LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 05

SECOND SEMESTER – April 2009

ST 2502 / 2501 – STATISTICAL MATHEMATICS – I

 

 

 

Date & Time: 23/04/2009 / 1:00 – 4:00 Dept. No.                                                   Max. : 100 Marks

 

 

SECTION A

 

Answer ALL questions.                                                                 (10 x 2 =20 marks)

 

1. Define a function.
2. What is a monotonic sequence?
3. State the comparison test for convergence of a series.
4. Define probability generating function of a random variable.
5. How is the variance of a random variable obtained from its moment generating function?
6. Define the derivative of a function at a point.
7. What do you mean by probability distribution function of a random variable?
8. Does the series      (x ≥ 1) converge?
9. Define rank of a matrix.
10. Define symmetric matrix and give an example.

 

SECTION B

 

Answer any FIVE questions.                                                         (5 x 8 =40 marks)

 

1. If      and  , then prove the following:

 

1. Prove that the sequence {a_n} defined by a_n =     is convergent.

 

1. Consider the experiment of tossing a biased coin with P (H) = ⅓, P (T) = ⅔ until a head appears. Let X = number of tails preceding the first head. Find moment generating function of X.

 

1. Show that if a function is derivable at a point, then it is continuous at that point.

 

1. Verify Lagrange’s mean value theorem for the following function:

f(x) = x² -3 x + 2 in [-2, 3]

 

1. When is a set of n vectors said to be linearly independent? Find whether the vectors (1, 0, 0), (4,1,2) and (2, -1, 4) are linearly independent or not.

 

 

 

1. A random variable has the following probability distribution:

X	0	1	2	3	4	5	6	7	8
p(x)	k	3k	5k	7k	9k	11k	13k	15k	17k

 

 

 

(i) Determine the value of k,  (ii) Find the distribution function of X.

 

1. Find the inverse of  A =

 

 

SECTION C

 

Answer any TWO questions.                                                (2 x 20 =40 marks)

 

1.  (i) Prove that a non decreasing sequence of real numbers which is bounded above is convergent.

(ii) Discuss the bounded ness of the sequence {a_n}where a_n is given by,   a_n =

 

1. State D’Alembert’s ratio test and hence discuss the convergence of the following series:

 

1. (i) State and prove Rolle’s Theorem.

(ii) Verify Rolle’s theorem for the following function: f (x) = x² – 6 x – 8 in [2, 4].

 

1. (i) A two-dimensional random variable has a bivariate distribution given by,

     P(X=x, Y=y) =     , for x = 0,1,2,3 and y = 0, 1. Find the marginal distributions

of X and Y.

(ii) If P(X=x, Y=y) =     , where x and y can assume only the integer values  0,

      1 and 2. Find the conditional distribution of Y given X = x.

 

 

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