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Mathematical Statistics - 2015 Past Year Paper - Question 1

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

where θ > 0 is an unknown parameter.

Then, the uniformly minimum variance unbiased estimator for

Mathematical Statistics - 2015 Past Year Paper - Question 2

Let X_{1},...,X_{100} be independent and identically distributed N(0, 1) random variables. The correlation between

Mathematical Statistics - 2015 Past Year Paper - Question 3

Consider the problem of testing H_{0} : θ = 0 against H_{1} : θ = 1/2 based on a single observation X from U(θ, θ + 1) population. The power of the test

Mathematical Statistics - 2015 Past Year Paper - Question 4

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

where k is a constant. The moment generating function M_{X}(t) is

Mathematical Statistics - 2015 Past Year Paper - Question 5

Suppose A and B are events with P(A) = 0.5, P(B) = 0.4 and = 0.2. Then P(Bc | A È B) is equal to

Mathematical Statistics - 2015 Past Year Paper - Question 6

Let X_{1},..., X_{n} be a random sample from a Gamma(α, β) population, where β > 0 is a known constant. The rejection region of the most powerful test for H_{0} : α = 1 against H_{1} : α = 2 is of the form

Mathematical Statistics - 2015 Past Year Paper - Question 7

Which of the following is NOT a linear transformation?

Mathematical Statistics - 2015 Past Year Paper - Question 8

If a sequence {x_{n}} is monotone and bounded, then

Mathematical Statistics - 2015 Past Year Paper - Question 9

be defined by f(x) = x(x – 1)(x – 2). Then

Mathematical Statistics - 2015 Past Year Paper - Question 10

Which of the following statements is true for all real numbers x ?

Mathematical Statistics - 2015 Past Year Paper - Question 11

Let X_{1} ,..., X_{n} be a random sample from a Poisson (θ) population, where q > 0 is unknown. The Cramer- Rao lower bound for the variance of any unbiased estimator of g(θ) = θe^{-θ} equals

Mathematical Statistics - 2015 Past Year Paper - Question 12

Let X and Y be two independent random variables such that X ~ U(0, 2) and Y ~ U(1, 3).Then P(X < Y) equals

Mathematical Statistics - 2015 Past Year Paper - Question 13

There are two boxes, each containing two components. Each component is defective with probability 1/4, independent of all other components. The probability that exactly one box contains exactly one defective component equals

Mathematical Statistics - 2015 Past Year Paper - Question 14

Consider a normal population with unknown mean m and variance σ^{2} = 9. To test H_{0} : μ = 0 against H_{1} : μ ≠ 0, a random sample of size 100 is taken. Based on this sample, the test of the form rejects the null hypothesis at 5% level of significance. Then, which of the following is a possible 95%confidence interval for μ ?

Mathematical Statistics - 2015 Past Year Paper - Question 15

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

where q > 0 is unknown. The maximum likelihood estimator of θ is

Mathematical Statistics - 2015 Past Year Paper - Question 16

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

where θ > 0 is unknown. Then, a consistent estimator for θ is

Mathematical Statistics - 2015 Past Year Paper - Question 17

Let the probability density function of a random variable X be given by

Mathematical Statistics - 2015 Past Year Paper - Question 18

Let X be a single observation from a population having an exponential distribution with mean 1/λ. Consider the problem of testing H_{0} : λ = 2 against H_{1} : λ = 4. For the test with rejection region X __>__ 3, let α = P(Type I error) and β = P(Type II error). Then

Mathematical Statistics - 2015 Past Year Paper - Question 19

Let Y be an exponential random variable with mean 1/θ, where q > 0. The conditional distribution of X given Y has Poisson distribution with mean Y. Then, the variance of X is

Mathematical Statistics - 2015 Past Year Paper - Question 20

2000 cashew nuts are mixed thoroughly in flour. The entire mixture is divided into 1000 equal parts and each part is used to make one biscuit. Assume that no cashews are broken in the process. A biscuit is picked at random. The probability that it contains no cashew nuts is

Mathematical Statistics - 2015 Past Year Paper - Question 21

Suppose X_{1},...,X_{n} are independent random variables and X_{k} ~ N(0, kσ^{2}), k = 1, ..., n, where σ^{2} is unknown. The maximum likelihood estimator for σ2 is

Mathematical Statistics - 2015 Past Year Paper - Question 22

Let X_{1} ,..., X_{10} be independent and identically distributed U(–5, 5) random variables. Then, the distribution of the random variable

Mathematical Statistics - 2015 Past Year Paper - Question 23

be a differentiable function so that f(x) f’(x) < 0 for all x. Then, which of the following is necessarily true?

