Model Test Paper, Mathematical Statistics - 2017

# Model Test Paper, Mathematical Statistics - 2017 - IIT JAM

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Model Test Paper, Mathematical Statistics - 2017 - Question 1

### If (xn) is a sequence of real numbers which converges to x then the sequence (sn) where

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 1

(xn) converges to x implies that for every ∈ > 0 there exists an N such that

|xn – x| < ∈ for all n ≥ N

We now divide the sum into those terms before N and those after N. We have now which requires that n > C/∈ .

Thus if we let N = max{N, C/∈} then for all n ≥ N we have |sn – x| < 2∈ which proves that the sequence (sn) converges to x.

Model Test Paper, Mathematical Statistics - 2017 - Question 2

### The general solution of the differential equation y”(x) – 4y’(x) + 8y(x) + 10ex cos x is

Model Test Paper, Mathematical Statistics - 2017 - Question 3

### If the probability that A and B will die within a year are p and q respectively, then the probability that only one of them will be alive at the end of the year is

Model Test Paper, Mathematical Statistics - 2017 - Question 4

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 4

2ydx + (2x – 3y)dy = 0 ...(1)

It is Homogeneous

M = 2y Model Test Paper, Mathematical Statistics - 2017 - Question 5

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

Model Test Paper, Mathematical Statistics - 2017 - Question 6

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)

Model Test Paper, Mathematical Statistics - 2017 - Question 7

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

Model Test Paper, Mathematical Statistics - 2017 - Question 8 Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 8

The above infinite series can be rewritten as follows,  Model Test Paper, Mathematical Statistics - 2017 - Question 9

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

Model Test Paper, Mathematical Statistics - 2017 - Question 10 Integration by Parts results in

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 10 Integration by parts or Model Test Paper, Mathematical Statistics - 2017 - Question 11

A function f is defined on interval (0, 1) as follows then which one of the following is true?

Model Test Paper, Mathematical Statistics - 2017 - Question 12

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

Model Test Paper, Mathematical Statistics - 2017 - Question 13

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

Model Test Paper, Mathematical Statistics - 2017 - Question 14

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, ..., Xn are characteristic random variables, find the mean and variance of Model Test Paper, Mathematical Statistics - 2017 - Question 15

In the regression line Y = a + bX:

Model Test Paper, Mathematical Statistics - 2017 - Question 16

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

Model Test Paper, Mathematical Statistics - 2017 - Question 17

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.

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 17 Thus the required probability is Model Test Paper, Mathematical Statistics - 2017 - Question 18

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].

Model Test Paper, Mathematical Statistics - 2017 - Question 19

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 :

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 19

Let the marks obtained per question be a random variable X.
Its probability distribution table is given below: Expected marks per question = E(x) Total marks expected for 150 questions = 1/16 × 150 = 75/8 marks per student

Total expected marks of 1000 students = 75/8 × 1000 = 9375 marks

Model Test Paper, Mathematical Statistics - 2017 - Question 20

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

Model Test Paper, Mathematical Statistics - 2017 - Question 21

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 21

Let us define the events :      In (1) let us take : a = 0, b = 1/2, ; c = 1/4, d = 3/4 s.t. a < b and c < d. Then using (2) we get :

F(b, d) = 1 ; F(b, c) = 1 ; F(a, d) = 1 ; F(a, c) = 0.

Substituting in (1) we get :

P(a < X ≤ b ∩ c < Y < d) =1 – 1 – 1 + 0 = –1

which is not possible since P( . ) ³ 0.
Hence F(x, y) defined in (2) cannot be the distribution function of variates X and Y.

Model Test Paper, Mathematical Statistics - 2017 - Question 22

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?

Model Test Paper, Mathematical Statistics - 2017 - Question 23

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 23  Model Test Paper, Mathematical Statistics - 2017 - Question 24

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.

Model Test Paper, Mathematical Statistics - 2017 - Question 25

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?

Model Test Paper, Mathematical Statistics - 2017 - Question 26

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 26

Let S be total number of successes.
Then Using Chebychev’s inequality, we get Model Test Paper, Mathematical Statistics - 2017 - Question 27

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

Model Test Paper, Mathematical Statistics - 2017 - Question 28

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;

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 28 Model Test Paper, Mathematical Statistics - 2017 - Question 29

In a test of H0 : μ = 100 against HA : 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:

Model Test Paper, Mathematical Statistics - 2017 - Question 30

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 30 *Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 31

A monotonically increasing sequence <xn> is convergent if

*Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 32

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?

