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Mathematical Statistics - 2016 Past Year Paper - Question 2

Let α , β,γ be real numbers such that β ≠ 0 and γ ≠ 0. Suppose

and P^{-1} =P. Then

Mathematical Statistics - 2016 Past Year Paper - Question 3

Let m > 1. The volume of the solid generated by revolving the region between the y-axis and the curve xy = 4, 1 __<__ y __<__ m, about the y-axis is 15π. The value of m is

Mathematical Statistics - 2016 Past Year Paper - Question 4

Consider the region S enclosed by the surface z = y^{2} and the planes z = 1,x = 0, x = 1, y = -1 and y = 1. The volume of S is

Mathematical Statistics - 2016 Past Year Paper - Question 5

Let X be a discrete random variable with the moment generating function

Then P (X __<__ 1) equals

Mathematical Statistics - 2016 Past Year Paper - Question 6

Let E and F be two independent events with

Then P(E) equals

Mathematical Statistics - 2016 Past Year Paper - Question 7

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

Then E (X^{2})

Mathematical Statistics - 2016 Past Year Paper - Question 8

The probability density function of a random variable X is given by

Then the distribution of the random variable Y = log_{e} X^{-2a} is

Mathematical Statistics - 2016 Past Year Paper - Question 9

Let X_{1} ,X_{2} ,..... be a sequence of i.i.d. N (0,1) random variables. converges in probability to

Mathematical Statistics - 2016 Past Year Paper - Question 10

Consider the simple linear regression model with n random observations Y_{i} = β_{0}+ β_{1}x_{i} + ε_{i}, i = 1,....,n, (n > 2). β_{0} and β_{1} are unknown parameters, x_{1},....,x_{n} are observed values of the regressor variable and ε_{1}... ,ε_{n} are error random variables with E (ε_{i}) = 0, i = 1,....,n, and for i, j = 1,...., n, For real constants is an

unbiased estimator of β_{1} , then

Mathematical Statistics - 2016 Past Year Paper - Question 11

Let (X, Y) have the joint probability density function

Then P(Y <1 | X = 3) equals

Mathematical Statistics - 2016 Past Year Paper - Question 12

Let X_{1},X_{2},..... be a sequence of i.i.d. random variables having the probability density function

If the distribution of converges to N ( 0,1) as

n → ∞, then a possible value of α is

Mathematical Statistics - 2016 Past Year Paper - Question 13

Let X_{1},...,X_{n} be a random sample from a population with the probability density function

If T_{n} = min {X_{1},...,X_{n}}, then

Mathematical Statistics - 2016 Past Year Paper - Question 14

Let X_{1},...,X_{n} be i.i.d. random variables with the probability density function

If X_{(n)} = max {X_{1},...,X_{n}},

Mathematical Statistics - 2016 Past Year Paper - Question 15

Let X and Y be two independent N ( 0,1) random variables. Then P (0 < X^{2} + Y^{2} < 4) equals

Mathematical Statistics - 2016 Past Year Paper - Question 16

Let X be a random variable with the cumulative distribution function

Then E (X) equals

Mathematical Statistics - 2016 Past Year Paper - Question 17

Let X_{1},....,X_{n} be a random sample from a population with the probability density function

For a suitable constant K, the critical region of the most powerful test for testing H_{0} : θ =1 against H_{1} : θ = 2 is of the form

Mathematical Statistics - 2016 Past Year Paper - Question 18

Let X_{1},...,X_{n}, X_{n+1}, X_{n+2},..., X_{n+m} (n > 4,m > 4) be a random sample from N (μ ,σ^{2});

then the distribution of the random variable

is

Mathematical Statistics - 2016 Past Year Paper - Question 19

Let X_{1},...,X_{n} (n > 1) be a random sample from a Poisson (θ) population, θ > 0, and Then the uniformly minimum variance unbiased estimator of θ^{2} is

Mathematical Statistics - 2016 Past Year Paper - Question 20

Let X be a random variable whose probability mass functions f(x | H_{0}) (under the null hypothesis H_{0}) and f (x |H_{1}) (under the alternative hypothesis H_{1}) are given by

For testing the null hypothesis H_{0} : X~f (x | H_{0}) against the alternative hypothesis H_{1} : X~f (x | H_{1}), consider the test given by: Reject

If α = size of the test and β = power of the test, then

Mathematical Statistics - 2016 Past Year Paper - Question 21

Let X_{1},..,X_{n} be a random sample from a N (2θ, θ^{2}) population, θ > 0. A consistent estimator for θ is

