Loyola College Question Paper 2009
Loyola College M.Sc. Statistics Nov 2008 Advanced Distribution Theory Question Paper PDF Download
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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
FIRST SEMESTER – November 2008
ST 1810 – ADVANCED DISTRIBUTION THEORY
Date : 08-11-08 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
SECTION – A
Answer ALL the questions (10 x 2 = 20 marks)
- Define a truncated distribution and give an example.
- Find the MGF of a power series distribution.
- Define lack of memory property for discrete random variable.
- If X is distributed as Lognormal, show that its reciprocal is also distributed as Lognormal.
- Let (X1, X2) have a bivariate Bernoulli distribution. Find the distribution of X1 + X2.
- Find the marginal distributions associated with bivariate Poisson distribution.
- Show that Marshall – Olkin bivariate exponential distribution satisfies bivariate lack of memory property.
- Define non-central chisquare – distribution and find its mean.
- Let X1, X2, X3, X4 be independent standard normal variables. Examine whether
X12 +3 X22 + X32 +4 X42 – 2 X1X2 + 6 X1X3 + 6 X2X4– 4 X3X4 is distributed as chi-square.
- Let X be N(q, 1), q = 0.1, 0.5. If q is discrete uniform, find the mean of the compound
distribution.
SECTION – B Answer any FIVE questions (5 x 8 = 40 marks)
- State and establish a characterization of Poisson distribution.
- Derive the pdf of a bivariate binomial distribution. Hence, show that the regressions are
linear.
- Let (X1, X2) follow a Bivariate normal distribution with V(X1) = V(X2). Examine
whether X1 + X2 and (X1 – X2)2 are independent.
- Show that the mean of iid Inverse Gaussian random variables is also Inverse Gaussian.
- Let (X1, X2) follow a Bivariate exponential distribution . Derive the distributions of Min{X1, X2} and Max{X1, X2}.
- Find the mean and the variance of a non-central F – distribution.
- Let X1, X2, X3,…, Xn be iid N(0, σ2), σ > 0 random variables.Find the MGF of X /AX/ σ2.
Hence find the distribution of X1X2.
- Illustrate the importance of the theory of quadratic forms in normal variables in ANOVA.
SECTION – C
Answer any TWO questions (2 x 20 = 40 marks)
- a) Let X1, X2, X3,…, Xn be iid non-negative integer-valued random variables. Show that X1
is geometric if and only if Min{X1, X2, X3,…, Xn} is geometric.
- b) State and establish the additive property of bivariate Poisson distribution.
- a) Let (X1, X2) have a bivariate exponential distribution of Marshall-Olkin. Find the
cov(X1, X2).
- b) Let (X1, X2) follow a bivariate normal distribution. State and establish any two of its
properties.
- a) Define non-central t – variable and derive its pdf.
- b) Let X be a random variable with the distribution function F given by
0 , x < 0
F(x) = (2x + 1)/4, 0 ≤ x < 1
1, x ³ 1.
Find the mean, median and variance of X.
- a) State and establish a necessary and sufficient condition for a quadratic form in normal variables to
have a chi-square distribution.
- b) Let (X1, X2) follow a trinomial distribution with index n and cell probabilities θ1 ,θ2. If the prior
distribution is uniform, find the compound distribution. Hence find the means of X1 and X2.
Loyola College M.Sc. Statistics April 2009 Testing Statistical Hypothesis Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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SECOND SEMESTER – April 2009
ST 2812 / 2809 – TESTING STATISTIACAL HYPOTHESIS
Date & Time: 22/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION A
Answer all questions. (10 x 2 = 20)
- Define level and power of a test.
- Let X be a random variable with pdf .
Obtain the Most Powerful Test of size for testing H0: θ = 1 Vs H1: θ = 2.
- Give the general form of (k+1) parameter exponential family of distributions.
- Define Uniformly Most Powerful Test.
- Let. Consider the test function
for testing H0: θ = 0.2 Vs H1: θ > 0.2.Obtain the value of power function at
θ = 0.4.
- What are the circumstances under which Locally Most Powerful test is used?
- What is meant by shortest length confidence interval?
- Define maximal invariant function.
- What is meant by nuisance parameter? Give an example.
- Define Likelihood Ratio Test.
SECTION B
Answer any FIVE questions. (5 x 8 = 40)
- Let denote a random sample fromDerive a Most Powerful test of level 0.05 for testing Vs. Also obtain the cut-off point.
- Show that the family of densities possesses MLR property.
- Let denote a random sample of size n from. Consider the problem of testing Vs. Show that UMP test of does not exist.
