Mathematical Statistics - 2015 Past Year Paper - IIT JAM MCQ

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