Mathematical Statistics - 2016 Past Year Paper - Question 22

An institute purchases laptops from either vendor V_{1} or vendor V_{2} with equal probability. The lifetimes (in years) of laptops from vendor V_{1} have a U (0, 4) distribution, and the lifetimes (in years) of laptops from vendor V_{2} have an Exp (12) distribution. If a randomly selected laptop in the institute has lifetime more than two years, then the probability that it was supplied by vendor V_{2} is

Mathematical Statistics - 2016 Past Year Paper - Question 23

Let y (x) be the solution to the differential equation

Mathematical Statistics - 2016 Past Year Paper - Question 24

Let a_{n} = e ^{-2n} sin n and b_{n} = e^{-n} n^{2} (sin n)^{2} for n __>__1. Then

Mathematical Statistics - 2016 Past Year Paper - Question 26

be a twice differentiable function. Further, let f (0) = 1, f(2) = 2 and f(4) = 3. Then

Mathematical Statistics - 2016 Past Year Paper - Question 27

Let f (x,y) = x^{2} - 400 x y^{2} for all (x,y) Then f attains its

Mathematical Statistics - 2016 Past Year Paper - Question 28

Let y (x) be the solution to the differential equation

Then y (1) equals

Mathematical Statistics - 2016 Past Year Paper - Question 29

The area between the curve y = g" (x) and the x-axis over the interval [0, 2] is

Mathematical Statistics - 2016 Past Year Paper - Question 30

singular matrix such that for a nonzero vectorand

Then

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 31

For two nonzero real numbers a and b, consider the system of linear equations

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 32

Which of the following statements is (are) TRUE?

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 33

Which of the following statements is (are) TRUE?

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 34

Which of the following statements is (are) TRUE?

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 35

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

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 36

Let X_{1},...,X_{n} (n > 1) be a random sample from a U (2θ -1, 2θ + 1) population, and Y_{1} = min {X_{1},...., X_{n}} , Y_{n} max {X_{1},...X_{n}}. Which of the following statistics is (are) maximum likelihood estimator (s) of θ ?

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 37

Let X_{1},...,X_{n} be a random sample from a N (0, σ^{2}) population, σ > 0. Which of the following testing problems has (have) the region as the most

powerful critical region of level α ?

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 38

Let X_{1},....X_{n} be a random sample from a N (0, 2θ^{2}) population, θ > 0. Which of the following statements is (are) TRUE?

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 39

Let X_{1},...,X_{n} be a random sample from a population with the probability density function

Which of the following is (are) 100 (1 -α) % confidence interval(s) for θ ?

*Multiple options can be correct

Mathematical Statistics - 2016 Past Year Paper - Question 40

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

Which of the following statements is (are) TRUE?

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 41

Let X_{1},....,X_{10} be a random sample from a N (3,12) population. Suppose distribution, then the value of α is _______________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 42

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

Then the upper bound of P (X - 2 >1) using Chebyshevs inequality is ________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 43

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

Then P ( X < Y) = _____________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 44

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

Then P (X > 0, Y< 0 ) = _________________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 45

Let Y be a Bin random variable. Using normal approximation to binomial distribution, an approximate value of P ( 22 __<__ Y __<__ 28) is ________________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 46

Let X be a Bin (2, p) random variable and Y be a Bin (4, p) random variable, 0 < p < 1. If

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 47

Consider the linear transformation

T(x, y,z) = (2x+ y + z, x+ z, 3x+2 y + z).

The rank of T is ________________________

*Answer can only contain numeric values

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 49

be defined by f (x) = x^{13} - e^{-x}+ 5 x +6.The minimum value of the function f on [0,13] is__________________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 50

Consider a differentiable function f on [ 0,1] with the derivative The arc length of the curve is ________________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 51

Let m be a real number such that m > 1. If

then m = ______________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 52

Let

The product of the eigen values of P^{-1} is ________________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 53

The value of the real number m in the following equation

is ________________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 54

converges to ______________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 55

Let X_{1},X_{2},... be a sequence of i.i.d. random variables with the probability density function

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 56

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

where c is a suitable constant. Then E (X ) = ________________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 57

Two points are chosen at random on a line segment of length 9 cm. The probability that the distance between these two points is less than 3 cm is ______________________

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 58

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

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 59

If X is a U ( 0,1) random variable, then

*Answer can only contain numeric values

Mathematical Statistics - 2016 Past Year Paper - Question 60

In a colony all families have at least one child. The probability that a randomly chosen family from this colony has exactly k children is (0.5)^{k} ; k = 1, 2,K . A child is either a male or a female with equal probability. The probability that such a family consists of at least one male child and at least one female child is _________

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