- For (k+1) parameter exponential family of densities, derive an unconditional UMPUT of level for testing Vs clearly stating the assumptions.
- State and prove the sufficient part of Generalized Neyman-Pearson lemma.
- Show that any test having Neyman structure is similar. Also show that the converse is true under certain assumptions (to be stated).
- Derive the Locally Most Powerful test for testing Vs based on a random sample of size n from, where and are known pdf’s.
- Find maximal invariant function under the group of i.) Location transformations and ii.) Scale transformations.
SECTION C
Answer any TWO questions. (2 x 20 = 40)
- a.) Derive a UMP test of level for testing Vs for the family of densities that possess MLR in T(x). Show that the power function of the above testing problem increases in
b.) Show that any UMP test is always UMPUT. (16+4)
- Consider a one parameter exponential family with density. Assume is strictly increasing in. Derive a UMP test of level for testing Vs.
- Let X and Y be independent Binomial variables with parameters and respectively, where m and n are assumed to be known. Derive a conditional UMPUT of size for testing Vs.
- Let anddenote independent random samples from and respectively. Derive the Likelihood Ratio Test for testing Vs.
Loyola College M.Sc. Statistics April 2009 Stochastic Processes Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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THIRD SEMESTER – April 2009
ST 3809 – STOCHASTIC PROCESSES
Date & Time: 23/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
PART-A
Answer all the questions: (10 X 2 = 20)
Define a stochastic process with an example.- Define a process with independent increments.
- Show that communication between two states i and j satisfies transitive relation.
- Define (i) transcient state (ii) recurrent state.
- Define a Markov process.
- Obtain the PGF of a Poisson process.
- Define a renewal function. What is the relation between a renewal function and the
distribution functions of inter occurrence times?
- When do you say that is a martingale with respect to ?
- What is a branching process?
- What is the relationship between Poisson process and exponential distribution?
PART-B
Answer any 5 questions: (5 X 8 = 40)
- State and prove Chapman – Kolmogorov equation for a discrete time Markov chain.
- Obtain the equation for in a Yule process with X(0) = 1.
- Let and be i.i.d random variables with mean 0 and variance.
Show that is a martingale with respect to .
- Show that the matrix of transition probabilities together with the initial distribution
completely specifies a Markov chain.
- Show that the renewal function satisfies
- Establish the relationship between Poisson process and Binomial distribution.
- Obtain the stationary distribution for the Markov chain having transition probability
matrix
- If a process has stationary independent increments and finite mean show that
where and .
PART-C
Answer any 2 questions: (2 X 20 = 40)
- a) State and prove the necessary and sufficient condition required by a state to be recurrent.
b.) Verify whether state 0 is recurrent in a symmetric random walk in three dimensions. (10+10)
- a) State the postulates of a Poisson process. Obtain the expression for.
b.) Obtain the distribution for waiting time of k arrivals for a Poisson process. (15+5)
- a) Obtain the generating function for a branching process. Hence obtain the mean and variance.
- b) Let be the probability that an individual in a generation generates k
off springs. If obtain the probability of extinction.
(15+5)
- a.) Obtain the renewal function corresponding to the lifetime density.
b.) Show that the likelihood ratio forms a martingale.
c.) Let be a martingale with respect to. If is a convex function with
show that is a sub martingale with respect to .
(10+5+5)
Loyola College M.Sc. Statistics April 2009 Statistical Process Control Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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FOURTH SEMESTER – April 2009
ST 4806 – STATISTICAL PROCESS CONTROL
Date & Time: 21/04/2009 / 9:00 – 12:00 Dept. No. Max. : 100 Marks
SECTION – A
Answer ALL the questions 10×2 =20
- Define quality improvement
- Explain six-sigma quality
- Discuss the statistical basis underlying the general use of 3 – sigma limits on control charts.
- How is lack of control of a process is determined by using control chart technique? .
- Write down the expression for process capability ratio (PCR) when only the lower specification is known.
- What information is provided by the OC curve of a control chart?
- Give an expression for AOQ for a single sampling plan.
- Write a short note on Multivariate Quality Control.
- Define a) Specification limits b) Natural tolerance limits.
- Explain double sampling plan.
SECTION- B
Answer any FIVE questions 5 x 8= 40
- What are the major statistical methods for quality improvement? .
- A normally distributed quality characteristic is monitored by a control chart with K sigma
control limits. Develop an expression for the probability that a point will plot outside the
control limits when the process is really in control .
