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Mathematical Statistics - 2017 Past Year Paper - Question 1

The imaginary parts of the eigenvalues of the matrix

are

Mathematical Statistics - 2017 Past Year Paper - Question 2

Let be such that u = (1 2 3 5)^{T} and v = (5 3 2 1)^{T}. Then the equation uv^{T} x = v has

Mathematical Statistics - 2017 Past Year Paper - Question 3

Which of the following statements is TRUE ?

Mathematical Statistics - 2017 Past Year Paper - Question 4

be a sequence defined as follows :

Which of the following statements is TRUE ?

Mathematical Statistics - 2017 Past Year Paper - Question 5

Let X be a continuous random variable with the probability density function

Mathematical Statistics - 2017 Past Year Paper - Question 6

Let X be a random variable with the moment generating function

Then P(X > 1) equals

Mathematical Statistics - 2017 Past Year Paper - Question 7

Let X be a discrete random variable with the probability mass function

p(x) = k(1 + |x|)^{2}, x = –2, –1, 0, 1, 2,

where k is a real constant. Then P(X = 0) equals

Mathematical Statistics - 2017 Past Year Paper - Question 8

Let the random variable X have uniform distribution on the interval . Then P(cos X > sin X) is

Mathematical Statistics - 2017 Past Year Paper - Question 9

be a sequence of i.i.d. random variables having common probability density function

Mathematical Statistics - 2017 Past Year Paper - Question 10

Let X_{1}, X_{2}, X_{3} be a random sample from a distribution with the probability density function

Which of the following estimators of θ has the smallest variance for all θ > 0 ?

Mathematical Statistics - 2017 Past Year Paper - Question 11

Player P_{1} tosses 4 fair coins and player P_{2} tosses a fair die independently of P_{1}. The probability that the number of heads observed is more than the number on the upper face of the die, equals

Mathematical Statistics - 2017 Past Year Paper - Question 12

Let X_{1} and X_{2} be i.i.d. continuous random variables with the probability density function

Using Chebyshev’s inequality, the lower bound of

Mathematical Statistics - 2017 Past Year Paper - Question 13

Let X_{1}, X_{2}, X_{3} be i.i.d. discrete random variables with the probability mass function

Let Y = X_{1} + X_{2} + X_{3}. Then P(Y __>__ 5) equals

Mathematical Statistics - 2017 Past Year Paper - Question 14

Let X and Y be continuous random variables with the joint probability density function

where c is a positive real constant. Then E(X) equals

Mathematical Statistics - 2017 Past Year Paper - Question 15

Let X and Y be continuous random variables with the joint probability density function

Mathematical Statistics - 2017 Past Year Paper - Question 16

Let X_{1}, X_{2}, ..., X_{m}, Y_{1}, Y_{2}, ..., Y_{n} be i.i.d. N(0, 1) random variables. Then

has

Mathematical Statistics - 2017 Past Year Paper - Question 17

be a sequence of i.i.d. random variables with the probability mass function

then possible values of m and M are

Mathematical Statistics - 2017 Past Year Paper - Question 18

Let x_{1} = 1.1, x_{2} = 0.5, x_{3} = 1.4, x_{4} = 1.2 be the observed values of a random sample of size four from a distribution with the probability density function

Then the maximum likelihood estimate of θ^{2} is

Mathematical Statistics - 2017 Past Year Paper - Question 19

be the observed values of a random sample of size four from a distribution with the probability density function

Then the method of moments estimate of θ is

Mathematical Statistics - 2017 Past Year Paper - Question 20

Let X_{1}, X_{2} be a random sample from an N(0, θ) distribution, where θ > 0. Then the value of k, for which the interval is a 95% confidence interval for θ, equals

Mathematical Statistics - 2017 Past Year Paper - Question 21

Let X_{1}, X_{2}, X_{3}, X_{4} be a random sample from N(θ_{1}, σ^{2}) distribution and Y_{1}, Y_{2}, Y_{3}, Y_{4} be a random sample from N(θ_{1}, σ^{2}) distribution, where θ_{1}, θ_{2} ∈ (-∞, ∞) and σ > 0. Further suppose that the two random samples are independent. For testing the null hypothesis H_{0} : θ_{1} = θ_{2} against the alternative hypothesis H_{1} : θ_{1} > θ_{2}, suppose that a test rejects H_{0} if and only if The power of the tes

Mathematical Statistics - 2017 Past Year Paper - Question 22

Let X be a random variable having a probability density function f ∈ {f_{0}, f_{1}}, where

For testing the null hypothesis against based on a single observation on X, the power of the most powerful test of size α = 0.05 equals

Mathematical Statistics - 2017 Past Year Paper - Question 25

Consider the function

f(x, y) = x^{3} – y^{3} – 3x^{2} + 3y^{2} + 7, x,

Then the local minimum (m) and the local maximum (M) of f are given by

Mathematical Statistics - 2017 Past Year Paper - Question 26

let the sequence be defined by

Then the values of c for which the seriesconverges are

Mathematical Statistics - 2017 Past Year Paper - Question 27

If for a suitable α > 0,

exists and is equal to

Mathematical Statistics - 2017 Past Year Paper - Question 28

Let

Which of the following statements is TRUE ?

