Mathematical Statistics - 2016 Past Year Paper - IIT JAM MCQ

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