- Sampes of n=6 items are taken from a manufacturing process at regular interval. A normally
distributed quality characteristic is measured and x-bar and S values are calculated for each
sample. After 50 subgroups have been analyzed, we have
- a) Calculate the control limits for the x-bar and S control charts.
- b) Assume that all the points on both charts plot within the control limits .What are the natural
tolerance limits of the process? .
- Write a detailed note on the moving average control chart.
- In designing a fraction non-conforming chart with CL at p =0.20 and 3-sigma control limits,
what is the simple size required to yield a positive LCL? What is the value of n necessary to
give a probability of 0.50 of detecting a shift in the process to 0.26?.
- Consider the single – sampling plan for which p1 = 0.01, a = 0.05, p2 = 0.10 and b = 0.10.
Suppose that lots of N = 2000 are submitted. Draw the AOQ curve and find the AOQL.
- What are acceptance and rejection lines of a sequential sampling plan for attributes?. How
are the OC and ASN values obtained for this plan? .
- What are chain samplings and skip-lot sampling plans?
SECTION- C
Answer any two questions 2 X 20 = 40
- a) Describe the procedure of obtaining the OC curve for a p-chart with an example .
- b) Explain process capability analysis with an illustration. ( 12+8 )
20.a) What are modified control charts?. Explain the method of obtaining control limits for these
charts.
- b) A control chart for non-conformities per unit uses 0. 95 and 0.05 probability limits .The
center line is at u=14. Determine the control limits if the size of the sample is 10. (14+6)
21.a) Discuss the purpose of cumulative sum chart .
- b) Outline the procedure of constructing V-mask. (8+12)
- a) Explain with an illustration the method of obtaining the probability of acceptance
for a triple sampling plan.
- b) What are continuous sampling plans?. Mention a few situations where these plans are
applied. (10 + 10)
Loyola College M.Sc. Statistics April 2009 Statistical Computing – III Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
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M.Sc. DEGREE EXAMINATION – STATISTICS
FOURTH SEMESTER – April 2009
ST 4808 – STATISTICAL COMPUTING – III
Date & Time: 25/04/2009 / 9:00 – 12:00 Dept. No. Max. : 100 Marks
Answer any three questions
- a) The data shown here are and R values for 24 samples of size n=5 taken from a process producing bearings. The measurements are made on the inside diameter of the bearing, with only the last three digits recorded.
| Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 34.5 | 34.2 | 31.6 | 31.5 | 35.0 | 34.1 | 32.6 | 33.8 | 34.8 | 33.6 | 31.9 | |
| R | 3 | 4 | 4 | 4 | 5 | 6 | 4 | 3 | 7 | 8 | 3 |
| Sample number | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
| 38.6 | 35.4 | 34 | 37.1 | 34.9 | 33.5 | 31.7 | 34 | 35.1 | 33.7 | 32.8 | |
| R | 9 | 8 | 6 | 5 | 7 | 4 | 3 | 8 | 4 | 2 | 1 |
| Sample number | 23 | 24 | |||||||||
| 33.5 | 34.2 | ||||||||||
| R | 3 | 2 |
(i). Sep up and R charts on this process. Does the process seem to be in statistical control? If necessary, revise the trial control limits.
(ii). If specifications on this diameter are 0.50300.0010, find the percentage of nonconforming bearings produced by this process. Assume that diameter is normally distributed.
b). In the semiconductor industry, the production of microcircuits involves many steps. The wafer fabrication process typically builds these microcircuits on silicon wafers and there are many microcircuits per wafer. Each production lot consists of between 16 and 48 wafers. Some processing steps treat each wafer separately, so that the batch size for that step is one wafer. It is usually necessary to estimate several components of variation: within-wafer, between-wafer, between-lot and the total variation. A critical dimension (measured in mm) is of interest to the process engineer. Suppose that five fixed position are used on each wafer (position 1 is the center) and that two consecutive wafers are selected of each batch. The data that results several batches are shown below.
(i) What can you say about over all process capability?
(ii) Can you construct control charts that allow within- wafer variability to be evaluated?
