Mathematical Statistics - 2019 Past Year Paper - IIT JAM MCQ

Let T: ℝ² → ℝ² be a linear transformation such that Suppose thatThen α + β + a + b equals

Two biased coins C₁ and C₂ have probabilities of getting heads 2/3 and  3/4 , respectively, when tossed. If both coins are tossed independently two times each, then the probability of getting exactly two heads out of these four tosses is 

Let X be a discrete random variable with the probability mass function

where c and d are positive real numbers. If P(|X| ≤ 1) = 3/4, then E(X) equals

Let X be a Poisson random variable and P(X = 1) + 2 P(X = 0) = 12 P(X = 2). Which one of the following statements is TRUE?

Let X₁, X₂, … be a sequence of i.i.d. discrete random variables with the probability mass function

If S_n = X₁ + X₂ + ⋯ + X_n , then which one of the following sequences of random variables converges to 0 in probability?

Let X₁, X₂, … , X_n be a random sample from a continuous distribution with the probability density function

If T = X₁ + X₂ + ⋯ + X_n , then which one of the following is an unbiased estimator of μ?

Let X₁, X₂, … , X_n be a random sample from a N(θ, 1) distribution. Instead of observing X₁, X₂, … , X_n, we observe Y₁, Y₂, … , Y_n , where Y_i = e^Xi , i = 1, 2, … , n. To test the hypothesis
H₀: θ = 1 against H₁: θ ≠ 1
based on the random sample Y₁, Y₂, … , Y_n , the rejection region of the likelihood ratio test is of the form, for some c₁ < c₂ ,

A possible value of  b ∈ ℝ for which the equation  x⁴ + bx³ + 1 = 0 has no real root is

Let the Taylor polynomial of degree  20 for Then a₁₅ is

The length of the curve from x = 1 to x = 8 equals

The volume of the solid generated by revolving the region bounded by the parabola x = 2y² + 4 and the line x = 6 about the line x = 6 is 

Let P be a 3 × 3 non-null real matrix. If there exist a 3 × 2 real matrix Q and a 2 × 3 real matrix R such that P = QR, then

If P =and 6P⁻¹ = aI₃ + bP − P², then the ordered pair (a, b) is

Let E, F and G be any three events with P(E) = 0.3, P(F|E) = 0.2, P(G|E) = 0.1 and P (F ∩ G|E) = 0.05. Then P(E − (F ∪ G)) equals

Let E and F be any two independent events with 0 < P(E) < 1 and 0 < P(F) < 1. Which one of the following statements is NOT TRUE?

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

Then the distribution of the random variable W = 2x² is

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

Then E (X) and P(X > 1), respectively, are

The lifetime (in years) of bulbs is distributed as an Exp(1) random variable. Using Poisson approximation to the binomial distribution, the probability (round off to 2 decimal places) that out of the fifty randomly chosen bulbs at most one fails within one month equals

Let X follow a beta distribution with parameters m (> 0) and 2. If P then  Var(X) equals

Let X₁, X₂ and X₃ be i.i.d. U(0,1) random variables. Then P(X₁ > X₂ + X₃) equals

Let X and Y be i.i.d. U(0, 1) random variables. Then E(X| > Y) equals

Let −1 and 1 be the observed values of a random sample of size two from N(θ, θ) distribution. The maximum likelihood estimate of θ is 

Let X₁ and X₂ be a random sample from a continuous distribution with the probability density function

where θ > 0. If X₍₁₎ = min{X₁, X₂} and then which one of the following statements is TRUE?

Let X₁, X₂, … , X_n be a random sample from a continuous distribution with the probability density function f(x). To test the hypothesis

the rejection region of the most powerful size α test is of the form, for some c > 0,

Let X₁, X₂, … , X_n be a random sample from a N(θ, 1) distribution. To test H₀: θ = 0 against H₁: θ = 1, assume that the critical region is given by Then the minimum sample size required so that P(Type I error) ≤ 0.05 is