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 32

(B,D) A binomial random variable Y with parameters n = 100 and p = 1/2
can be written as : where X1, ..., X100 are mutually independent Bernoulli random variables with parameter p
= 1/2
Thus: we know that the distribution of X100 can be approximated by a normal distribution: Therefore, the distribution of Y can be approximated by: Thus, Y can be approximated by a normal distribution with mean μ = 50 and variance
σ2 = 25.

*Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 33

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 33    *Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 34

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

*Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 35

Let {Xn} be a sequence of independent Bernoulli random variables with parameter p = 1/2
generic term Xn of the sequence has support  Rn = { 0, 1} and probability mass function: Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 35

(B,C) The sequence {Xn} is and IID sequence. The mean of a generic term of the sequence is The variance of a generic term of the sequence can be derived thanks to the usual formula for computing the variance (Var[X] – E[X2] – E[X]2) : Therefore, the sequence {Xn} satisfies the conditions of Lindeberg-Levy Central Limit Theorem (IID, finite mean, finite variance). Using the Central Limit Theorem to approximate its distribution, we obtain: or *Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 36

Cauchy’s nth Root Test

Let ∑un be a positive term series such that *Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 37

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 37 Now for maximum or minimum of z, dz/dx = 0 Therefore z is maximum and minimum at x = 1/2 and x = 1 respectively

Hence y is also maximum and minimum at x = 1/2 and x = 1 respectively..

Maximum value of y is and minimum value of y is
In(3 – 2 – 6 + 6 + 1) i.e., In 2

*Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 38

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 ?

*Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 39

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 39

A pair of dice can fall in 36 ways. A seven can occur in 6 ways, an eleven in 2 ways.
Then P(seven) = 6/36 = 1/6
= and P(eleven) = 2/36 = 1/18
= . Therefore, in 360 tosses we would expect 1/6 (360) = 60 sevens and 1/18 (360) = 20 elevens, so that with a P value of 0.044. we would be inclined to reject the hypothesis that the dice are fair. Using Yates’ correction, however, we find with a P value of 0.056.

*Multiple options can be correct
Model Test Paper, Mathematical Statistics - 2017 - Question 40

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

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 41

Evaluate the following : *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 42

The following nonlinear differential equation can be solved exactly by separation of variables. The value of θ(100) most nearly is

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 43

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 43

Writing the same matrix side by side with the unit matrix of order 3, we have  Hence the inverse of the given matrix is  *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 44

In the integral change the order of integration, and evaluate the integral.

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 45

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

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 46

The solution for what should be the value of a?

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 46  *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 47

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

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 48

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?

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 48

The total number of flaws is given by: (0 × 4) + (1 × 3) + (2 × 5) + (3 × 2) + (4 × 4) + (5 × 1) + (6 × 1) = 46
So the average number of flaws for the 20 sheets is given by: *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 49

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?

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 50

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?

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 51 Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 51 Which is five times the geometric series with a = 56 and r = 2/7. Since r < 1, then the geometric series converges to a/(1- r). Thus, *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 52

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

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 53

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

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 54

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?

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 55

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.

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 55

The average number of failures per week is “Not more than one failure” means we need to include the probabilities for “0 failures” plus “1 failure”. *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 56 *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 57

The series converges absolutely to the limit__.

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 57 (Notice between the two middle steps, we used L’Hopital rule) Conclusion: Since 2/e < 1, the series converges.

*Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 58

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

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 58

On the hypothesis that the flower colour is independent for the flatness of leaves, the theoretical frequency for the plants having white flowers and flat leaves = 135 x 119/160= 100
approx.
Other theoretical frequencies follow from the fact that the border frequencies for the rows and columns remain unchanged. The expected frequencies are shown in the following 2 × 2 contingency table. *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 59

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 ?

Detailed Solution for Model Test Paper, Mathematical Statistics - 2017 - Question 59

If the player scores one goal from her first two attempts, then the sequence of events could be either a goal first (G) followed by a miss (M) or a miss first (M) followed by a goal (G). Thus : Pr(one goal from first two attempts) = Pr(GM) + Pr(MG) *Answer can only contain numeric values
Model Test Paper, Mathematical Statistics - 2017 - Question 60

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.

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