Mathematical Statistics - 2015 Past Year Paper - IIT JAM

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

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

Let X₁,..., 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 H₀ : α = 1 against H₁ : α = 2 is of the form

Which of the following is NOT a linear transformation?

If a sequence {x_n} is monotone and bounded, then

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

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

Let X₁ ,..., 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

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

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

Consider a normal population with unknown mean m and variance σ² = 9. To test H₀ : μ = 0 against H₁ : 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 ?

Let X₁ ,..., 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

Let X₁,..., 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

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

Let X be a single observation from a population having an exponential distribution with mean 1/λ. Consider the problem of testing H₀ : λ = 2 against H₁ : λ = 4. For the test with rejection region X > 3, let α = P(Type I error) and β = P(Type II error). Then

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

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

Suppose X₁,...,X_n are independent random variables and X_k ~ N(0, kσ²), k = 1, ..., n, where σ² is unknown. The maximum likelihood estimator for σ2 is

Let X₁ ,..., X₁₀ be independent and identically distributed U(–5, 5) random variables. Then, the distribution of the random variable

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

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

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

Then, the number of roots of f is

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

is singular is

 is a continuous function.

Then h’(1) is equal to

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 x₀ + V = {x₀ + v : v ∈ V}. Then

 a differentiable function whose derivative is continuous. Then

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?

Consider the ordinary differential equation

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

 be a continuous function such that

Then

g(x) = f(x) (f(x) + f(–x)).

Then

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

Let X₁,..., 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?

Let X₁,..., X_n be a random sample from a N(θ, θ²) distribution, where θ > 0 is unknown. Let

Which of the following statements is (are) correct?

Suppose X and Y are independent and identically distributed random variables with finite variance σ². Which of the following expressions is (are) equal to σ²?

Let X₁, X₂, ... be a sequence of independent and identically distributed random variables with mean 2 and variance 4. Which of the following statements is (are) true?

Let X₁,..., X_n (assume n > 2) be a random sample from an N(μ, σ²) population where -∞ < μ < ∞ and σ² > 0 are unknown. Which of the following statements is (are) true?

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) = __________.

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

X is a random variable with density

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 __________.

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 __________.

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 __________.

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 __________.

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

The number of real solutions of the equation x³ + 3x² + 3x + 7 = 0 is __________.

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

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

Then E(X) = __________.

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

Let X₁,..., X_n be independent and identically distributed random variables with U(0, 1) distribution.
Then

Based on 20 observations (x₁, y₁), ..., (x₂₀, y₂₀), 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 __________.

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

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

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

The length of the curve 

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 __________.

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

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

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