(iii) What control charts would you establish to evaluate variability between wafers? Set
up these charts and use them to draw conclusions about the process.
| Lot No. | Wafer No. | Position | |||||||
| 1 | 2 | 3 | 4 | 5 | |||||
| 1 | 1 | 2.15 | 2.13 | 2.08 | 2.12 | 2.10 | |||
| 2 | 2.13 | 2.10 | 2.04 | 2.08 | 2.05 | ||||
| 2 | 1 | 2.02 | 2.01 | 2.06 | 2.05 | 2.08 | |||
| 2 | 2.03 | 2.09 | 2.07 | 2.06 | 2.04 | ||||
| 3 | 1 | 2.13 | 2.12 | 2.10 | 2.11 | 2.08 | |||
| 2 | 2.03 | 2.08 | 2.03 | 2.09 | 2.07 | ||||
| 4 | 1 | 2.04 | 2.01 | 2.10 | 2.11 | 2.09 | |||
| 2 | 2.07 | 2.14 | 2.12 | 2.08 | 2.09 | ||||
| 5 | 1 | 2.16 | 2.17 | 2.13 | 2.18 | 2.10 | |||
| 2 | 2.17 | 2.13 | 2.10 | 2.09 | 2.13 | ||||
| 6 | 1 | 2.04 | 2.06 | 2.00 | 2.10 | 2.08 | |||
| 2 | 2.03 | 2.10 | 2.05 | 2.07 | 2.04 | ||||
| 7 | 1 | 2.04 | 2.02 | 2.01 | 2.00 | 2.05 | |||
| 2 | 2.06 | 2.04 | 2.03 | 2.08 | 2.10 | ||||
| 8 | 1 | 2.13 | 2.10 | 2.10 | 2.15 | 2.13 | |||
| 2 | 2.10 | 2.09 | 2.13 | 2.14 | 2.11 | ||||
| 9 | 1 | 2.00 | 2.03 | 2.08 | 2.07 | 2.08 | |||
| 2 | 2.01 | 2.03 | 2.06 | 2.05 | 2.04 | ||||
| 10 | 1 | 2.04 | 2.08 | 2.09 | 2.10 | 2.01 | |||
| 2 | 2.06 | 2.04 | 2.07 | 2.04 | 2.01 | ||||
(17 +17)
- a). Bath concentrations are measured hourly in a chemical process. Data (in PPM) for the
last 32 hours are shown below (read down from left).
| 160 | 186 | 190 | 206 |
| 158 | 195 | 189 | 210 |
| 150 | 179 | 185 | 216 |
| 151 | 184 | 182 | 212 |
| 153 | 175 | 181 | 211 |
| 154 | 192 | 180 | 202 |
| 158 | 186 | 183 | 205 |
| 162 | 197 | 186 | 197 |
The process target is =175 PPM.
(i). Estimate the process standard deviation.
(ii). Construct a tabular cusum for this process using standardized values of h = 5 and
k = .
b). A product is shipped in lots of size N = 2000. Find a Dodge-Romig single-sampling plan for which the LTPD = 1%, assuming that the process average is 0.25% defective. Draw the OC curve and ATI curve for this plan. What is the AOQL for this sampling plan? (20+14)
3) (a) Analyze the following 32 factorial design (24)
Replicate I Replicate II
| a0b0
20 |
a1b0
32 |
a0b2
40 |
| a1b1
60 |
a0b1
48 |
a2b0
55 |
| a2b1
60 |
a1b2
31 |
a2b2
51 |
| a1b1
42 |
a1b2
60 |
a0b1
40 |
| a2b0
25 |
a0b0
62 |
a1b0
45 |
| a2b2
61 |
a2b1
31 |
a0b2
42 |
(b) Construct BIBD using the following :
V = 7, b =7, r = 3, k = 3, λ=1 (10)
4) (a) Analyze the following 23 factorial experiment in blocks of 4 plots, involving three fertilizers N,
P and K each at two levels. (17)
Replicate I Replicate II
| Block 1 | np
88 |
npk
90 |
(1)
115 |
k
75 |
| Block 2 | p
101 |
n
111 |
pk
75 |
nk
55 |
| Block 3 | (1)
115 |
npk
95 |
nk
90 |
p
80 |
| Block 4 | np
125 |
k
95 |
pk
80 |
n
100 |
Replicate III
| Block 5 | pk
53 |
nk
76 |
(1)
65 |
np
82 |
| Block 6 | n
75 |
npk
100 |
P
55 |
k
92 |
(b) Use the Kuhn-Tucker conditions to solve the following Non-Linear Programming Problem:
Maximize z = 2x1 + x2 -x12
Subject to the constraints:
2x1+ 3x2 ≤ 6,
5x1+ 2x2 ≤ 10
x1, x2 ≥ 0 (17)
5) (a) Use Penalty method to solve the following L.P.P:
Minimize = 9x1 + 10x2
Subject to the constraints:
2x1 + 4x2 ≥ 50,
4x1 + 3x2 ≥ 24,
3x1 + 2x2 ≥ 60
x1, x2 ≥ 0 (17)
(b) Use Beale’s method to solve the following Q.P.P:
Minimize z = 6- 6x1 + 2x12 – 2x1x2 + 2x22
Subject to x1 + x2 ≤ 2
x1, x2 ≥ 0 (17)
Loyola College M.Sc. Statistics April 2009 Statistical Computing – I Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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FIRST SEMESTER – April 2009
ST 1812 – STATISTICAL COMPUTING – I
Date & Time: 04/05/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
Answer ALL the questions. Each carries THIRTY FOUR marks
- a). In a population containing 490 live birds of same weight and age, the birds were divided into 70 equal groups. They were then given a stimulus to increase the growth rate. The following data gives the frequency distribution of birds with significant weights at the end of 6 weeks. Fit a truncated binomial distribution and test the goodness of fit.
