LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 04

SECOND SEMESTER – APRIL 2007

ST 2500 – STATISTICAL MATHEMATICS – I

 

 

 

Date & Time: 20/04/2007 / 1:00 – 4:00 Dept. No.                                                Max. : 100 Marks

 

 

SECTION A

Answer ALL questions. Each carries 2  marks                                         [10×2=20]

1. Define a bounded function and give an example.
2. Define a monotonic sequence and give an example.
3. What is ‘permutation of indistinguishable objects’? State its value in factorial notation.
4. Define the limit of a sequence and give an example.
5. If A and B are independent events, show that A and B^c are independent.
6. What are the supremum and infinum of the function f(x) = x – [x], x €R ?
7. Find n_oÎN, such that |(n/n+2) -1| < 1/3 .
8. Define probability mass function (p.m.f) of a discrete random variable and state its properties.
9. Define Moment Generating Function and state its uses.
10. Is the following a P.G.F:  F(s) = s/ (3-s) ?

SECTION B

Answer any FIVE   questions.  Each carries 8  marks                          [5×8=40]

1. State and prove the Addition Theorem of Probability for two events. Extend it for three events.
2. Discuss the convergence of the series
3. A and B play a game in which their chances of winning are in the ratio 3 :2    Find A’s chance of winning at least 2 games out of 5 games.
4. Find the sum of the series .
5. State and prove Baye’s theorem.
6. Investigate the extreme values of the function f(x) = 2x⁵ -10X⁴ +10x³ + 8
7. Identify the  distribution for which  is the P.G.F
8. Let X be a random variable with the probability mass function

x:    1           2              3

P(x): 1/2        1/4          1/4         Find P.G.F and hence Mean and Variance

 

 

 

 

 

 

SECTION C

Answer any TWO  questions.  Each carries 20 marks                         [2×20=40]

19 [a] Let S_n = 1 + , n=1,2,3,…prove that Lim S_n exists  and lies between

2 and 3.                                                                                                           [12]

[b] Let S_n =  and   t_n =    , verify that the limit of the difference   of

           two convergent sequences is the difference  of their limits.                              [8]

20 Discuss the convergence of the Geometric series

for  possible variations in x                                                                             [12]

[b] Let X be a random variable with the following distribution:

x  :    -3      6      9

P(x):    1/6    ½    1/3

Find [i] E(X)     [ii] E(X²)        [iii] E(2X+1)²              [iv] V(X)                     [8]

 

 

21 [a] What is the expectation of the number of failures preceding  the first success

in an infinite series of independent trials with constant probability ‘p’ of

success in each trial.

 

[b] Let X be a random variable with the probability mass function

x:   0    1          2          3          ….

P(x): ½   (½)²   (½)³     (½)⁴           …

Find the  M.G.F and hence Mean and Variance

22 [a] Let X be a random variable which denotes the product  of the  numbers on

the upturned faces, when two dice are rolled:

[i] Construct the probability distribution

[ii] Find the probability for the following :

1) the product of the faces is more than 30

(2) the product of the faces is maximum 10

(3) the product of the faces is at least 9 and  at most 24

[iii] Find E(X)

[b]Define Poisson distribution. Find the moment generating function (m.g.f.).Hence find the

mean and the variance.

 

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