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QUESTION: 1

Let {x_{n} }n __>__ 1 be a sequence of positive real numbers. Which one of the following statements is always TRUE?

Solution:

QUESTION: 2

Consider the function f (x, y) = x^{3} − 3xy^{2}, x,y ∈ ℝ. Which one of the following statements is TRUE?

Solution:

QUESTION: 3

If F(x) = for x ∈ ℝ, then F′(1) equals

Solution:

QUESTION: 4

Let T: ℝ^{2} → ℝ^{2} be a linear transformation such that Suppose thatThen α + β + a + b equals

Solution:

QUESTION: 5

Two biased coins C_{1} and C_{2} 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

Solution:

QUESTION: 6

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

Solution:

QUESTION: 7

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?

Solution:

QUESTION: 8

Let X_{1}, X_{2}, … be a sequence of i.i.d. discrete random variables with the probability mass function

If S_{n} = X_{1} + X_{2} + ⋯ + X_{n} , then which one of the following sequences of random variables converges to 0 in probability?

Solution:

QUESTION: 9

Let X_{1}, X_{2}, … , X_{n} be a random sample from a continuous distribution with the probability density function

If T = X_{1} + X_{2} + ⋯ + X_{n} , then which one of the following is an unbiased estimator of μ?

Solution:

QUESTION: 10

Let X_{1}, X_{2}, … , X_{n} be a random sample from a N(θ, 1) distribution. Instead of observing X_{1}, X_{2}, … , X_{n}, we observe Y_{1}, Y_{2}, … , Y_{n} , where Y_{i} = e^{Xi} , i = 1, 2, … , n. To test the hypothesis

H_{0}: θ = 1 against H_{1}: θ ≠ 1

based on the random sample Y_{1}, Y_{2}, … , Y_{n} , the rejection region of the likelihood ratio test is of the form, for some c_{1} < c_{2} ,

Solution:

QUESTION: 11

Solution:

QUESTION: 12

Solution:

QUESTION: 13

A possible value of b ∈ ℝ for which the equation x^{4} + bx^{3} + 1 = 0 has no real root is

Solution:

QUESTION: 14

Let the Taylor polynomial of degree 20 for Then a_{15} is

Solution:

QUESTION: 15

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

Solution:

QUESTION: 16

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

Solution:

QUESTION: 17

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

Solution:

QUESTION: 18

If P =and 6P^{−1} = aI_{3} + bP − P^{2}, then the ordered pair (a, b) is

Solution:

QUESTION: 19

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

Solution:

QUESTION: 20

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?

Solution:

QUESTION: 21

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

Then the distribution of the random variable W = 2x^{2} is

Solution:

QUESTION: 22

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

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

Solution:

QUESTION: 23

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

Solution:

QUESTION: 24

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

Solution:

QUESTION: 25

Let X_{1}, X_{2} and X_{3} be i.i.d. U(0,1) random variables. Then P(X_{1} > X_{2} + X_{3}) equals

Solution:

QUESTION: 26

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

Solution:

QUESTION: 27

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

Solution:

QUESTION: 28

Let X_{1} and X_{2} be a random sample from a continuous distribution with the probability density function

where θ > 0. If X_{(1)} = min{X_{1}, X_{2}} and then which one of the following statements is TRUE?

Solution:

QUESTION: 29

Let X_{1}, X_{2}, … , 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,

Solution:

QUESTION: 30

Let X_{1}, X_{2}, … , X_{n} be a random sample from a N(θ, 1) distribution. To test H_{0}: θ = 0 against H_{1}: 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

Solution:

*Multiple options can be correct

QUESTION: 31

Let be a sequence of positive real numbers such that the series converges. Which of the following statements is (are) always TRUE?

Solution:

*Multiple options can be correct

QUESTION: 32

Let f : ℝ → ℝ be continuous on ℝ and differentiable on (−∞, 0) ∪ (0, ∞). Which of the following statements is (are) always TRUE?

Solution:

*Multiple options can be correct

QUESTION: 33

Let P be a 2 × 2 real matrix such that every non-zero vector in ℝ^{2} is an eigenvector of P. Suppose that λ_{1} and λ_{2} denote the eigenvalues of P and P for some t ∈ ℝ. Which of the following statements is (are) TRUE?

Solution:

*Multiple options can be correct

QUESTION: 34

Let P be an n x n non-null real skew-symmetric matrix, where n is even. Which of the following statements is (are) always TRUE?

Solution:

*Multiple options can be correct

QUESTION: 35

Let X be a random variable with the cumulative distribution function

Which of the following statements is (are) TRUE?

Solution:

*Multiple options can be correct

QUESTION: 36

Let X and Y be i.i.d. Exp (λ) random variables. If Z = max{X − Y, 0}, then which of the following statements is (are) TRUE?

Solution:

*Multiple options can be correct

QUESTION: 37

Let the discrete random variables X and Y have the joint probability mass function

Which of the following statements is (are) TRUE?

Solution:

*Multiple options can be correct

QUESTION: 38

Let X_{1}, X_{2}, … be a sequence of i.i.d. continuous random variables with the probability density function

If S_{n} = X_{1} + X_{2} + ⋯ + X_{n} and = s_{n }/n, then the distributions of which of the following sequences of random variables converge(s) to a normal distribution with mean 0 and a finite variance?

