University of Hyderabad (UoH) 2007 M.Sc Statistics Entrance - Question Paper
University of Hyderabad, Entrance Examination, 2000 M.Sc. (Statistics - OR)
Hall Ticket No.
Time: 2 hours Part A: 25
Max. Marks: 75 Part B: 50
Instructions
1. The OMR sheet contains space for answers to 100 questions. Answer Part A in
1 to 25 and Part B in 26 to 50. Ignore the remaining spaces.
2. Fill in your hall ticket number in the space provided in both the OMR sheet and on this page.'
3. Calculators are not allowed.
4. Each correct answer in Part A carries 1 mark and each wrong answer carries Qjmark. So do not gamble.
5. Each correct answer in Part B carries 2 marks and each wrong answer carries
O'bk ramark'
6. There will be no penalty if a question is unanswered.
7. Answers are to be given on the OMR sheet provided. .
8. The appropriate answer should be coloured in by either -a) black ball point/pen qljx black sketch -pen-- DO NOT USE A PENCIL.
3T-2-
1. Seventeen teams take part in the Foot Ball Championship of a
country. In how many ways can the gold, silver and bronze medals
be distributed among the teams?
3 17i 17'
(A) 17J (B) (C) (D) 14!
2. How many subsets does a set with n elements have? (The empty set is considered as a subset of every set)
(A) 2n (B) n! (C) 2n (D) n(n-l)
3. Let N(A) be the number of elements in a set A. Then N(A u B) is
(A) N(A) + N(B) (B) N(A) + N(B) - N(A n B)
(C) NCA) - N(A n B) (D) N(A) + N(B) + N(A n B)
4. The probabilities of events A,B and A n B are known. What is the probability of the event A v B?
(A) P(A) - P(A n B) (B) P{B) - P(A a B)
(C) P(A) + P(B) (D) 1 - P(A n B)
5. If X is a random variable with normal distribution with mean 1 and variance 2 denoted by N(l,2), what is the distribution of 2X?
(A) N{2,2) (B) N(2,8) (C) N(2,4) (D) N(2,2)
6. Let X,,...,X be a random sample from a normal distribution with
1 n
mean and variance 25. Which one of the following hypotheses is simple.
(A) Hq: < 17 (B) Hq: 17 (C) HQ: = 17 (D) none of these
7. Suppose we subdivide the population into at least two subgroups (such as by marital status) and then draw a random sample from each of the groups. This type of sampling scheme is called
(A) aggregate sampling (B) cluster sampling
(C) stratified sampling (D) none of these
8. 10 books numbered 1,2,...,10 are to be arranged in a row; the probability that book number 7 has books number 3 and 4 on either side is
9. The random variable X has the following probability distribution P(X = 1) = P(X = -1) = -5-, P(X = 2) = P(X = -2) =
(A) The median is positive (B) The mean is positive
(C) The median is zero and the mean is negative
(D) The mean and median are both zero
n-1
_2 1 3
then P(X i 10) is 11
10. Let P(X = n) =
, n = 1,2,..
a 10
f 2_ 3
(B) I I1
(D)
CC)
(A)
11. A and B are independent events, then
CB) P(A|B) = P(A IB) (D) P(A) = P{B)
(A) P(A|B) = P(A]B ) (C) P(A|BC) = P(A|B)
12. The number of arrangements of letters in the word EXCESS is (A) 720 (B) 360 (C) 180 (D) 90
n!
n
13. lim
is
n > co n
(A) 1 (B) 0 (C) oo CtS e 14. The covariance between two random variables X and Y is c
then
xy
covariance between aX and bY is
(B) |ab|c
CC)
(D)
(A) ab c
r c a | xy
b| cxy
xy
15. T and are two unbiased estimators for a function g(e) of the 12
parameter , based on a sample of size n, we will prefer T if
(B) ET < ET
(D) E|TXI < E|T2!