Mathematical Statistics - 2015 Past Year Paper - Question 24

Let M be the matrix Which of the following matrix equations does M satisfy?

Mathematical Statistics - 2015 Past Year Paper - Question 25

If the determinant of an n × n matrix A is zero, then

Mathematical Statistics - 2015 Past Year Paper - Question 27

The number of distinct real values of x for which the matrix

is singular is

Mathematical Statistics - 2015 Past Year Paper - Question 28

is a continuous function.

Then h’(1) is equal to

Mathematical Statistics - 2015 Past Year Paper - Question 29

Let A be a 5 × 3 real matrix of rank 2. be a non- zero vector that is in the column space of A. Let S = Define the translation of a subspace V of as the set x_{0} + V = {x_{0} + v : v ∈ V}. Then

Mathematical Statistics - 2015 Past Year Paper - Question 30

a differentiable function whose derivative is continuous. Then

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 31

Suppose {a_{n}}, {b_{n}} are sequences such that a_{n} > 0, b_{n} > 0 for all n __>__ 1. Given that converges and diverges, which of the following statements is (are) necessarily FALSE?

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 32

Consider the ordinary differential equation

Which of the following is (are) solution(s) to the above?

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 33

be a continuous function such that

Then

*Multiple options can be correct

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 35

Which of the following matrices can be the variance- covariance matrix of a random vector

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 36

Let X_{1},..., X_{n} be a random sample from a N(θ, 1) population, where -∞ < θ < ∞ is unknown. Which of the following statistics is (are) sufficient for θ?

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 37

Let X_{1},..., X_{n} be a random sample from a N(θ, θ^{2}) distribution, where θ > 0 is unknown. Let

Which of the following statements is (are) correct?

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 38

Suppose X and Y are independent and identically distributed random variables with finite variance σ^{2}. Which of the following expressions is (are) equal to σ^{2}?

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 39

Let X_{1}, X_{2}, ... be a sequence of independent and identically distributed random variables with mean 2 and variance 4. Which of the following statements is (are) true?

*Multiple options can be correct

Mathematical Statistics - 2015 Past Year Paper - Question 40

Let X_{1},..., X_{n} (assume n __>__ 2) be a random sample from an N(μ, σ^{2}) population where -∞ < μ < ∞ and σ^{2} > 0 are unknown. Which of the following statements is (are) true?

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 41

Let X and Y be independent exponentially distributed random variables with means 1/4 and 1/6 respectively. Let Z = min{X, Y}. Then E(Z) = __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 42

Let X be a Geom(0.4) random variable. Then P(X = 5|X __>__ 2) = __________ .

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 43

X is a random variable with density

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 44

X is a single observation from a Bin(1, p) population, where p ∈ [1/5, 4/5] is unknown. If the observed value of X is 0, then the maximum likelihood estimator of p is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 45

A system comprising of n identical components works if at least one of the components works. Each of the components works with probability 0.8, independent of all other components. The minimum value of n for which the system works with probability at least 0.97 is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 46

Let X be a normal random variable with mean 2 and variance 4, and g(a) = P(a __<__ X __<__ a + 2). The value of a that maximizes g(a) is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 47

The volume of the solid formed by revolving the curve y = x between x = 0 and x = 1 about the x- axis is equal to __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 48

Let [x] be the greatest integer less than or equal to

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 49

The number of real solutions of the equation x^{3} + 3x^{2} + 3x + 7 = 0 is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 50

be a non- constant, three times differentiable function. for all integers n, then f”(1) = __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 51

Let Y ~ U(0, 1). The conditional probability density function of X given Y is

Then E(X) = __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 52

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

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 53

Let X_{1},..., X_{n} be independent and identically distributed random variables with U(0, 1) distribution.

Then

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 54

Based on 20 observations (x_{1}, y_{1}), ..., (x_{20}, y_{20}), the following values are obtained.

For X = 1, the predicted value of Y based on a least squares fit of a linear regression model of Y on X is __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 55

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

Then P(X = 0 | 0 __<__ X < 1) = ____________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 56

The probability density function f(x) of a random variable X is symmetric about 0. Then

*Answer can only contain numeric values

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 58

The system of equations

x + y + 2z = 2

2x + 3y – z = 5

4x + 7y + cz = 6

does NOT have a solution. Then, the value of c must be equal to __________.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 59

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

y” – 2y’ + y = 0, y(0) = 1 and y’(0) = 1.

*Answer can only contain numeric values

Mathematical Statistics - 2015 Past Year Paper - Question 60

The area of the region in the first quadrant enclosed by the curves y = 0, y = x and is equal to __________.

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