       (C) 3/10             (D) 1/12

7. A random variable X distributed uniformly is such that P(X < 5) =1/2 and

P(X > 7) = 1/6. What is the support of X)?

(A) (5,7)            (B) (0,8)         (C) (1,9)                (D) (2,8)

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

(A) –              (B)             (C)                (D)

9. If X₁ and X₂ are independent random variables each having the distribution function G, then the distribution function of max(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) = 3, then E(X) is

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

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 variance of X is

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

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)  constraints can involve non-linear term.

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

(C) The objective function can involve non-linear terms.

(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 cells in any IBF solution for a TP is

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

1. Consider the following LPP:

Max Z = X₁ +X₂

Subject to  X₁ 3

X₁, 2X₂4

X₁,X₂, 0

The solution is

(A) (0,2)          (B) (3;0)          (C) (3, )        (D) (4,0)

18. An LPP has 6 variables and 4 constraints. How many sets of basic variables are

possible ?

(A) 10        (B) 6                (C) 20              (D) 15

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

(A) strictly increasing in      (B) strictly decreasing in

(C) periodic in                      (D) constant.

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)

1. Stratified sampling is advantageous over SRS

(A) If the within stratum variances more than the population variance.

(B) If the within stratum variances are less than the population variance.

(C) Without any condition.

(D) If the averages within all the strata are equal.

1. The variance of a random vairable in terms of conditional expectations and

conditional

variance is given by

(A) V(Y) = V –

(B) V(Y) = V +

(C) V(Y) = V +

(D) V(Y) = V +

24. If ₣ is the trivial sigma field, than E(X l ₣) is

(A) Constant   (B) E (X)         (C) X   (D) none of the above

25. The inclusion probability of any two specified units in a SRSWOR is

(A)

(B)

(C)

(D)

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 – 3| is NOT differentiable at

(A) 0                (B) +3              (C) –3              (D) f is differentiable everywhere

29.A tosses a fair coin twice 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)  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 could be the 1^st and 2^nd raw moments for a Poisson distribution?

(A) 2, 8                        (B) 4, 12          (C) 5, 30          (D) 6, 40

• Let X and Y be random variables with identical means and variances. and if X and Y are un correlated then ca (X+Y, X-Y) is

(A)  (X)                (B)  0               (C)  2V(Y)      (D)  1

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

(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 theorem which provides an improved unbiased estimator is

(A) Factorization theorem (B) Basis theorem (C) Rao-Blackwell theorem

(D) Gauss-Markor theorem.

• The upper control limit of a c- chart is 48. The lower control limit is

(A) 0                (B) 8                (C) 4                (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 X₁ and X₂ be independent Poisson random variables. Show that the conditional distribution of X₁ given X₁+ X₂= n is binomial.
42. State Liapunov form of central limit theorem. Give an application.
43. State and establish any two properties of a Poisson process.
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. Derive UMP level test for testing  based on a random

sample from

46. Given a random sample from , find the moment estimators of

.

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

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

the random variables are X₁ = 9, X₂ = 8, X₃ =9, find the MLEs of the

parameters.

(b) Obtain the MLE of q based on a random sample of size 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,), find the distribution of the

range.

(b)Given a random sample from , find Bhattacharyya lower bound of order 2 for estimating .                                                                                                    (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(r out of k samples give a point out of control limits).

(b) If the mean is normally distributed and the control limits are 3-sigma limits, find the above probability with r=2, k=4.                      (4 + 6)

 

 

 

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