(A) V(T ) > V(T2) CC) cov(TrT2) > 0
16. Let events A and B be mutually exclusive subsets of S. Which of the following statements is true concerning A and B?
(B) (A n B) = (A u B) (D) (A u B) = S
(A) (A n B) = B CC) (A r\ B) = A
17. Which of the following statements is always true for the normal distribution?
(A) P(X 6) = 1-P(X < 5) (C) P(X & 2) = 1-P(X 1)
(B) P(X 6) = P(X > 6) (D) PCX > 2) = PCX > 3)
|
(A) the boundary of a circle (C) is a closed disc |
(B) the interior of a circle (D) the exterior of a circle |
19. The function f(x) = 5X-4X-3X+2X, then for f(x) = 0
(A) x = 0 and x = 1 are solutions CB) x = 0 is the only solution
(C) x = 1 and x = 2 are solutions CD) has no solution
20. The random variable X has pdf f (x) = |x|, -1<x<1. Then
(A) E(X) > 0 CB) E(X) < 0 ,2
(D) P(X>0) >
(C) EX = 0
21. The value of
dx
x + 1
-1
(A) does not exist
(B) is 0
CD) is > 0
CC) is < 0
Answer Questions 22 and 23 based on the joint probability distribution of X and Y given below.
V2
|
\Y X |
-1 |
0 |
1 | |||||
|
0 |
0 |
1/4 |
1/4 | |||||
|
1 |
1/6 |
1/6 |
1/6 | |||||
|
22. |
PCX |
= 0) is | ||||||
|
(A) |
1/6 |
CB) |
1/3 |
CC) |
5/12 CD) |
1/2 | ||
|
23. |
PCX |
= 1 1 Y |
= |
0) |
is | |||
|
CA) |
1/5 |
CB) |
2/5 |
CC) |
3/5 CD) |
4/5 | ||
|
24. |
X |
N(0,1) |
0<a |
l<a2 |
suppose |
PCX a = |
c; PCX | |
|
PC- |
a2 < X |
< |
al |
) is | ||||
|
CA) |
a2l |
(B) a |
ra2 |
CC) ot. +a. |
CD) | |||
25. The moment generating functions of two independent random variables X and X are MCt) and M (t) respectively. The MGF of X:+2X2 is
(A) M (t) + 2M2(t) CC) 2M (t)M (t)
(B) M (t) + M (2t) CD) M (t)M (2t)
26. X is a random variable with Poisson distribution, P(X=1) = 2P(X=0), then P(X=2) is
(A) equal to P(X=1) (B) twice P(X=1)
(C) equal to P(X=0) (D) 4 times P(X-O)
27. From n distinct objections, the number of subsets of size 3 is twice the number of subsets of size 2. Therefore
(A) n = 10 (B) n = 12 (C) n - 8
CD) information given is not sufficient to determine n.
28. A group of 7 friends, 3 girls and 4 boys go to watch a film, they have tickets to seat numbers 7 to 13 (all in a row), they decide that only boys will sit on seats 7 and 13. In how many ways can these 7 friends be seated with the given condition?
(A) 720 (B) 5047 (C) 24 CD) 1440
29. P(AC v B) = 0.7, P(A) = 0.4, P(B) = 0.5, then P(A u B)
(A) is 0.9 (B) cannot be obtained from the information given
CC) is 0.6 (D) is 1
30. aj~a2 = ai~a3 = a2~a3 = this information on a2a2 and a
(A) both mean and variance can be obtained
(B) variance can be obtained but not the mean
(C) mean can be obtained but not variance
CD) neither mean nor variance can be obtained
31. There are 10 slips numbered 1,...,10 in a bag, two slips are drawn, the set of all possible outcomes S is
CA) {1,----10} (B) {(i, j); i, je{ 1_____10}}
(C) < ( i,j ); i*j {1,. . .,10}} (D) < { i,j > ; i*j e -lx|
32. fv(x) = ce -co<x<oo, what should c be so that f is a
X X
probability density function?
