Tamil Nadu Open University (TNOU) 2009-2nd Year M.Sc Mathematics Tamilnadu open university Maths Mathematical statistics - Question Paper
M.Sc. DEGREE exam —
JUNE, 2009.
(AY 2005–06 and CY–2006 batches only)
Second Year
Mathematics
MATHEMATICAL STATISTICS
Time : three hours Maximum marks : 75
PART A — (5 x five = 25 marks)
ans any 5 ques..
every ques. carries five marks.
1.Find the value of the constant if is a p.d.f of a random
variable .
2.A factory manufacturing television has 4 units . The units manufacture 15%, 20%, 30%, 35% of the total output respectively. It was obtained that out of their outputs 1%, 2%, 2%, 3% are defective. A television is chosen at random from the output and obtained to be defective. What is the probability that it came from unit ?
3.Let be a random variable with probability distribution . obtain the moment generating function and hence obtain its mean and variance.
4.Let be mutually stochastic independent random variables having respectively, the normal distribution . Let where are constant. Prove that is normally distributed with mean and .
5.Let be . obtain the limiting distribution of the random variable .
6.Let be a random sample from the exponential distribution with the density function obtain the maximum likelihood estimator of .
7.In a certain political campaign, 185 out of 351 voters favour particular candidate. obtain the 95% confidence interval for the fraction of the voting population who favour this candidate.
8.Define (a) consistent estimator and (b) unbiased estimator.
PART B — (5 x 10 = 50 marks)
ans any 5 ques..
every ques. carries 10 marks.
9.State and prove Chebychev’s inequality.
10.If is a continuous random variable with p.d.f. obtain the correlation coefficient ranging from and .
11.In a normal distribution, 31% of the items are under 45 and 8% are over 64. obtain the mean and variance of the distribution.
12.Derive t–distribution.
13.Let denote the distribution function of a random variable whose distribution depends upon the positive integer . Let be a constant which does not depend upon . The random variable converges stochastically to the constant if , . Prove.
14.State and prove Neymann–Factorisation theorem.
15.State and prove Rao–cramer inequality.
16.State and prove Neymann–Pearson theorem.
——–––––––––
M.Sc.
DEGREE EXAMINATION
JUNE, 2009.
(AY 200506 and CY2006 batches only)
Second Year
Mathematics
MATHEMATICAL STATISTICS
Time : 3 hours Maximum marks : 75
PART A (5 5 = 25 marks)
Answer any FIVE questions.
Each question carries 5 marks.
1.
Find the value of the constant 
if
is a p.d.f of a random
variable 
.
2.
A factory manufacturing television has four units 
. The units manufacture 15%,
20%, 30%, 35% of the total output respectively. It was found that out of their
outputs 1%, 2%, 2%, 3% are defective. A television is chosen at random from the
output and found to be defective. What is the probability that it came from
unit 
?
3.
Let be a random variable with
probability distribution 
. Find
the moment generating function and hence find its mean and variance.
4.
Let 
be mutually stochastic
independent random variables having respectively, the normal distribution 
. Let 
where 
are constant. Prove that 
is normally distributed
with mean 
and 
.
5.
Let 
be 
. Find the limiting
distribution of the random variable 
.
6.
Let be a random sample from
the exponential distribution with the density function 
Find the maximum likelihood
estimator of 
.
7.
In a certain political campaign, 185 out of 351 voters favour
particular candidate. Find the 95% confidence interval for the fraction 
of the voting population
who favour this candidate.
8. Define (a) consistent estimator and (b) unbiased estimator.
PART B (5 10 = 50 marks)
Answer any FIVE questions.
Each question carries 10 marks.
9. State and prove Chebychevs inequality.
10. If
is a continuous random
variable with p.d.f. 
find the
correlation coefficient between and .
11. In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and variance of the distribution.
12. Derive tdistribution.
13. Let
denote the distribution
function of a random variable 
whose
distribution depends upon the positive integer 
.
Let 
be a constant which does
not depend upon . The random
variable converges stochastically to
the constant if 
, 
. Prove.
14. State and prove NeymannFactorisation theorem.
15. State and prove Raocramer inequality.
16. State and prove NeymannPearson theorem.
| Earning: Approval pending. |