Solution:

*Multiple options can be correct

QUESTION: 39

Let X_{1}, X_{2}, … , X_{n} be a random sample from a U(θ, 0) distribution, where θ < 0.

If T_{n} = min{X_{1}, X_{2}, … , X_{n} }, then which of the following sequences of estimators is (are) consistent for θ?

Solution:

*Multiple options can be correct

QUESTION: 40

Let X_{1}, X_{2}, … , be a random sample from a continuous distribution with the probability density function, for λ > 0,

To test the hypothesis H_{0}: λ = 1/2 against H_{1}: λ = 3/4 at the level α (0 < α < 1), which of the following statements is (are) TRUE?

Solution:

*Answer can only contain numeric values

QUESTION: 41

(round off to 2 decimal places) equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 42

Let f : [0, 2] → ℝ be such that |f(x) − f(y)| ≤ |x − y|^{4/3} for all x, y ∈ [0, 2].

If

Solution:

*Answer can only contain numeric values

QUESTION: 43

The value (round off to 2 decimal places) of the double integral

equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 44

If is a real orthogonal matrix, then a^{2} + b^{2} + c^{2} + d^{2} equals ________

Solution:

*Answer can only contain numeric values

QUESTION: 45

Two fair dice are tossed independently and it is found that one face is odd and the other one is even. Then the probability (round off to 2 decimal places) that the sum is less than 6 equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 46

Let X be a random variable with the moment generating function

Using Chebyshev’s inequality, the upper bound for P

Solution:

*Answer can only contain numeric values

QUESTION: 47

In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 48

Let X be the number of heads obtained in a sequence of 10 independent tosses of a fair coin. The fair coin is tossed again X number of times independently, and let Y be the number of heads obtained in these X number of tosses. Then E(X + 2Y) equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 49

Let 0, 1, 0, 0, 1 be the observed values of a random sample of size five from a discrete distribution with the probability mass function P(X = 1) = 1 − P(X = 0) = 1 − e ^{−λ} , where λ > 0. The method of moments estimate (round off to 2 decimal places) of λ equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 50

Let X_{1}, X_{2}, X_{3} be a random sample from N(μ1, σ^{2} ) distribution and Y_{1}, Y_{2}, Y_{3} be a random sample from N(μ_{2}, σ^{2}) distribution. Also, assume that (X_{1}, X_{2}, X_{3}) and (Y_{1}, Y_{2}, Y_{3}) are independent. Let the observed values of and be 1 and 5, respectively. The length (round off to 2 decimal places) of the shortest 90% confidence interval of μ_{1} − μ_{2} equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 51

Solution:

*Answer can only contain numeric values

QUESTION: 52

For any real number y, let [y] be the greatest integer less than or equal to y and let {y} = y − [y]. For n = 1, 2, …, and for x ∈ ℝ, let

Then

Solution:

*Answer can only contain numeric values

QUESTION: 53

The volume (round off to 2 decimal places) of the region in the first octant (x ≥ 0, y ≥ 0, z ≥ 0 ) bounded by the cylinder x^{2} + x^{2} = 4 and the planes z = 2 and y + z = 4 equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 54

If ad − bc = 2 and ps − qr = 1, then the determinant of equals _________.

Solution:

*Answer can only contain numeric values

QUESTION: 55

In an ethnic group, 30% of the adult male population is known to have heart disease. A test indicates high cholesterol level in 80% of adult males with heart disease. But the test also indicates high cholesterol levels in 10% of the adult males with no heart disease. Then the probability (round off to 2 decimal places), that a randomly selected adult male from this population does not have heart disease given that the test indicates high cholesterol level, equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 56

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

where a and b are positive real numbers. If E (X) = 1, then E(X^{2}) equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 57

Let X and Y be jointly distributed continuous random variables, where Y is positive valued with E(Y^{2}) = 6. If the conditional distribution of X given Y = y is y(1 − y, 1 + y), then Var(X) equals __________

Solution:

*Answer can only contain numeric values

QUESTION: 58

Let X_{1}, X_{2}, … , X_{10} be i.i.d. N(0, 1) random variables. If T = , then

Solution:

*Answer can only contain numeric values

QUESTION: 59

Let X_{1}, X_{2}, X_{3} be a random sample from a continuous distribution with the probability density function

Let X_{(1)} = min{X_{1}, X_{2}, X_{3}} and c > 0 be a real number. Then (X_{(1)} − c, X_{(1)}) is a 97% confidence interval for μ, if c (round off to 2-decimal places) equals __________.

Solution:

*Answer can only contain numeric values

QUESTION: 60

Let x1, X_{2}, X_{3}, X_{4} be a random sample from a discrete distribution with the probability mass function P(X = 0) = 1 − P( = 1) = 1 − p, for 0 < p < 1. To test the hypothesis

consider the test:

Reject H_{0} if X_{1} + X_{2} + X_{3} + X_{4} > 3.

Let the size and power of the test be denoted by α and y, respectively. Then α + y(round off to 2 decimal places) equals __________.

Solution:

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