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)

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.

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

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

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

If X₁ and X₂are 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

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

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)

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)

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)

If X is an exponential random variable with E (e^X) = 3, then E(X) is

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

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)

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

(A) 0 (B)-1 (C) ½ (D) 1

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)

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

(D) All variables must be nonnegative

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

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

Consider the following LPP:

Max Z = X₁+X₂

Subject to X₁3

X₁, 2X₂4

X₁,X₂, 0

The solution is

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

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

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

(A) (B)

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

of all

(A) Unbiased tests (B) Similar tests

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.

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

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 +

(D) V(Y) = V +

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

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

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

(A)

(B)

(C)

(D)

Choose the correct statement,

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

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

(D) Proportional allocation is better then optimum allocation

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

(A) (B)

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

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)

Let X₁ and X₂be independent Poisson random variables. Show that the conditional distribution of X₁given X₁+ X₂= n is binomial.
State Liapunov form of central limit theorem. Give an application.
State and establish any two properties of a Poisson process.
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

Derive UMP level test for testing based on a random

sample from

Given a random sample from , find the moment estimators of

(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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