LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2011

ST 2502/ST 2501 – STATISTICAL MATHEMATICS – I

 

 

 

Date : 08-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

 

Answer ALL questions                                                                                             (10×2 =20 Marks)

 

1. Define least upper bound of a set.
2. Define convergent sequence.
3. Define cumulative distribution function and state any two of its properties.
4. Give an example for a monotonic sequence.
5. Define absolute convergence and conditional convergence for a series of real numbers.
6. Define M.G. F of a random variable.
7. State Roll’s theorem.
8. Define Taylor’s expansion of a function about x = a.
9. Define rank of a matrix.
10. Define symmetric matrix. Give an example.

 

PART – B

 

Answer any FIVE questions                                                                                                        (5×8=40 Marks)

 

1. Show that every convergent sequence is bounded. Is the converse true? Justify your answer.
2. Obtain the c.d.f. of the total number of heads occurring in three tosses of a fair coin.
3. Establish the convergence of (a)  ; (b) .
4. Show that if a function is derivable at a point, then it is continuous at that point.
5.  If two random variables X and Y have the joint probability density       

       

Find the marginal densities.

 

1. Find the Lagrange’s and Cauchy’s remainder after n^th term in the Taylor’s series expansion of log_e(1+ x).

 

1.  Verify whether or not the following sets of vectors form linearly independent sets:

       (a) (1, 2, 3), (2, 2, 0)

 

       (b) (3, 1, -4), (2, 2, -3)

1. Find the inverse of a matrix  .

 

PART – C

 

Answer any TWO questions                                                                                          (2×20=40 Marks)

 

1. (a) Prove that a non-increasing sequence of real numbers which is bounded below is convergent.

 

(b)Prove that the sequence  given by  is convergent.

 

1. (a) State and Prove Rolle’s Theorem

 

(b) Find a suitable c of Rolle’s Theorem for the function   

 

                 .

 

1. A random variable X has the following probability function

 

x	1	2	3	4	5	6	7
P(x)	0	k	2k	2k	k2	2k2	7k2+k

 

• Find k
• Evaluate (a) (b)  (c)  
• If , find the minimum value of k 
• Determine the distribution function of X.

 

1.  (a) If  

            is the joint p.d.f. of X and Y, find the marginal p.d.f.’s. Also, evaluate                 

            P[ (X < 1)  (Y < 3) ]

(b) Find the rank of the matrix .

 

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