LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3500/STA502 – STATISTICAL MATHEMATICS – II

04.11.2003                                                                                                           Max:100 marks

9.00 – 12.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

1. If P^* is a partition of [a , b] finer than the partition P, state the inequality governing the upper sums lower sums of a function f corresponding to P and P^*.
2. Find .
3. State the first Fundamental Theorem of Integral calculus.
4. Solve: .
5. “The function f(x,y) =   xy/(x²+y²) ,    (x,y)  ¹(0,0)

 

0                ,   (x, y) = (0, 0)

does not have double limit as (x, y)   – verify.

6. State the rule for the partial derivative of a composite function of two variables.
7. Define Gamma distribution.
8. Write down the Beta integral with integrand involving Sine and Cosine functions.
9. Define a symmetric matrix.
10. Find the rank of the matrix .

 

SECTION-B

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

 

11. Evaluate (a) . (4+4)

(b)

12. If f(x) = kx² , 0 < x¸< 2 , is the probability density function (p.d.f) of X, find (i) k

(ii) P[X<1/4],  (iii) P,  (iv) P[X >1].

13. Solve the non-homogeneous differential equation:

(y – x – 3) dy = (2x + y +6) dx

14. For the function          xy(x² – y²) / (x² + y²)  ,    (x,y)  ¹(0,0)

f(x,y) =

0                                ,      (x, y) = (0, 0)

Show that f_x (x, 0) = f_y (0,y) = 0 , f_x (0, y) = -y , f_y (x, 0) = x.

15. Find the mean and variance of Beta distribution of II kind stating the conditions for their existence.
16. If f(x,y) = e^-x-y , x > 0, y > 0, is the joint p.d.f of (x, y),  find the joint c.d.f. of (x, y).  Verify that the second order mixed derivative of the joint c.d.f.is indeed the joint p.d.f.
17. Establish the reversal law for Transpose of product of matrices. Show that the operations of Inversion and Transpositions are commutative.
18. Find the inverse of using Cayley – Hamilton Theorem.

 

SECTION-C

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

 

19. a) Show that, if fÎ R [a, b] then f² Î R [ a, b].
20. b) If f(x) = c.e^-x, x > 0, is the p.d.f. of X, find (i) c (ii) E(X), (iii) Var (X).
21. c) Discuss the convergence of (8+6+6)
22. a) Investigate for extreme values of the function

f (x, y) = x³ + y³ – 12x – 3y + 5, x, y Î R.

1. b) Define joint distribution function for bivariate case and state its properties. Establish

the property which gives the probability P[x₁ < X £ x_2,  y₁ < Y £ y₂] in terms of the

joint distribution function of (X, Y).                                                               (10+10)

21. If x + y ,   0 < x, y < 1

f (x, y) =

0        ,   otherwise

is the joint p.d.f of (x, y),  find the means and variances  of X and Y and covariance

between X and Y.  Also find  P [ Y < X] and the marginal p.d.f’s of X and Y.

22. a) By partitioning into 2 x 2 submatrices find the inverse of
23. b) Find the characteristic roots and any characteristic vectors for the matrix

 

(10 + 10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2004

ST 3500/STA 502 – STATISTICAL MATHEMATICS – II

21.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION -A

 

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

 

1. Define a Skew-Symmetric matrix and give an example.
2. Define an Orthogonal matrix. What can you say about its determinant?
3. Find the rank of .
4. State a necessary and sufficient condition for R-integrability of a function.
5. Is convergent?
6. If f(x) = C x², 0 < x < 1, is a probability density function (p.d.f), find ‘C’.
7. Give an example of a homogeneous differential equation of first order.
8. Distinguish between ‘double’ and ‘repeated’ limits.
9. State any two properties of a Bivariate distribution function.
10. State the rule of differentiation of a composite function of two variables.

 

SECTION -B

 

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

 

11. Define ‘upper triangular matrix’. Show that the product of two upper triangular matrices is an upper triangular matrix.
12. Find the inverse of A = using Cayley- Hamilton theorem.
13. Find a)     b)
14. State and prove first Fundamental Theorem of Integral Calculus.
15. If X has p.d.f f(x) = x²/18, -3 £ x £ 3,  find the c.d.f of X.  Also, find P(< 1),

P (X < -2)

16. Solve: .

 

 

 

17. Show that the mixed derivative of the following function at the origin are different:

 

 

f (x, y) =

 

 

18. Define Gamma integral and Gamma distribution.

find the mean and variance of the distribution.

 

SECTION – C

 

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

 

19. a) Find the inverse of using sweep-out process or partitioning

method.

1. b) Find the characteristic roots and any characteristic vector associated with them for the

matrix.