| No of birds | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Frequency | 6 | 15 | 21 | 17 | 8 | 2 | 1 |
b). The following are the marks of 135 students of B.Com in a city college.
| Marks | No. of Students |
| 0-10 | 5 |
| 10-20 | 15 |
| 20-30 | 30 |
| 30-40 | 40 |
| 40-50 | 25 |
| 50-60 | 10 |
| 60-70 | 6 |
| 70-80 | 4 |
Fit a normal distribution by ordinate method and test the goodness of fit at 5% level of significance.
(OR)
c). Fit a truncated Poisson distribution to the following data and test the goodness of fit at 1% level of significance.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| f | 86 | 52 | 26 | 8 | 6 | 1 |
d). Fit a distribution of the form, where
, x=0,1,2,…, > 0 and
; x=1,2,…, 0<p<1,
to the following data:
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| f | 72 | 113 | 118 | 58 | 28 | 12 | 4 | 2 | 2 |
- a). Find the inverse of the following symmetric matrix using partition method.
b). Obtain the characteristic roots and vectors for the matrix
(OR)
c). Find for the following matrices:
i).
ii).
d). Draw a random sample of size 10 from the exponential distribution having the density function
.
Also find the mean and variance of the sample observations.
- a) The following data were collected on a simple random sample of 10 patients with hypertension.
| Serial No. | mean arterial blood pressure (mm/Hg) | weight (kg) | heart beats / min |
| 1 | 105 | 85 | 63 |
| 2 | 115 | 94 | 70 |
| 3 | 116 | 95 | 72 |
| 4 | 117 | 94 | 73 |
| 5 | 112 | 89 | 72 |
| 6 | 121 | 99 | 71 |
| 7 | 121 | 99 | 69 |
| 8 | 110 | 90 | 66 |
| 9 | 110 | 89 | 69 |
| 10 | 114 | 92 | 64 |
i). Fit a regression model and estimate effect of all variables / unit of measurement, taking blood pressure as the dependent variable.
ii). Find R2 and comments on it
(OR)
- b) The following table explains a company monthly income based on their advertisement
on V-Slicer product.
| Serial No. | Monthly Income on sales ($’000) | Advertisement on TV ($ ‘000) | Advertisement on News Paper ($ ‘000) |
| 1 | 5.5 | 0.2 | 0.1 |
| 2 | 6.7 | 0.5 | 0.2 |
| 3 | 8.0 | 1.2 | 0.8 |
| 4 | 10.1 | 2.0 | 0.9 |
| 5 | 15 | 3.0 | 1.4 |
| 6 | 18.0 | 4.0 | 2.0 |
| 7 | 23 | 5.0 | 2.5 |
| 8 | 28 | 6.2 | 3.8 |
| 9 | 32 | 8.0 | 4.1 |
| 10 | 35 | 10.0 | 5.2 |
- Draw a scatter diagram for the above data.
- Fit a regression mode taking TV advertisement and News paper advertisement as independent variables and estimate monthly income when TV advertisement is 15 and News paper advertisement is 7 in 1000 dollars.
Loyola College M.Sc. Statistics April 2009 Multivariate Analysis Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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THIRD SEMESTER – April 2009
ST 3808 – MULTIVARIATE ANALYSIS
Date & Time: 16/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
PART – A
Answer all the questions. (10 X 2 = 20)
- Give an example in the bivariate situation that the marginal distributions are normal but the bivariate distribution is not.
- Let X, Y and Z have trivariate normal distribution with null mean vector and covariance matrix
2 3 4
3 2 -1
4 -1 1 ,
find the distribution of Y+X.
- Mention any two properties of multivariate normal distribution.
- Explain the use of partial and multiple correlation coefficients.
- Define Hotelling’s T2 – statistics. How is it related to Mahlanobis’ D2?
- Outline the use of discriminant analysis.
- What are canonical correlation coefficients and canonical variables?
