Exam Details
| Subject | statistics | |
| Paper | paper 1 | |
| Exam / Course | indian economic service and indian statistical service examination (ies/iss) | |
| Department | ||
| Organization | union public service commission | |
| Position | ||
| Exam Date | 2014 | |
| City, State | central government, |
Question Paper
1. Answer all of the following:
A positive integer X is selected at random from the first 50 natural numbers. Calculate
p(X 96/X 50).
Suppose that all the four outcomes 03 and 04 of an experiment are equally likely. Define A 01, B 02, 04} and C 03, What can you say about the pairWise independence and mutually independence of the events Band C
The marginal distributions of X and Y are given in the following table:
y x 1 2 Total
3 1/4
4 3/4
Total 1/2 1/2 1
If the covariance between X and Y is zero, find the cell probabilities and see whether X and Y are independent.
Show that for 40,000 throws of a balanced coin, the probability is at least 0·99 that the proportion of heads will fall between 0·475 and 0·525.
If be the probability density function of a distribution, then show that
integral xfx(x) dx =µ -IT
whereL'=L-J.l, U -J.l and and
are the probability density function' and distribution function of the standard normal distribution respectively.
X1,X2,...,XN are independently, identically distributed random variables. Define SN Xl X2 ... XN, where N is a random variable independent of i ... N.
Show that the moment generating function of SN is
MSN(t) MN(log
where is the mgf of a random variable Y.
Hence find the mgf of SN when N follows a Poisson distribution with parameter A and Xi follows an exponential distribution with mean parameter l(l)N.
If be the probability density function of a lognormal distribution, show that
fu 1
q.l+_k2a2
xkfx dx 2
L
-logL -Jl kd
h .&.Ij{ -cr an
were
u
log U
-kcr and is the distribution function of the standard normal distribution. Hence find and
In a lottery 1000 tickets are sold and the cost of a ticket is Rs 10. The lottery offers a first prize of 1,000, two second prizes of Rs 500 each, and three third prizes of Rs 100 each. A person purchases a ticket. If X denotes the amount he may get, find and
n balls are distributed among r cells at random, each cell being free to receive any number of balls. Calculate the probability that a particular cell contains k balls when
balls are distinguishable, and
balls are non-distinguishable.
Write down the probability mass function of a trinomial distribution. Obtain the moment generating function of the distribution. Show that the correlation coefficient of any two variables is negative. Why?
Prove that two events cannot be incompatible and independent simultaneously.
is a sequence of independent random variables. Show that Xn X Xn X
where X is a random variable. I.s the converse true?
There are three identical bags U., U2 and U3 . U1 contains 3 red and 4 black balls; U2 contains 4 red and 5 black balls; U3 contains 4 red and 4 black balls. One bag is chosen at random; a ball is drawn at random from the chosen bag and it is found to be red.· Find the probability that the first bag is chosen.
Let be the probability generating function associated with a non-negative, integer valued random variable X. Show that
E
The runs scored by two batsmen A and B in five cricket matches were as follows:
BatsmanA: 50 60 100 70 20
Batsman B 120 100 30 20 40
Discuss the consistency and efficiency of the batsmen.
Given the following correlation matrix of order 3 x 3
1 0·5708 0·6735
1 0·4487
1
Calculate r12.3 r1.23·
Consider two samples as follows:
Sample 1 16, 17, 22, 24
Sample 2 10, 12, 18, 20, 26, 28, 32
Test whether the examples have come from the same population by Wilcoxon-Mann-Whitney test. [Given value of Z for a 0·05 1.645, where Z is
Define power of a test and discuss its role in selecting the best test. Describe a test procedure for testing equality of means of two independent normal populations, when standard deviations are equal but unknown for small samples.
Given the following values of the function y evaluate and also find x for which =25.
10, 15, 42.
Describe a test procedure for testing the null hypothesis Ho Jlx -Jly 00 particular value) for a bivariate normal population
22 .22
BN(J.lx' J.ly cry when G y and pare known; and p are unknown.