33. Two distinct numbers are selected from the probability that the larger of the two is more than 50 is
(A) at most 1/4
(B) more than 1/4 but not more than 2/3
(C) more than 1/2 but not more than 2/3
(D) more than 2/3
34. The first and second raw moments of a random variable X are 12 and 100 respectively. Which of the following is correct.
(A) This can never happen.
(B) This can happen if X has binomial distribution
2 2
(C) This can happen if X N(jn, cr ) for some choices of (jul, cr )
(D) This is true for X P(12)
35. X and Y are two random variables taking values 1,2,3,4 with the following distributions PCX = i) = i = 1,2,3,4; P(Y = 1) = 5/16, P(Y = 2) = 1/32, P(Y = 3) = 11/32, P(Y - 4) = 5/16 then
(A) EX = EY CB) EX > EY (C) P(X>1) < P(Y>1) (D) EX < EY
36. The correlation coefficient between X and Y denoted by p is 0,
XY
which of the following statements is always true
P-X.-Y > 0 (B) p 2 2 = 0 (C) P-X,-Y = 0 (D) P-X,-Y <
X , Y
37. The probability that among k randomly selected digits, the digits
0 and 1 are not there.
(A) 9k/10k (B) 8k/10k (C) 2k/10k (D) l/10k
38. In every scanning cycle, a radar tracking a space object detects the object with constant probability p. What is the probability of detecting the object in n cycles?
(A) pn (B) l-pn (C) (l-p)n (D) 1 - (l-p)n
39. An unbiased coin is tossed until a head is obtained. If N denotes the number of tosses required, what is P(N > 1)?
(A) 1/2 (B) 1 (C) 1/4 (D) 1/8
1
40. Let X be a random variable such that P{X = i) = -=-- for
2n+l
1 = -n, -n+1.....-l,0,l,...n. What is V(X)?
(A) n(n+1-> (B) n2(+1) (Q 2-(ntl) (p) nl<n+1>.2
41. Let and be independent identically distributed random variables with variance one, what is the covariance between X+X and X -X ?
(A) 2 (B) -2 (C) 1 (D) 0
42. For a random variable X with EX = 3 and EX = 13. P(-2<X<8) is
13 7 1
"25 25 15 2
43. X , . . . ,X is a random sample from a distribution with probability density function,
f(x,0) = 0 x0 *, 0<x<l for e > 0 =0, o.w
n
(A) V X. is a sufficient statistic for 0
1
i=l
n
(B) TT X. is a sufficient statistic for i=l 1
1 n
(C) 7 X. is sufficient statistic for n .u i
i=l
(D) none of the above is correct
44. Let P(A) = 0.2 and P(B|A) = 0.7, then
(A) P(A v B) = 0.76
(B) P(A v B) - 0.24
(C) PCA u B) = 0.56
(D) PCA u B) cannot be determined with the information given
45. X and X independent Poisson random variables with parameter A,
2
an unbiased estimator for \ is
X+X X +X X*+X*-(X +X9)
(A) (B) - (C) max(X. ,X0) (D)
2 2 1*2' 2
46. {x:|x-2| > 2} n {x:|x3| < 2} is
(A) an open interval
(B) a closed set
(C) a non-empty finite set
(D) an open set which is not an interval
| |||||||||||||||
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rows of matrix are not linearly independent? |
|
is | ||||||||||||||||||||||||
- - 1
49. X and Y are random variables satisfying log Y = X N(0,1), EY is
1/2
(A) e'
(B) e
(C) e
(D) none of these is correct
50. The set of vectors (1,0,0); (2,3,0); (4,5,6), (7,8,9) in [R are
3
(A) A basis for [R .
(B) Linearly dependent but not spanning R .
3
(C) Linearly dependent and spanning (R .
(D) Linearly independent.
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Attachment: |
| Earning: Approval pending. |
