Mathematical Statistics - 2017 Past Year Paper - IIT JAM

 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

where I is the k × k identity matrix. Then which of the following statements is (are) TRUE ?

Let   and  be sequence of real numbers such that  is increasing and   is decreasing. Under which of the following conditions, the sequence   is always convergent ?

Let f : [0, 1] → [0, 1] be defined as follows :

Which of the following statements is (are) TRUE ?

Let f(x) be a non- negative differentiable function on such that f(a) = 0 = f(b) and |f’(x)| < 4. Let L₁ and L₂ be the straight lines given by the equations y = 4(x – a) and y = –4(x – b), respectively. Then which of the following statements is (are TRUe ?

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

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

Which of the following statements is (are) TRUE ?

Let X₁, X₂ be a random sample from a distribution with the probability mass function

Which of the following is (are) unbiased estimator(s) of θ?

Let X₁, X₂, X₃ be a random sample from a distribution with the probability density function

is an unbiased estimator of θ, which of the following CANNOT be attained as a value of the variance of

be a random sample from a distribution with the probability density function

Which of the following statistics is (are) sufficient but NOT complete ?

Let X₁, X₂, X₃, X₄ be a random sample from an N(θ, 1) distribution, where θ ∈ (-∞, ∞). Suppose the null hypothesis H₀ : θ = 1 is to be tested against the hypothesis H₁ : θ < 1 at α = 0.05 level of significance. For what observed values of the uniformly most powerful test would reject H₀ ?

Let the random variable X have uniform distribution on the interval (0, 1) and Y = –2 log_e X. Then E(Y) equals ________.

If Y = log₁₀ X has N(μ, σ²) distribution with moment generating function , t ∈ (-∞, ∞), then P(X < 1000) equals ________.

Let X₁, X₂, X₃, X₄, X₅ be independent random variables with X₁ ~ N(200, 8), X₂ ~ N(104, 8), X₃ ~ N(108, 15), X₄ ~ N(120, 15) and X₅ ~ N(210, 15). Let U 

Then P(U > V) equals _________.

Let X and Y be discrete random variables with the joint probability mass function.

Then P(Y = 1 | X = 1) equals _________.

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

Then 9Cov(X, Y) equals __________.

Let X₁, X₂, X₃, Y₁, Y₂, Y₃, Y₄ be i.i.d. N(μ, σ²) random variables. 

 has t_v distribution, then (v – k) equals __________.

where. Let f have a local minimum at 

The area bounded between two parabolas y = x² + 4 and y = –x² + 6 is __________.

For j = 1, 2, ..., 5, let P_j be the matrix of order 5 × 5 obtained by replacing the j^th column of the identity matrix of order 5 × 5 with the column vector v = (5 4 3 2 1)^T. Then the determinant of the matrix product P₁ P₂ P₃ P₄ P₅ is __________

Let a unit vector v = (v₁ v₂ v₃)^T be such that Av = 0 where

Then the value of

Then the number of roots of F(x) = 0 in the interval (0, 4) is __________.

A tangent is drawn on the curve   at the point which meets the x- axis
at Q. Then the length of the closed curve OQPO, where O is the origin, is __________.

The volume of the region

is __________.

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

where k is a real constant. Then P(1 < X < 5) equals __________.

Let X₁, X₂, X₃ be independent random variables with the common probability density function

Let Y = min {X₁, X₂, X₃}, E(Y) =

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

Then E(X | Y = –1) equals __________.

Let X and Y be discrete random variables with 

Then 3P(Y = 1) – P(Y = 0) equals __________.

Let X₁, X₂, ..., X₁₀₀ be i.i.d. random variables with E(X₁) = 0, E(X₁²) = σ², where s > 0. Let S =  If an approximate value of P(S < 30) is 0.9332, then σ² equals __________.

Let X be a random variable with the probability density function

If E(X) = 2 and Var(X) = 2, then P(X < 1) equals __________.