Let Xl, X2, ... Xn be n independently identically distributed random variables following an exponential distribution with mean parameter 6. Obtain the distributions of
Maximum X2, •••
Minimum X2, ••• xn) and
Median for n 2m 0).
A medicine supposed to have effect in preventing TB was treated on 500 individuals and their records were compared with the records of 500 untreated individuals as follows. Study the effectiveness of medicine by calculating
Yule's coefficient of association
Yule's coefficient of colligation.
NO-TB TB
Treated 252 248
Untreated 224 276
Distinguish between correlation and intra-class correlation.
The height (in em) of three brothers belonging to each of three families are recorded below. Compute the intra-class correlation coefficient.
Family Heights of brothers (in cm)
I 158.0 161·5 161·3
170·7 169·4 III 163.6 163·3 164·1
Discuss the role of asymptotic relative efficiency in judging the relative preference of two tests. Following are the yields (in kg) of a crop recorded from an experiment with median 20:
15·4 16-4 17-3 18·2 19·2 20.9 22.7 23.6 24·5
Test the null hypothesis
HO: M =20 against H1 M 20 at a =0·05.
[You are given P 0.50]
The force of mortality IS defined as Jlx dlx/dx when lx is the number of persons at exact age x (in years)· in any year of time. Given the following table, find a value of J.1 50.
Age 50 51 52 53
Ix 73,499 72,724 71,753 70,599
Explain the procedure for testing the hypothesis of equality of variances of two independent normal populations when population means are unknown. Write down the sampling distribution of the statistic. A sample of size 10 is drawn from each of two uncorrelated normal populations. Sample means and variances are:
1st population: mean variance 26
2nd population: mean variance 10
Test at level of significance whether the first population has greater standard deviation than that of the second population. [Given FO.05; 9,9 3·18]
Let X follow a binomial distribution P). Explain the test procedure for
Ho P Po against H1: P Po when the sample size is
small
large.
It is desired to use sample proportion p as an estimator of the population proportion with probability·0·95 or higher, that p is within 0·05 ofP. How large should sample size be?
Using Euler's method, compute the values of y correct upto 4 places of decimal for the differential equation dy/dx x y with initial condition Xo Yo= taking h =0.05.
A positive integer X is selected at random from the first 50 natural numbers. Calculate
p(X 96/X 50).
Suppose that all the four outcomes 03 and 04 of an experiment are equally likely. Define A 01, B 02, 04} and C 03, What can you say about the pairWise independence and mutually independence of the events Band C
The marginal distributions of X and Y are given in the following table:
y x 1 2 Total
3 1/4
4 3/4
Total 1/2 1/2 1
If the covariance between X and Y is zero, find the cell probabilities and see whether X and Y are independent.
Show that for 40,000 throws of a balanced coin, the probability is at least 0·99 that the proportion of heads will fall between 0·475 and 0·525.
If be the probability density function of a distribution, then show that
integral xfx(x) dx =µ -IT
whereL'=L-J.l, U -J.l and and
are the probability density function' and distribution function of the standard normal distribution respectively.
X1,X2,...,XN are independently, identically distributed random variables. Define SN Xl X2 ... XN, where N is a random variable independent of i ... N.
Show that the moment generating function of SN is
MSN(t) MN(log
where is the mgf of a random variable Y.
Hence find the mgf of SN when N follows a Poisson distribution with parameter A and Xi follows an exponential distribution with mean parameter l(l)N.
If be the probability density function of a lognormal distribution, show that
fu 1
q.l+_k2a2
xkfx dx 2
L
-logL -Jl kd
h .&.Ij{ -cr an
were
u
log U
-kcr and is the distribution function of the standard normal distribution. Hence find and
In a lottery 1000 tickets are sold and the cost of a ticket is Rs 10. The lottery offers a first prize of 1,000, two second prizes of Rs 500 each, and three third prizes of Rs 100 each. A person purchases a ticket. If X denotes the amount he may get, find and
n balls are distributed among r cells at random, each cell being free to receive any number of balls. Calculate the probability that a particular cell contains k balls when
balls are distinguishable, and
balls are non-distinguishable.