(10+10)

20. a) Test the convergence of: (i) (ii)   (iii) .
21. b) Define Lower and Upper sum in the context of Riemann integration. Show that lower

sums increase as partitions become finer.                                                           (12+8)

21. a) Investigate the maximum and minimum of

f(x,y) = 21x – 12x² – 2y² + x³ + xy²

1. b) If f(x,y) = e^-x-y, x,y > 0, is the p.d.f of (X, Y),  find the distribution function.     (12+8)
2. a) Change the order of integration and evaluate: .
3. b) Define Beta distributions of I and II kinds.

Find the mean and variance of Beta distribution of I kind                                  (10+10)

 

 

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                        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AB 08

THIRD SEMESTER – NOV 2006

ST 3500 – STATISTICAL MATHEMATICS – II

(Also equivalent to STA 502)

 

 

Date & Time : 02-11-2006/9.00-12.00     Dept. No.                                                   Max. : 100 Marks

 

 

 

SECTION A

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

 

1. Define Hermitian matrix and give an example of 3×3 Hermitian matrix
2. Write the formula for finding the determinant by using partioning of matrices
3. Define Riemann integral
4. Evaluate
5. Define repeated limits and give an example
6. Define improper integral of I^st   kind
7. What are the order and degree of the differential equation ?
8. Define continuity of functions of two variables
9. Evaluate
10. If X, Y are random variables with joint distribution  function F(x,y), express          Pr[k₁< X ≤ k_{2 ,}   m₁ < Y ≤ m₂] in terms of

 

 

SECTION B

Answer any FIVE questions                                                                    (5×8 =40)

 

11. State and prove the first fundamental theorem of integral calculus
12. Find the inverse of the matrix A =by using 2×2 partitioning
13. Discuss the convergence of following improper integrals:

[a]     [b]

14. Define Gamma distribution and hence derive its mean and variance
15. Solve the differential eqation

 

 

 

 

16. Investigate the existence of the repeated limits and double limit at the origin of the

function  f(x,y) =

17. Investigate for extreme values of f(x,y) = (y-x)⁴ + (x-2)², x, y Î R.

 

18 . Define Beta distribution of 2^nd  kind.  Find its mean and variance by stating the

conditions for their existence.

SECTION C

Answer any TWO questions                                                             (2 x 20 =40)        

19. [a] Find the rank of the matrix A=

[b] Find the characteristic roots of the following matrix. Also find the inverse of A

using  Cayley-Hamilton theorem, where

A=

20. [a] Test if    converges absolutely

[b] Compute mean, mode and variance for the following p.d.f

 

21. a] Find maximum or minimum of f(x,y) = (x-y)² + 2x – 4xy, x, y Î

b] Show that the mixed derivatives of the following function at the origin are

different:

22.a] Let f(x,y) =    be  the joint p.d.f of (x,y).

Find the co-efficient of correlation between X and Y.

[b]  Change the order of integration and evaluate

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

AC 06

DEGREE EXAMINATION –STATISTICS

THIRD SEMESTER – APRIL 2007

ST 3500 – STATISTICAL MATHEMATICS – II

 

 

Date & Time: 21/04/2007 / 1:00 – 4:00            Dept. No.                                                     Max. : 100 Marks

 

 

SECTION A

 

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

 

1. Define Skew-Hermitian matrix and give an example of 3×3 skew Hermitian

matrix.

2. State Cayley-Hamilton theorem.
3. Evaluate the primitive integral : .
4. Define order and degree of differential equations and give an example
5. What is meant by double limit? Give an example
6. Define improper integral of second kind
7. Define Integrability and Integral of a function
8. Solve (2- 4x²)dy = (6x-xy) dx
9. If f(x) =     is a p.d.f. , find the value of K
10. Evaluate

 

SECTION B

Answer any FIVE questions                                                                    (5×8 =40)

11. Find the inverse of the matrix A =by using 2×2 partitioning
12. State and prove a necessary and sufficient condition for integrability of a

function

13. Evaluate :
14. Define Beta distribution of 1^st kind and hence find its mean and variance by stating the conditions for their existence.

 

15. Show that double limit at the origin may not exist but repeated limits exist for

the following function :

f(x,y) =

 

 

16. Investigate the extreme values of f(x,y) = (y-x)⁴ + (x-2)², x, y Î R.

 

17. Prove that

 

 

 

18. Compute mean and variance for the following p.d.f

 

SECTION C

Answer any TWO questions                                                             (2 x 20 =40)        

19.a] Find the rank of the matrix A=

 

[b] Find the characteristic roots of the following matrix. Also find the

inverse using  Cayley-Hamilton theorem:

A=

20. a] Test if    converges absolutely

b] Solve the differential equation

21. a] If f(x,y) =

is the joint p.d.f of (X,Y), find the joint d.f. F(x,y).