- Write down any four similarity measures used in cluster analysis.
- Write the c.f. of X where
X~N2 { , }.
10.Write a short note on data mining.
PART B
Answer any FIVE questions. (5 X 8 = 40)
- Obtain the maximum likelihood estimator S of p-variate normal distribution with
mean vector known.
- Let X1, X2,…, X n be independent N( 0 , 1 ) random variables. Show that X’ A X
is chi-square if A is idempotent, where X= ( X1,X2,…,X n )’.
- 13. How will you test the equality of covariance matrices of two multivariate normal
distributions on the basis of independent samples drawn from two populations?.
- Let (Xi, Yi)’ , i = 1, 2, 3 be independently distributed each according to bivariate
normal with mean vector and covariance matrix as given below. Find the joint
distribution of six variables. Also find the joint distribution of and .
Mean vector: (m, t)’, covariance matrix:
- Outline single linkage and complete linkage clustering procedures with an
example.
- Giving suitable examples explain how factor scores are used in data analysis.
- Consider a multivariate normal distribution of X with
m = , S = .
Find i ) the conditional distribution of ( X1, X3 ) / ( X2, X4 )
- ii) s42
- a) Define i ) Common factor ii) Communality iii) Total variation
b)Explain classification problem into two classes and testing problem.
PART C
Answer any two questions. (2 X 20 = 40)
- a) Derive the distribution function of the generalized T2 – statistic.
- b) Test at level 0.05 ,whether µ = ( 0 0 )’ in a bivariate normal population with
σ11 = σ22= 10 and σ12= -4 , by using the sample mean vector= (7 -3)‘ based
on a sample size 20. (15 + 5)
- a) What are principal components?. Outline the procedure to extract principal
components from a given covariance matrix.
- b) Define partial correlation between Xi and Xj .Also prove that
______ ______
r12.3= ( r12-r13r23)/ {Ö(1-r223) Ö(1-r213)}. ( 12+8)
21.a) Consider the two data sets
X1= and X2 =
for which .
1) Calculate the linear discriminant function.
2) Classify the observation x0‘= ( 2 7 ) as population π1 or population π2 using
the decision rule with equal priors and equal costs.
- b) Explain how the collinearity problem can be solved in the multiple regression.
( 14+6)
22.a) Explain the method of extracting canonical correlations and their variables
from a dispersion matrix.
- b) Prove that under some assumptions (to be stated), variance and covariance can
be written as S = LL’ + y in the factor analysis model. Also discuss the effect
of an orthogonal transformation. (8 + 12)
Loyola College M.Sc. Statistics April 2009 Fuzzy Theory And Applications Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
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M.Sc. DEGREE EXAMINATION – STATISTICS
THIRD SEMESTER – April 2009
ST 3875 – FUZZY THEORY AND APPLICATIONS
Date & Time: 27/04/2009 / 9:00 – 12:00 Dept. No. Max. : 100 Marks
| SECTION – A |
Answer ALL the Questions (10 x 2 = 20 marks)
- Write the axiomatic skeleton of fuzzy t-conorm.
- Define Archimedean t-norm.
- Define Drastic fuzzy intersection.
- Write a short note on fuzzy number.
- If , find
- Define a fuzzy variable and give an example.
- Define scalar cardinality of a fuzzy set.
- State the role of a ‘knowledge engineer’ in constructing fuzzy sets.
- Distinguish between direct methods and indirect methods of constructing membership functions.
- Define an artificial neural network.
| SECTION – B |
Answer any FIVE Questions (5 x 8 = 40 marks)
- Define & and prove that ,.
- Prove that , where and denote drastic and yager class of t-norm.
- Let the triple be a dual generated by an increasing generator. Prove that fuzzy operations satisfy the law of excluded middle and the law of contradiction. Also prove that does not satisfy distributive law.
- Let and B. Find the 4 basic operations for the fuzzy numbers A and B.
- Prove under usual notations: (i) α(Ac) = ( (1 – α) +A)c (ii) = α+ ()
- State the axiomatic skeleton and desirable requirements for fuzzy complements. Prove that if the monotonic and involutive axioms are satisfied, then the boundary and continuity conditions are satisfied.
