UPSC conducts exam for Indian Statistical Service every year across various exam centers in the country. Below you can read and download full ISS 2011, Indian Statistical Service question paper with answers. You can download this ISS previous year paper.

STATISTICS- I

Time Allowed : .Three  hrs                                                       Maximum Marks : 200

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INSTRUCTIONS

Candidates should attempt FIVE questions in ALL including Questions No. 1 and 5 which are compulsory. The remaining THREE questions
should be answered by choosing at least ONE question each from Section A .and Section B. The number of marks carried by each question
is indicated against each. Answers must be written only in. ENGLISH. (Symbols and abbreviations. are as usual) Any essential data assumed by candidates for answering questions must be clearly stated. A normal distribution table and a ‘t’ table are attached with this question paper.

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SECTION A

1. Answer any five of the following :
(a) Verify•the following identities :
ISS Question 1(A)

(b) A fair die is thrown until a 6 appears. Specify .the sample space. What is the probability that it must be thrown at least 3 times ?

(c) P(A) = 1/3 and P(B^c) = 1/4. Can A and B be disjoint ? Explain.

(d) Show that E(X – a)^2 is minimized for a = E(X), assuming that the· first- 2 moments of X exist.

(e) Let
ISS Question 1(E)
Show that f(x) is a probability density ‘function . Obtain V(X).

(f) Show that f(x)
ISS Question 1(F)
probability density function for an appropriate value of a. Upto what order do the moments of this p.d.f. exist ?

2.
ISS Question 2(A)

(b) Show that the sum of two independent Poisson random variables with parameters ƛ and μ respectively is a Poisson random variable with parameter ƛ + μ.

( c) Let the joint p.d.f. of (X, Y) be
ISS Question 2(C)
Obtain the probability ISS Question 2(C) 2

(d)
ISS Question 2(d)

3. (a) Let X₁and X₂be two independent exponentially distributed random variables with the same mean e. Define V = max (X₁X₂) and W = min (X₁X₂). Show that V -W and 2W are independent and identically distributed random variables.

(b) Let X be a positive valued random variable. Prove that
ISS Question 3 (B)
Hence deduce the Chebychev’s inequality.

(c) Let X have the continuous c.d.f. F(x). Define U = F(x). Show that both – log U arid – log (1 – U) are exponential random variables:

(d) Obtain the median and the quartiles of the Cauchy distribution with p.d.f.
ISS Question 3 (d)

4.
ISS Question 4 (A)

(b) Let (X, Y) have a bivariate distribution with ,finite moments up to order 2. Show that
(i) E(E(XIY)) = E(X), and
(ii) ISS Question 4 (b)

(c) Show. that a. c.d.f. can. have at mo~t a .countable number of jumps.

(d) Consider the .following bivariat~ p.m.f .. of (X, Y) :

p(O, 10) = p(O, 20) =2/18

p(1, 10) = p(1, 30) =3/18

p(1, 20) =4/18; p(2, 30) =4/18

Obtain the conditional mass functions p(y/x = 2),and p(y/x = 1).


SECTION B


5. Answer any five of the following :

(a) Tabulate the exact null distribution of Wilcoxon rank sum’ statistic for n₁=n₂=3

(b) Let (X, Y) have the uniform distribution over the range 0 < y· < x < 1. Obtain the conditional mean and variance of X given Y = y.

(c) Let the temperature before and after administration of aspirin be
ISS Question 5(C)

Test by the sign test; whether aspirin is effective in reducing temperature. What is the p-value of the calculated statistic ?

(d) Use Simpson’s rule withdrew ordinates to compute an approximation to 7t with the help of the integration of the function (1 + x^2)-^1 from 0 to 1.

(e) Use mathematical induction to prove

ISS Question 5(e)

(f)
ISS Question 5(F)
6. (a) Let X and Y be two random variables with correlation coefficient 0·9. Can a third random variable Z have correlation coefficient – 0·9 with both X and Y ? Give reasons for your answer.

(b) Show that the square of the one sample t-statistic has the F-distribution. What are its degrees of freedom ?

(c) Define the correlation ratio n^1 of X on Y. If p is the usual correlation coefficient between X and Y, then show that ISS Question 6(C)

(d) Prove that the sum of two independent chi-squared random variables is also chi-squared.

7. (a) Apply the Wilcoxon two-sample test to the following data on the first breakdown. times of two brands of computers :
Brand A  98, 102, 47, 85, 99, 140, 130
Brand B  95, 125, 160, 155, 148.
Use 1·96 as the critical point for the appropriate test.

(b) Describe a .test of independence of two normal random variables based on r, the sample
correlation coefficient using the t-distribution. If n = 10 and r = 0·9, then carry out the test.

(c) Show that the best predictor of Y, in terms of minimum MSE, is linear in X, if (X, Y) have bivariate normal distribution .

(d) Explain the Wald – Wold-fowitz run test for randomness in a sequence of two types of symbols. Find E^Ho (R) where R denotes the number of runs of elements of one kind.

8. (a) The following are the frequencies in the given intervals :
ISS Question 8 (A)

Draw the histogram of this data. Calculate the mean of the data from the frequency table.

(b)
ISS Question 8 (B)

(c)
ISS Question 8 (C)

(d)
ISS Question 8 (d)

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