Model Test Paper, Mathematical Statistics - 2017 - IIT JAM

The differential equation 2ydx – (3y – 2x)dy = 0 is

What should be the value of z used in a 93% confidence interval?

Let X be a discrete random variable with values x = 0, 1, 2 and probabilities P(X = 0) = 0.25, P(X = 1) = 0.50, and P(X = 2) = 0.25, respectively.Find E(X²⁾

If two dice are thrown, what is the expected value of sum of the face values?

Let a_n be a sequence such that a₁ = a, a₂ = b and an = (a + a_n–1)/2 for n > 2. Calculate the limit?

  Integration by Parts results in

A function f is defined on interval (0, 1) as follows

then which one of the following is true?

Let f : R → R be s.t. f(x – f(y) ≤ (x – y)² then which of the following is TRUE ?

If two events A and B are such that P(A^c) = 0.3, P(B) = 0.4, P(A ∩ B^c) = 0.5, then P(B/A ∪ B^c) =

 A die is thrown (n + 2) times. After each throw a ‘+’ is recorded for 4, 5 or 6 and ‘–’ for 1, 2 or 3, the signs forming an ordered sequence each, except the first and the last sign, is attached a characteristic random variable which takes the value 1 if both the neighbouring signs differ from the one between them and 0 otherwise. If X₁, X₂, ..., X_n are characteristic random variables, find the mean and variance of

In the regression line Y = a + bX:

What would a chi- square significance value of P > 0.05 suggest?

A random variable X has the density function f(x) = c/(x² + 1). where  –∞ < x < ∞. Find the probability that X² lies between 1/3 and 1.

Y is an exponential random variable with parameter λ = 0.2. Given the event A = {Y < 2}.
Find the conditional expected value, E [Y | A].

An examination paper has 150 multiple- choice questions of one mark each, with each question having four choice. Each incorrect answer fetches- 0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is :

If probability density function of a random variable X is

f(x) = x² for –1 < x < 1, and

      = 0 for any other value of x

then, the percentage probability     is

Let F(x, y) be the d.f. of X and Y

if : F(x, y) = 1, for x + 2y ≥ 1

F(x, y) = 0, for x + 2y < 1,
then

If the product moment of X and Y is 3 and the mean of X and Y are both equal to 2, then what is the covariance of the random variables 2X + 10 and – 5/2Y + 3?

If X ~ N(μ,  σ²) and X₁, X₂, ..., X_n be a random sample from the population X, then

Which of the following are true statements?
I. The are under the curve of the t- distribution between ± 1 standard deviation is greater when d.f. = 5 than when d.f. = 10.
II. There is less are in the tails, beyond ± 3 standard deviations, of t-distribution when d.f. = 5 than when d.f. = 10.
III. For a given α, the critical t- value increases as d.f. decreases.

American Airlines claims that the average number of people who pay for in- flight moves, when the plane is fully loaded, is 42 with a standard deviation of 8. A sample of 36 fully loaded planes is taken. What is the probability that fewer than 38 people paid for the in- flight moves?

A symmetric die is thrown 600 times. Find the lower hound for the probability of getting 80 to 120 sixes.

An estimator T₁ = t₁(x₁, x₂, ..., x_n) for q is said to be admissible if for any other estimator T₂ = t₂(x₁, x₂, ..., x_n) for q, the relation is of the type:

A sample of 3 observations, (X₁ = 0.4, X₂ = 0.7, X₃ = 0.9) is collected from a continuous distribution with density

Estimate θ by the method of moments;

In a test of H₀ : μ = 100 against H_A : μ  ≠  100, a sample of size 10 produces a sample mean of 103 and a p- value of 0.08. Thus, at the 0.05 level of significance:

Suppose n = 100. Then the probability of type II error is :

A monotonically increasing  sequence <x_n> is convergent if

Let Y be a binomial random variable with parameters n = 100 and p = 1/2 Using the Central LimitTheorem which of the following are correct?

The distribution has two parameters. Given X₁,....,X_n is a random sample from the N(μ, σ²) distribution, find the method of moments estimates of μ and σ².

If f(x) = x sin x for all x ∈ R then f : R → R is

Let {X_n} be a sequence of independent Bernoulli random variables with parameter p = 1/2
generic term X_n of the sequence has support  R_n = { 0, 1} and probability mass function:

Cauchy’s nth Root Test

Let ∑u_n be a positive term series such that

Which of the following statements holds in (0, 2) if the function y = In (3x⁴ – 2x³ – 6x² + 6x + 1)

The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by,

then which of the following holds ?

In 360 tosses of a pair of dice, 74 “sevens” and 24 “elevens” are observed then which of the following are correct?

Given that lim a_n = a, lim b_n = b, and <Sn> and <Tn> are two sequences, where S_n = max {a_n, b_n} and T_n = min {a_n, b_n}. then which of the following are true?

Evaluate the following :

The following nonlinear differential equation can be solved exactly by separation of variables.

The value of θ(100) most nearly is

Using the Gauss- Jorden reduction method, if we find the inverse of the matrix    a₁₁ of the inverted matrix is _______

In the integral

change the order of integration, and evaluate the integral.

If X follows a binomial distribution with parameters n = 100 p = 1/3, then P(X = r) is maximum when r =

The solution for   what should be the value of a?

Let f:R → R be continuous such that f(x) = x² + 5 for all x ∈ Q then the value of f ( √2 ) is

Twenty sheets of aluminum alloy were examined for surface flaws. The frequency of the number of sheets with a given number of flaws per sheet was as follows :

What is the probability of finding a sheet chosen at random which contains 3 or more surface flaws?

You know the population mean for a certain test score. You select 10 people from the population to estimate the standard deviation. How many degrees of freedom does your estimation of the standard deviation have?

A waiter believes that his tips from various customers have a slightly right skewed distribution with a mean of 10 dollars and a standard deviation of 2.50 dollars. What is the probability that the average of 35 customers will be more than 13 dollars?

In psychology research, it is conventional to reject the null hypothesis if the probability value is lower than what number?

The distribution only has one parameter. Given X₁, X₂, ..., X_n is a random sample from a U(0, θ) distribution, find the coefficient of method of moments estimator (MMFE) of θ.

You do not know the population mean of a different test score. You select 15 people from the population and use this sample to estimate the mean and standard deviation. How many degrees of freedom does your estimation of the standard deviation have?

If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty weeks,calculate the probability that there will not be more than one failure during a particular week.

The series    converges absolutely to the limit__.

From the following table showing the number of plants having certain character, test the hypothesis that the flower colour is independent of flatness of leaves.

You may use the following table giving the value of c2 for one degree of freedom, for different values of P.

Calculate the value of χ² for the above table

Suppose that a netball player has a probability of 1/2 of scoring a goal each time. What is the
probability that she will score one goal from her first two attempts ?

Suppose you observe a sample of 100 independent draws from a normal distribution having known mean μ = 0 and unknown variance σ²  Denote the 100 draws by X₁, ..., X₁₀₀. Suppose that:

Find an upper limit of  confidence interval for s², using a set estimator of s² having 99% coverage probability.