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 :

(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

Show that f(x) is a probability density ‘function . Obtain V(X).

(f) Show that f(x)

probability density function for an appropriate value of a. Upto what order do the moments of this p.d.f. exist ?

2.

(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

Obtain the probability

(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

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.

4.

(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)

(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

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

(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

(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 :

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

(b)

(c)

(d)

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