- Let X ={x1, x2 ,x3} be a universal set and suppose two experts E1 and E2 have specified the valuations of these three as elements of two fuzzy sets A and B as
given in the following table:
Membership in A Membership in B
| Element | E1 | E2 |
| x1 | 0.6 | 0.5 |
| x2 | 0.2 | 0.3 |
| x3 | 0.8 | 0.6 |
| Element | E1 | E2 |
| x1 | 0.2 | 0.4 |
| x2 | 0.9 | 0.7 |
| x3 | 0.6 | 0.3 |
Assuming that for set A, the experts have to be given weights as c1 = 0.7 and c2 = 0.3 and that for set B, the weights are c1 = 0.2, c2 = 0.8, find the degree of membership of the three elements in A and in B. Also, find the degree of membership in AUB by bounded sum operator.
- State the three different classes of network architectures and briefly describe any one of them with a diagram.
| SECTION -C |
Answer any TWO Questions (2 x 20 = 40 marks)
- (a) Let i be a t-norm and strictly increasing and continuous function in (0,1) such that g(0)=0, g(1)=1. Prove that the function , where denotes pseudo inverse of g is a t-norm.
(b) Prove that the triples and are dual with
respect to any fuzzy complement. (15+5)
- Let MIN and MAX be binary operations on the set of all fuzzy numbers. Prove that for any fuzzy numbers A, B, C the following properties hold:
(a) MIN(A,MIN(B,C))=MIN(MIN(A,B),C)
(b) MAX(A,MAX(B,C))=MAX(MAX(A,B),C)
(c) MIN(A,MAX(A,B))=A
(d) MAX(A,MIN(A,B))=A
(e) MIN(A,MAX(B,C))=MAX(MIN(A,B), MIN(A,C))
(f) MAX(A,MIN(B,C))=MIN(MAX(A,B), MAX(A,C))
- (a)Explain the indirect method of constructing a membership function with one expert.
(b) State the role of ‘activation function’ in neural networks. Describe the three basic types of activation functions. (10 + 10)
- (a) Briefly explain the three practical issues in ‘Pattern Recognition’.
(b) State the problem of ‘Fuzzy Clustering’ and present the Fuzzy c-means
algorithm. (6 + 14)
Loyola College M.Sc. Statistics April 2009 Reliability Theory Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
|
SECOND SEMESTER – April 2009
ST 2957 / ST 2955 – RELIABILITY THEORY
Date & Time: 27/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION –A (10 x 2 = 20)
Answer any TEN questions. Each question carries TWO marks
- Define the following: (i) Mean time before failure(MTBF)
(ii) Steady state availabilty
- If the hazard function r(t)=3t2, t>0, obtain the corresponding probability distribution of
time to failure
- Obtain the reliability of a parallel system consisting of n components, when the
reliability of each component is known. Assume that the components are non-repairable.
- Explain in detail an n unit standby system.
- What is meant by reliability allocation?
- Define a coherent structure and give two examples.
- If x is a path vector and y ≥ x, show that y is also a path vector.
- Write down the structure function for a two out of three system.
- Let h ( p ) be the system reliability of a coherent structure. Show that h(p)is strictly
increasing in each pi, whenever 0<pi<1, i=1,2,3,…,n .
- Give an example of a distribution, which is IFR as well as DFR.
SECTION-B (5×8 =40 marks)
Answer any FIVE questions. Each question carries EIGHT marks
- Obtain the reliability function, hazard rate and the system MTBF for the following
failure time density function
f(t) = 12 exp(-4 t3)t2, t>0.
- What is a series system? Obtain the system failure time density function for a series system with
n independent components. Suppose each of the n independent components has an exponential
failure time distribution with the parameter λi, i= 1,2,…,n. Find the system reliability.
- Find the system MTBF for a (k,n) system, when the lifetime distribution is
exponential with the parameter λ. Assume that the components are non-repairable.
- Assuming that the components are non-repairable and the components have identical
constant failure rate λ, obtain the MTBF of the series-parallel system.
- Let Φ be a coherent structure. Show that
Φ(x .y ) ≤ Φ(x ) Φ(y )
Show that the equality holds for all x and y if and only if the structure is series.
- Let h be the reliability function of a coherent system. Show that
h( p Ц p‘) ≥ h( p ) Ц h( p‘) for all 0 ≤ p , p‘ ≤ 1.
Also, show that the equality holds if and only if the system is parallel.
- If two sets of associated random variables are independent, show that their union is
the set of associated random variables.
- Show that Wiebull distribution is a DFR distribution.