[b] Let  , x, y Î R. Find  f_xx , f_yy , f_xy f_yx

22. [a] Let f(x,y) = be the joint p.d.f of (x,y). Find the

co-efficient of  correlation    between X and Y.

[b]  Change the order integration and evaluate

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

NO 12

THIRD SEMESTER – APRIL 2008

ST 3501 / 3500 – STATISTICAL MATHEMATICS – II

 

 

 

Date : 26/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

 

1. Define Upper and Lower Sums of a function corresponding to a given partition of a closed interval.
2. State the Second Mean Value theorem for integrals.
3. If f(x) = C x² (1 – x), for 0 < x < 1 is a probability density function (p.d.f.), find the value of C.
4. Define improper integral of second kind.
5. Show that the improper integral  converges (where a > 0)
6. Find L^–1
7. State the general solution for the linear differential equation  + Py = Q
8. State the postulates of a Poisson Process.

 

1. State the Fundamental Theorem on a necessary and sufficient condition for the consistency of a system of equations  A+ = .
2. If λ is a characteristic root of A, show that λ² is a characteristic root of A².

 

SECTION – B

Answer any FIVE Questions                                                                  (5 x 8 = 40 marks)

 

1. Let P_n = {0, 1/n, 2/n, ….., (n – 1)/n, 1} be a partition of [0, 1]. For the function f(x) = x, 0 ≤x ≤ 1, find U[P_n, f] and L[P_n, f]. Comment on the integrability of the function.
2. Evaluate: (i)  (ii)
3. Show that the integral (where a > 0) converges for p > 1 and diverges for p ≤ 1.

 

1. Define Beta integrals of First kind and Second kind. Show that one can be obtained from the other by a suitable transformation.

(P.T.O)

1. Solve:  =

 

1. Solve: (D² + 4D + 6) y = 5 e^{– 2 x}

 

1. Establish the relationship between the characteristic roots and the trace and determinant of a matrix.

 

1. Give a parametric form of solution to the following system of equations:

4x₁ – x₂ + 6x₃ = 0

2x₁ + 7x₂ +12x₃ = 0

x₁ – 4x₂ – 3x₃ = 0

5x₁ – 5x₂ +3x₃ = 0

 

SECTION – C

 

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

 

1. (a) State and prove the First Fundamental Theorem of Integral Calculus.

(b) Evaluate (i)  (ii)                                                      (12+8)

 

1. (a) Show that mean does not exist for the distribution with p.d.f.

f(x) = , – ∞ < x < ∞

(b)L[f(t)] = F(s), show that L[t f(t)] = – F(s). Using this result find L[t² e^{– 3 t}]

(8+12)

 

1. Evaluate: (a) ∫ ∫ x² y² dx dy over the circle x² + y² ≤ 1.

(b) ∫ ∫ y dx dy over the region between the parabola y = x² and the line x + y = 2

(10+10)

 

1. (a) State and Prove Cayley-Hamilton Theorem.

(b) If P is a non-singular matrix and A is any square matrix, show that A and      P^–1AP have the same characteristic equation. Also, show that if ‘x’ is a characteristic vector of A, then P^–1 x is a characteristic vector of P^–1AP.      (12+8)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 07

THIRD SEMESTER – November 2008

ST 3500 – STATISTICAL MATHEMATICS – II

 

 

 

Date : 06-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

 

1. Define lower sum of a function corresponding to a partition of an interval [a, b].
2. State the linearity property of Riemann integrals.
3. Define probability density function (p.d.f.).
4. What is an improper integral? Give an example.
5. Define absolute convergence of an improper integral.
6. Is the integral  dx convergent or divergent?
7. Change the order of integration in the double integral
8. Define a symmetric matrix.
9. Show that inverse of a non-singular matrix is unique.
10. Define characteristic root and vector of a matrix.

 

SECTION – B

Answer any FIVE Questions                                                                  (5 x 8 = 40 marks)

 

1. If P  is any partition of [a, b], show under usual notations that

m ( b – a) ≤ L (P , f ) ≤ U(P , f ) ≤ M ( b – a)

 

1. Show that if f is R-integrable on [a, b], then | f | is integrable on [a, b].

 

1. If f(x) = cx , 0 < x < 1, is a p.d.f. find ‘c’ and the mean and variance of the distribution.
2. Discuss the convergence of the improper integral  dx (where a > 0) by varying ‘p’.

 

1. Evaluate ∫ ∫ xy dy dx over the positive quadrant of the circle x² + y² = a²

 

1. Define moment generating function of a bivariate distribution. Show how the means, variances and covariance can be found from it.

 

1. If λ is the characteristic root of a matrix A, show that λⁿ is a characteristic root of Aⁿ and both have the same associated characteristic vector. Also, show that one can always find a normalized (unit) characteristic vector associated with a characteristic root.

