Mathematical Statistics - 2017 Past Year Paper - IIT JAM MCQ

 be a sequence defined as follows :

Which of the following statements is TRUE ?  

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

Let X be a random variable with the moment generating function

Then P(X > 1) equals

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

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

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

Let X₁, X₂, X₃ 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 ?

Player P₁ tosses 4 fair coins and player P₂ tosses a fair die independently of P₁. The probability that the number of heads observed is more than the number on the upper face of the die, equals

Let X₁ and X₂ be i.i.d. continuous random variables with the probability density function

Using Chebyshev’s inequality, the lower bound of

Let X₁, X₂, X₃ be i.i.d. discrete random variables with the probability mass function

Let Y = X₁ + X₂ + X₃. Then P(Y > 5) equals

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

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

Let X₁, X₂, ..., X_m, Y₁, Y₂, ..., Y_n be i.i.d. N(0, 1) random variables. Then

has

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

 then possible values of m and M are

Let x₁ = 1.1, x₂ = 0.5, x₃ = 1.4, x₄ = 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 θ² is

 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

Let X₁, X₂ 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

Let X₁, X₂, X₃, X₄ be a random sample from N(θ₁, σ²) distribution and Y₁, Y₂, Y₃, Y₄ be a random sample from N(θ₁, σ²) distribution, where θ₁, θ₂ ∈ (-∞, ∞) and σ > 0. Further suppose that the two random samples are independent. For testing the null hypothesis H₀ : θ₁ = θ₂ against the alternative hypothesis H₁ : θ₁ > θ₂, suppose that a test rejects H₀ if and only if   The power of the tes

Let X be a random variable having a probability density function f ∈ {f₀, f₁}, 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

are

Consider the function

f(x, y) = x³ – y³ – 3x² + 3y² + 7, x,

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

 let the sequence be defined by

Then the values of c for which the seriesconverges are

If for a suitable α > 0,

exists and is equal to

Let

Which of the following statements is TRUE ?

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

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