LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – STATISTICSSECOND SEMESTER – APRIL 2008ST 2500 – STATISTICAL MATHEMATICS – I
Date : 23/04/2008 Dept. No. Max. : 100 Marks Time : 1:00 – 4:00 SECTION – A
Answer ALL the Questions (10 x 2 = 20 marks)
1. Define an r-permutation of n objects.2. Define mutually exclusive events.3. An urn contains 6 white, 4 red and 9 black balls. If 2 balls are drawn at random find the probability that both are red. 4. Define probability mass function (p.m.f.) of a random variable.5. State any two properties of a distribution function.6. Write down the p.m.f. of a Poisson distribution.7. Define an oscillating sequence and give an example.8. Define a monotonic sequence with an example.9. Define ‘Sequence of Partial Sums’ for a series.10. State Cauchy’s Root test.
SECTION – BAnswer any FIVE Questions (5 x 8 = 40 marks)
11. State the ‘Addition Theorem of Probability’. Applying the theorem, find the probability that a number chosen at random from 1 to 40 is a multiple of 2 or 5.12. Two digits are chosen one by one at random without replacement from {1, 2, 3, 4, 5}. Find the probability that (i) an odd digit is selected in the first draw; (ii) an odd digit is selected in the second draw, (iii) odd digits are selected in both draws.13. Show that the sequence an = converges and that its limit lies between 2 and 3.14. If sn = L and tn = M, show that (sn+ tn) = L + M and s¬n.tn = L. M15. Two fair dice are rolled. Let X = Sum of the two numbers that show up. Obtain the distribution function of X.16. State the D’Alembert’s Ratio Test. Applying it, test for convergence of the series .17. State the ‘Limit form of Comparison test’. Applying it, test whether the series + + + + ………. converges or diverges.18. Define a Binomial distribution. If 10 fair coins are thrown, find the probability of getting (i) Exactly 5 heads, (ii) At least 7 heads.
(P.T.O)
SECTION – CAnswer any TWO Questions (2 x 20 = 40 marks)
19. (a) State and Prove Baye’s Theorem. (b) A factory produces a certain type of outputs by three machines. The daily production figures by the three machines are 3000 units, 2500 units and 4500 units. The percentages of defectives produced by the three machines are 1 %, 1.2 % and 2 %. An item is drawn at random from a day’s production and is found to be defective. What is the probability that the defective came from (i) machine 1, (ii) Machine 2, (iii) Machine 3? (10 +10)
20. (a) Show that the sequence sn = + + ……. + is convergent. State the result that you use here.(b) A random variable X has the following probability distribution: x : –3 6 9 p(x): 1/6 1/2 1/3 Find E(X), Var(X) and E(2 X+1) (10 +10) 21. Discuss the convergence of the following series: (a) (b) (10 +10)
22. (a) Give the logarithmic series and show that it is convergent for |x | < 1. (b) Discuss the convergence of the geometric series . (10+10)
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