Mathematical Statistics - 2017 Past Year Paper - Question 29

Let Q, A, B be matrices of order n × n with real entries such that Q is orthogonal and A is invertible. Then the eigenvalues of Q^{T} A^{–1} BQ are always the same as those of

Mathematical Statistics - 2017 Past Year Paper - Question 30

be the curve defined by

Let L be the length of the arc of this curve from the origin to the point P on the curve at which the tangent is perpendicular to the x- axis. Then L equals

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 31

where I is the k × k identity matrix. Then which of the following statements is (are) TRUE ?

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 32

Let and be sequence of real numbers such that is increasing and is decreasing. Under which of the following conditions, the sequence is always convergent ?

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 33

Let f : [0, 1] → [0, 1] be defined as follows :

Which of the following statements is (are) TRUE ?

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 34

Let f(x) be a non- negative differentiable function on such that f(a) = 0 = f(b) and |f’(x)| __<__ 4. Let L_{1} and L_{2} be the straight lines given by the equations y = 4(x – a) and y = –4(x – b), respectively. Then which of the following statements is (are TRUe ?

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 35

Let E and F be two events with 0 < P(E) < 1, 0 < P(F) < 1 and P(E) + P(F) __>__ 1. Which of the following statements is (are) TRUE ?

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 36

The cumulative distribution function of a random variable X is given by

Which of the following statements is (are) TRUE ?

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 37

Let X_{1}, X_{2} be a random sample from a distribution with the probability mass function

Which of the following is (are) unbiased estimator(s) of θ?

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 38

Let X_{1}, X_{2}, X_{3} be a random sample from a distribution with the probability density function

is an unbiased estimator of θ, which of the following CANNOT be attained as a value of the variance of

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 39

be a random sample from a distribution with the probability density function

Which of the following statistics is (are) sufficient but NOT complete ?

*Multiple options can be correct

Mathematical Statistics - 2017 Past Year Paper - Question 40

Let X_{1}, X_{2}, X_{3}, X_{4} be a random sample from an N(θ, 1) distribution, where θ ∈ (-∞, ∞). Suppose the null hypothesis H_{0} : θ = 1 is to be tested against the hypothesis H_{1} : θ < 1 at α = 0.05 level of significance. For what observed values of the uniformly most powerful test would reject H_{0} ?

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 41

Let the random variable X have uniform distribution on the interval (0, 1) and Y = –2 log_{e} X. Then E(Y) equals ________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 42

If Y = log_{10} X has N(μ, σ^{2}) distribution with moment generating function , t ∈ (-∞, ∞), then P(X < 1000) equals ________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 43

Let X_{1}, X_{2}, X_{3}, X_{4}, X_{5} be independent random variables with X_{1} ~ N(200, 8), X_{2} ~ N(104, 8), X_{3} ~ N(108, 15), X_{4} ~ N(120, 15) and X_{5} ~ N(210, 15). Let U

Then P(U > V) equals _________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 44

Let X and Y be discrete random variables with the joint probability mass function.

Then P(Y = 1 | X = 1) equals _________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 45

Let X and Y be continuous random variables with the joint probability density function

Then 9Cov(X, Y) equals __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 46

Let X_{1}, X_{2}, X_{3}, Y_{1}, Y_{2}, Y_{3}, Y_{4} be i.i.d. N(μ, σ^{2}) random variables.

has t_{v} distribution, then (v – k) equals __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 47

where. Let f have a local minimum at

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 48

The area bounded between two parabolas y = x^{2} + 4 and y = –x^{2} + 6 is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 49

For j = 1, 2, ..., 5, let P_{j} be the matrix of order 5 × 5 obtained by replacing the j^{th} column of the identity matrix of order 5 × 5 with the column vector v = (5 4 3 2 1)^{T}. Then the determinant of the matrix product P_{1} P_{2} P_{3} P_{4} P_{5} is __________

*Answer can only contain numeric values

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 51

Let a unit vector v = (v_{1} v_{2} v_{3})^{T} be such that Av = 0 where

Then the value of

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 52

Then the number of roots of F(x) = 0 in the interval (0, 4) is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 53

A tangent is drawn on the curve at the point which meets the x- axis

at Q. Then the length of the closed curve OQPO, where O is the origin, is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 54

The volume of the region

is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 55

Let X be a continuous random variable with the probability density function

where k is a real constant. Then P(1 < X < 5) equals __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 56

Let X_{1}, X_{2}, X_{3} be independent random variables with the common probability density function

Let Y = min {X_{1}, X_{2}, X_{3}}, E(Y) =

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 57

Let X and Y be continuous random variables with the joint probability density function

Then E(X | Y = –1) equals __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 58

Let X and Y be discrete random variables with

Then 3P(Y = 1) – P(Y = 0) equals __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 59

Let X_{1}, X_{2}, ..., X_{100} be i.i.d. random variables with E(X_{1}) = 0, E(X_{1}^{2}) = σ^{2}, where s > 0. Let S = If an approximate value of P(S __<__ 30) is 0.9332, then σ^{2} equals __________.

*Answer can only contain numeric values

Mathematical Statistics - 2017 Past Year Paper - Question 60

Let X be a random variable with the probability density function

If E(X) = 2 and Var(X) = 2, then P(X < 1) equals __________.

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