Mathematical Statistics - 2016 Past Year Paper - IIT JAM

Consider the region S enclosed by the surface z = y²  and the planes z = 1,x = 0, x = 1, y = -1 and y = 1. The volume of S is  

Let X be a discrete random variable with the moment generating function 

Then P (X < 1) equals 

Let E and F be two independent events with 

Then P(E) equals 

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

Then E (X²)

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

Then the distribution of the random variable Y = log_e X^-2a is 

Let X₁ ,X₂ ,.....  be a sequence of i.i.d. N (0,1) random variables.  converges in probability to  

Consider the simple linear regression model with n random observations Y_i = β₀+ β₁x_i + ε_i, i = 1,....,n, (n > 2). β₀  and β₁ are unknown parameters, x₁,....,x_n are observed values of the regressor variable and ε₁... ,ε_n are error random variables with E (ε_i) = 0, i = 1,....,n, and for i, j = 1,...., n,   For real constants is an 
unbiased estimator of β₁ , then

Let (X, Y) have the joint probability density function 

Then P(Y <1 | X = 3) equals 

Let X₁,X₂,..... be a sequence of  i.i.d. random variables having the probability density function 

If the distribution of converges to N ( 0,1) as 
n → ∞, then a possible value of α is 

Let X₁,...,X_n be a random sample from a population with the probability density function 

If T_n = min {X₁,...,X_n}, then

Let X₁,...,X_n be i.i.d. random variables with the probability density function 

If X_(n) = max {X₁,...,X_n},   

Let X and Y be two independent N ( 0,1) random variables. Then P (0 < X² + Y² < 4) equals 

Let X be a random variable with the cumulative distribution function 

Then E (X) equals 

Let X₁,....,X_n be a random sample from a population with the probability density function 

For a suitable constant K, the critical region of the most powerful test for testing H₀ : θ =1 against H₁ : θ = 2 is of the form 

Let X₁,...,X_n, X_n+1, X_n+2,..., X_n+m (n > 4,m > 4) be a random sample from N (μ ,σ²);

 then the distribution of the random variable 

is

Let X₁,...,X_n (n > 1) be a random sample from a Poisson (θ) population, θ > 0, and Then  the uniformly minimum variance unbiased estimator of θ² is   

Let X be a random variable whose probability mass functions f(x | H₀) (under the null hypothesis H₀)  and f (x |H₁) (under the alternative hypothesis H₁)  are given by 

For testing the null hypothesis H₀ : X~f (x | H₀)  against the alternative hypothesis H₁ : X~f (x | H₁),  consider the test given by: Reject

If α = size of the test and β = power of the test, then 

Let X₁,..,X_n be a random sample from a N (2θ, θ²) population, θ > 0. A consistent estimator for θ is

An institute purchases laptops from either vendor V₁  or vendor V₂ with equal probability. The lifetimes (in years) of laptops from vendor  V₁  have a U (0, 4) distribution, and the lifetimes (in years) of laptops from vendor V₂ have an Exp (12) distribution.  If a randomly selected laptop in the institute has lifetime more than two years, then the probability that it was supplied by vendor V₂ is  

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

Let a_n = e ^-2n sin n and b_n = e^-n n² (sin n)² for n >1. Then

Let 

Then 

 be a twice differentiable function. Further, let f (0) = 1, f(2) = 2  and f(4) = 3.   Then

Let f (x,y) = x² - 400 x y² for all (x,y)  Then f attains its 

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

Then y (1) equals   

The area between the curve y = g" (x) and the x-axis over the interval [0, 2] is 

singular matrix such that for a nonzero vectorand

Then

For two nonzero real numbers a and b, consider the system of linear equations 

Which of the following statements is (are) TRUE? 

Which of the following statements is (are) TRUE? 

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 | F) > P (E).   Which of the following statements is (are) TRUE? 

Let X₁,...,X_n (n > 1) be a random sample from a U (2θ -1, 2θ + 1) population,  and Y₁ = min {X₁,...., X_n} , Y_n max {X₁,...X_n}. Which of the following statistics is (are) maximum likelihood estimator (s)  of θ ?

Let X₁,...,X_n be a random sample from a N (0, σ²) population, σ > 0. Which of the following testing problems has (have) the region  as the most 
powerful critical region of level α ?  

Let X₁,....X_n be a random sample from a N (0, 2θ²) population, θ > 0. Which of the following statements is (are) TRUE? 

Let X₁,...,X_n be a random sample from a population with the probability density function 

Which of the following is (are) 100 (1 -α) % confidence interval(s) for θ ?

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

Which of the following statements is (are) TRUE? 

Let X₁,....,X₁₀ be a random sample from a N (3,12) population. Suppose  distribution, then the value of α is _______________ 

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

Then the upper bound of P (X - 2 >1) using Chebyshevs inequality is ________________ 

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

Then P ( X < Y) = _____________________ 

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

Then P (X > 0, Y< 0 ) = _________________________ 

Let Y be a Bin random variable.  Using normal approximation to binomial distribution, an approximate value of P ( 22 < Y < 28) is ________________________ 

Let X be a Bin (2, p) random variable and Y be a Bin (4, p) random  variable, 0 < p < 1.  If 

Consider the linear transformation 

T(x, y,z) = (2x+ y + z, x+ z, 3x+2 y + z).

The rank of T  is ________________________ 

be defined by f (x) = x¹³ - e^-x+ 5 x +6.The minimum value of the function f on [0,13] is__________________________

Consider a differentiable function f on [ 0,1] with the derivative  The arc length of the curve   is ________________________ 

Let m be a real number such that m > 1. If  

then m = ______________________ 

Let 

The product of the eigen values of P^-1 is ________________________ 

The value of the real number m in the following equation  

is ________________________ 

converges to ______________________ 

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

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

where c is a suitable constant. Then E (X ) = ________________________ 

Two points are chosen at random on a line segment of length 9 cm.  The probability that the distance between these two points is less than 3 cm is ______________________ 

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

If X is a U ( 0,1) random variable, then 

In a colony all families have at least one child. The probability that a randomly chosen family from this colony has exactly k children is (0.5)^k ; k = 1, 2,K . A child is either a male or a female with equal probability. The probability that such a family consists of at least one male child and at least one female child is _________