Indian Institute of Technology Guwahati (IIT-G) 2007 JAM Maths Statistics (M.ScEntrance) - Question Paper
JAM Maths Statistics (M.Sc. Entrance)
Full ques. Paper in attachment
Mathematical Statistics Paper
2007
IMPORTANT NOTE FOR CANDIDATES
Attempt ALL the 25 questions.
Questions 1-15 (objective questions) carry six marks each and questions 16-25 (subjective questions) carry twenty one marks each.
Write the answers to the objective questions in the Answer Table for Objective Questions provided on page 7 only.
Let the random variable X have binomial distribution with parameters 3 and 6. A test of hypothesis H0 : 6 = 3/4 against H1 : Q = 1/4 rejects H0 if X < 1. Then the test has
1.
(B) size = 5/32, power = 18/32 (D) size = 1/32, power = 31/32
(A) size = 5/32, power = 27/32
(C) size = 15/32, power = 27/32
2. Let X be a random variable having probability density function
axr.
, x > xc x < xn
f(x;x0,a) =
x
0,
then P(Y > 3) is
(C) e~3a
Vxo y
-3a
(D) 1-e
where a > 0, x0 > 0 . If Y = In
v
(A) e~3ax (B) 1 - e~3ax<>
dimension of the null space of T is (A) 0 (B) 1
(C) 2
(D) 3
Space for rough work
4. Let X1,X2,...,X2n be random variables such that V(Xi) = 4, i = l,2,...,2n and Cov (XitXj) = 3, 1 < i * j < 2n . Then VCX - X2 + X3 - X4 + + X2_1 - X2n ) is
(A) n (B) 2 n (C) Sn-2 (D) n +1
5. Let Xx and X2 be independent random variables, each having exponential distribution with parameter X. Then, the conditional distribution of XY given X1 + X2 = 1 is
(A) Exponential with mean 2 (B) Beta with parameters A / 2 and X / 2
(C) Uniform on the interval (0,1) (D) Gamma with mean 2X
6. Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,0). Then the uniformly minimum variance unbiased estimator (UMVUE) of 0 is
n +1
(A)
(C) 2X
(D) XM
L(n)
(n)
n
(B) X(1) +
Space for rough work
Let A be a 4 x 4 nonsingular matrix and B be the matrix obtained from A by adding to its third row twice the first row. Then det(2 A~lB) equals
(A) 2 (B) 4 (C) 8 (D) 16
Independent trials consisting of rolling a fair die are performed. The probability that
2 appears before 3 or 5 is
Let X1,X2,...,X6 be independent random variables such that
P{X, =-l) = P(Xi =l) = i =1,2,3,...,6.
z
Then P
is
1=1
10. Let 1, x and x2 be the solutions of a second order linear non-homogeneous differential equation on -1 < x < 1. Then its general solution, involving arbitrary constants Cl and C2, can be written as
(A) C1(l-x)+C2(x -x2) + l (B) Cxx+C2x2 + 1
(C) C1(1 + x) + C2(1 + x2) + 1 (D) Cx+C2x+x2
11. Let
|
Then (A) f'(x) is continuous at x = 0 (C) f'( 0) exists |
x sin , x * 0 x fix) = 0, x = 0. (B) f"(x) is continuous at x = 0 (D) f{0) exists |
12. Let E and F be two events such that 0 < P(E) < 1 and P(E \ F) + P(E | Fc) = 1. Then (A) E and F are mutually exclusive (B) P(EC | F) + P(EC |FC)=1
(C) E and F are independent (D) P(E | F) + P(EC \ Fc) = 1
13. Let X1,X2,...,Xn be a random sample from an exponential distribution with mean 1/A. The maximum likelihood estimator of the median of the distribution is
(A) (B) X(ln2) (C) (D)
ln(2X)
(In 2) X
. 1. 1 2 + 3 4 + 5- 6 + ...+ (2n) ,
14. lim-:y- -, - equals
yin2 +1 +4n2 -1
(A) - (B) 1/2 (C) 0 (D)
-1/2
15. By changing the order of integration, the integral
i <>*
J Jf(x,y)dydx
o 1
can be expressed as
1 lny 1 lny
(A) J jf(x,y)dxdy (B) J f{x,y)dxdy
0 1 0 0
e 1
(C) | f(x,y)dxdy (D) J j f(x,y)dxdy
11 1 lny
16. (a) Let fix) - x3 + 3x - 2,x e R. Show that the equation f(x) - 0 has only one real root.
Also, find x0 in the interval (0,1) such that the tangent to the curve y = fix) at the point (x0,f(x0)) is parallel to the line joining the points (0,-2) and (1,2). (9)
(b) Let T : R2 > R3 be a linear transformation with
T(l,l) = (0,0,1) and 7X1,2) = (0,1,1).
Then find the linear transformation T{x,y). Also, find the associated matrix referred to the standard bases. (12)
17. (a) Find the volume of the solid whose base is the region in the xy -plane that is bounded by the parabola y = 2 - x2 and the line y = x , while the top of the solid is bounded by the plane z = x + 2. (9)
(b) Find all the values of x for which the series
OO
n+1 n
(-l)n+1x
converges. (12)
0, x < 0 x + k
, k < x < k + 1, k = 0,1,2
F(x) =
1, * > 3.
Find :
(a) P(X = j) for all non-negative integers j
(b) PCX > 2)
(c) P(-l < X < 1).
Let X1,...,Xn be independent random variables with Xk having normal distribution with mean kd and variance cr2 for k =1,2Find the maximum likelihood estimator of 0 based on X1,Xn . Show that it is an unbiased and consistent estimator of 0. (21)
e-1
P(X -- m,Y = n) =-, m - 0,1,2n = 0,1,2,...
(n -m)\m\2n
Find the marginal probability mass functions of X and Y. Also, find the conditional probability mass function of X given Y = 5, and that of Y given X = 6. (21)
Let Xl,...,Xn (n > 2) be a random sample from a distribution having the probability mass function
P{X =x) = 0(1 - 6Y, x = 0,1,2,...
n
where 0 < 0 < 1. Show that T = Xt is a complete sufficient statistic. Find the uniformly
i=i
minimum variance unbiased estimator (UMVUE) of 6. (21)
dy
- + y = g(x), 0 < X < oo; ax
y(0) = 2,
where
3, 0 < x < 7t 12 g(x) =
Icosx, x > n 12.
23. Let X1,X2,X3,... be a sequence of independent and identically distributed random variables each with mean 4 and variance 4. Show that for large n ,
0.5 < P
< 0.9.
(21)
n
16/i - 124n X2iX2i_x < 16n + 12Vra
j=i
An urn contains ten balls of which M (an unknown number) are white. To test the hypothesis H0 : M = 3 against Hx : M = 7, three balls are drawn at random from the urn without replacement. If X is the number of white balls drawn, show that the most powerful test rejects H0 if X > k, where k is a constant. Find the power, if the size of this test is 11/60. (21)
25. (a) Evaluate the integral JJV* +y )/2 dxdy , where R is the region bounded by the lines
R
y = 0 and y = x , and the arcs of the circles x2 + y2 =1 and x2 + y2 = 2 . (9)
(b) Let
, (x,y)*(0,0)
3 , 3
x +y
f(x,y) =
0, (x,y) = (0,0).
Determine whether the function is continuous and differentiable at (0,0).
(12)
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Attachment: |
| Earning: Approval pending. |