Write down the probability mass function of a trinomial distribution. Obtain the moment generating function of the distribution. Show that the correlation coefficient of any two variables is negative. Why?
Prove that two events cannot be incompatible and independent simultaneously.
is a sequence of independent random variables. Show that Xn X Xn X
where X is a random variable. I.s the converse true?
There are three identical bags U., U2 and U3 . U1 contains 3 red and 4 black balls; U2 contains 4 red and 5 black balls; U3 contains 4 red and 4 black balls. One bag is chosen at random; a ball is drawn at random from the chosen bag and it is found to be red.· Find the probability that the first bag is chosen.
Let be the probability generating function associated with a non-negative, integer valued random variable X. Show that
E
The runs scored by two batsmen A and B in five cricket matches were as follows:
BatsmanA: 50 60 100 70 20
Batsman B 120 100 30 20 40
Discuss the consistency and efficiency of the batsmen.
Given the following correlation matrix of order 3 x 3
1 0·5708 0·6735
1 0·4487
1
Calculate r12.3 r1.23·
Consider two samples as follows:
Sample 1 16, 17, 22, 24
Sample 2 10, 12, 18, 20, 26, 28, 32
Test whether the examples have come from the same population by Wilcoxon-Mann-Whitney test. [Given value of Z for a 0·05 1.645, where Z is
Define power of a test and discuss its role in selecting the best test. Describe a test procedure for testing equality of means of two independent normal populations, when standard deviations are equal but unknown for small samples.
Given the following values of the function y evaluate and also find x for which =25.
10, 15, 42.
Describe a test procedure for testing the null hypothesis Ho Jlx -Jly 00 particular value) for a bivariate normal population
22 .22
BN(J.lx' J.ly cry when G y and pare known; and p are unknown.
Let Xl, X2, ... Xn be n independently identically distributed random variables following an exponential distribution with mean parameter 6. Obtain the distributions of
Maximum X2, •••
Minimum X2, ••• xn) and
Median for n 2m 0).
A medicine supposed to have effect in preventing TB was treated on 500 individuals and their records were compared with the records of 500 untreated individuals as follows. Study the effectiveness of medicine by calculating
Yule's coefficient of association
Yule's coefficient of colligation.
NO-TB TB
Treated 252 248
Untreated 224 276
Distinguish between correlation and intra-class correlation.
The height (in em) of three brothers belonging to each of three families are recorded below. Compute the intra-class correlation coefficient.
Family Heights of brothers (in cm)
I 158.0 161·5 161·3
170·7 169·4 III 163.6 163·3 164·1
Discuss the role of asymptotic relative efficiency in judging the relative preference of two tests. Following are the yields (in kg) of a crop recorded from an experiment with median 20:
15·4 16-4 17-3 18·2 19·2 20.9 22.7 23.6 24·5
Test the null hypothesis
HO: M =20 against H1 M 20 at a =0·05.
[You are given P 0.50]
The force of mortality IS defined as Jlx dlx/dx when lx is the number of persons at exact age x (in years)· in any year of time. Given the following table, find a value of J.1 50.
Age 50 51 52 53
Ix 73,499 72,724 71,753 70,599
Explain the procedure for testing the hypothesis of equality of variances of two independent normal populations when population means are unknown. Write down the sampling distribution of the statistic. A sample of size 10 is drawn from each of two uncorrelated normal populations. Sample means and variances are:
1st population: mean variance 26
2nd population: mean variance 10
Test at level of significance whether the first population has greater standard deviation than that of the second population. [Given FO.05; 9,9 3·18]
Let X follow a binomial distribution P). Explain the test procedure for
Ho P Po against H1: P Po when the sample size is
small
large.
It is desired to use sample proportion p as an estimator of the population proportion with probability·0·95 or higher, that p is within 0·05 ofP. How large should sample size be?
Using Euler's method, compute the values of y correct upto 4 places of decimal for the differential equation dy/dx x y with initial condition Xo Yo= taking h =0.05.