SECTION-C (2X20=40 marks)
Answer any two questions. Each question carries TWENTY marks
- a) Obtain the reliability function, hazard rate and the system MTBF for exponential
failure time distribution with the parameter λ. (8 marks)
- b) Obtain the system failure time density function for a (m, n) system. Assume that
the components are non-repairable. (12 marks)
20.a) Define the terms: (i) Hazard rate and (ii) Interval reliability (4 marks)
- b) For a simple 1 out of 2 system with constant failure rate λ and constant repair rate
μ, obtain the system of differential-difference equations. Also, obtain
the system reliability and system MTBF. (16 marks)
21.a) Define: (i) Dual of a structure (ii) Minimal path vector and (iii) Minimal cut vector
(6 marks)
- b) Let h be the reliability function of a coherent system. Show that
h( p . p‘) ≤ h( p ) . h( p‘) for all 0 ≤ p , p‘ ≤ 1. (10 marks)
Also show that the equality holds if and only if the system is series.
- c) If X1, X2, …, Xn are associated binary random variables, show that
(1-X1), (1-X2),…,(1-Xn) are also associated binary random variables.(4 marks)
22.a) If the probability density function of F exists, show that F is an IFR
distribution iff r(t)↑t. (10 marks)
- b) Examine whether Gamma distribution G(λ, α) is IFR or DFR. (10 marks)
Loyola College M.Sc. Statistics April 2009 Probability Theory Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
|
M.Sc. DEGREE EXAMINATION – STATISTICS
SECOND SEMESTER – April 2009
S 815 – PROBABILITY THEORY
Date & Time: 04/05/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION-A (10 × 2=20)
Answer ALL the questions.
- With reference to tossing a regular coin once and noting the outcome, identify
completely all the elements of the probability space (Ω, A, P).
∞
- If P(An ) =1, n=1,2,3,… , evaluate P( ∩ An ) .
n=1
- Show that the limit of any convergent sequence of events is an event.
- Define a random variable and its probability distribution.
- Calculate E(X), if X has a distribution function F(x), where
F(x) = 0 if x<0
x/2 if o≤ x<1
- if x ≥ 1.
- If X1 and X2 are independent random variables and g1 and g2 are Borel functions, show
that g1(X1) and g2(X2 )are independent.
- State Glivenko-Cantelli theorem.
- Φ is the characteristic function (CF) of a random variable X, find the CF of (2X+3).
- State Kolmogorov’s strong law of large numbers(SLLN).
- State Lindeberg-Feller central limit theorem.
SECTION-B (5 × 8 = 40)
Answer any FIVE questions.
- Define the distribution function of a random variable X. State and establish its
defining properties.
- Explain the independence of two random variables X and Y. Is it true that if X and
Y are independent, X2 and Y2 are independent? What about the converse?
- State and prove Borel zero –one law.
- State and prove Kolmogorov zero-one law for a sequence of independent random
variables.
- Define convergence in probability. Show that convergence in probability implies
convergence in distribution.
- a) Define “Convergence in quadratic mean” for a sequence of random variables.
- b) X is a random variable, which takes on positive integer values. Define
Xn = n+1 if X=n
n if X=(n+1)
X otherwise
Show that Xn converges to X in quadratic mean. (2+6)
- Establish the following:
(a) If Xn → X with probability one, show that Xn → X in probability.
(b) Show that Xn → X almost surely if and only if for every є >0,
P [lim sup │ Xn – X│> є ]= 0
- Let { Xn ,n ≥1} be a sequence of independent random variables such that Xn has
uniform distribution on (-n, n). Examine whether the central limit theorem holds for
the sequence { Xn, n≥1}.
SECTION-C (2 x 20 = 40 marks)
Answer any TWO questions
19.a) Show that the probability distribution of a random variable is determined by its
distribution function.
- b) Show that the vector X =(X1, X2,…, Xp ) is a random vector if and only if Xj,
j =1, 2, 3… p is a real random variable.
- c) If X is a random variable with continuous distribution function F, obtain the
probability distribution of F(X). (6+8+6)
20.a) State and prove Kolmogorov’s inequality. (10 marks)
- b) State and prove Kolmogorov three series theorem for almost sure convergence of
the series Σ Xn of independent random variables. (10 marks)
21.a) If Xn and Yn are independent for each n, if Xn → X, Yn → Y, both in
distribution, prove that (Xn2 + Yn2) → (X2+Y2) in distribution. (10 marks)
- b) Let { Xn } be a sequence of independent random variables with common frequency
function f(x) = 1/x2 , x=1,2,3,… Show that Xn /n does not converge to zero with
probability one. (10 marks)
22.a) State and prove Levy continuity theorem for a sequence of characteristic
functions.(12 marks).
b)Let {Xn} be a sequence of normal variables with E (Xn) = 2 + (1/n) and
var(Xn) = 2 + (1/n2), n=1, 2, 3…Examine whether the sequence converges in
distribution.(8 marks).