 

1. Define (i) Hermitian matrix, (ii) Idempotent matrix, (iii) Scalar matrix,    (iv) Orthogonal matrix

 

 

 

 

 

SECTION – C

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

 

1. State and prove the First and Second Fundamental Theorems of Integral Calculus.

 

1. (a) State and prove the Comparison Test for convergence of an improper integral of any one kind.

(b) Test the convergence of the integrals: (i)  dx  (ii)  dx          

(10 +10)

 

1. Let f(x, y) = x + y, 0 < x, y < 1, be the joint p.d.f. of (X, Y). Find the joint distribution function. Also, find the means, variances and covariance.

 

1. (a) State and prove Cayley-Hamilton Theorem.

(b) Find the inverse of the following matrix using the above theorem:

(10 +1 0)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 08

THIRD SEMESTER – November 2008

ST 3501 – STATISTICAL MATHEMATICS – II

 

 

 

Date : 06-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

 

 

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

 

1. Define lower sum of a function for a given partition of a closed interval [a, b].
2. Give an example of a non-monotonic function that is integrable.
3. Define probability density function (p.d.f.) of a random variable (r.v).
4. Give one example each for improper integrals of I and II kinds.
5. Show that the improper integral  converges absolutely.
6. Find the value of
7. If L[f(t)] = F(s), express L[ tⁿ F(t)] in terms of F.
8. State the general solution for the linear differential equation  + Py = Q
9. Define linearly dependent vectors.
10. Define a positive definite quadratic form.

 

SECTION – B

 

Answer any FIVE Questions                                                                      (5 x 8 = 40 marks)

 

1. For any partition P of [a, b], show (under usual notations) that

m (b – a ) ≤ L(P , f)  ≤ U(P , f) ≤ M (b – a )

 

1. State (without proof) a necessary and sufficient condition for R-integrability of a function in a closed interval [a, b]. Using this condition, show that if g R[a, b] and if g is bounded away from 0, then 1/g  R[a, b]
2. If a r.v. X has p.d.f. f(x) = ,  – ∞ < x < ∞, find the moment generating function of X. Hence find its mean and variance.
3. Show that: (a)  is divergent (b)   is convergent
4. Find L[ cos at] and hence find L[ sin at] using the result for Laplace transform of the derivative of a function.

 

1. Solve: (x + y)² d x = 2 x² d y.

 

1. Find any non-trivial solution which exists for the following system of equations:

2 x₁ +3 x₂ – x₃ + x₄ = 0

3 x₁ + 2 x₂ – 2 x₃ + 2 x₄ = 0

5 x₁ – 4 x₃ + 4 x₄ = 0

 

1. Show that if λ is a characteristic root of A, then λⁿ is a characteristic vector of Aⁿ with the same associated characteristic vector. Establish a similar result for A^{– 1}.

 

SECTION – C

 

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

 

1. (a) State and prove the First and Second Fundamental Theorems of Integral Calculus.

 

(b) If f(x) = c x², 0 < x < 2, is the p.d.f. of a r.v X, find ‘c’ and P( 1 < X < 2)

(16+4)

1. (a) Show that  is divergent

(b) Discuss the convergence of gamma integral.                                                         (8+12)

 

1. (a) Evaluateover the region between the parabola y = x² and the line x+y =2.

(b) Evaluate  over the half circle x² + y² ≤ a² with y ≥ 0.

(10+10)

 

1. (a) State and prove Cayley-Hamilton Theorem.

 

(b) Show that a polynomial of degree ‘n’ has (n +1) distinct real roots if and  only

if all its coefficients are zero.                                                                                        (12+8)

 

 

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    LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 12

THIRD SEMESTER – April 2009

ST 3501 / ST 3500 – STATISTICAL MATHEMATICS – II

 

 

 

Date & Time: 06/05/2009 / 9:00 – 12:00     Dept. No.                                                       Max. : 100 Marks

 

 

 

SECTION A

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

 

1. Define upper sum and lower sum of a function.
2. When do you say that a function is Riemann integrable on [a,b]?
3. Define moment generating function of a continuous random variable.
4. Define improper integrals.
5. State the μ test for convergence of integrals.
6. Define Laplace transform of a function.
7. Let f(x) =    cx ², 0 < x < 2

0, otherwise

If this is a probability density function, find c.

1. State the second fundamental theorem of integral calculus.
2. State any two properties of Riemann integral.
3. Define absolute convergence of a function.

 

SECTION B

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

 

1. For any partition P  on [a, b], prove that m (b – a) ≤ L (P, f) ≤ U (P, f) ≤ M (b – a).

Where m =  , M = and f is a bounded function on [a,b].

1. Let f(x) = x ², 0 ≤ x ≤ 1. By considering partitions of the form  P _n =

Show that  U (P_n, f) =  L (P_n, f) = 1/3

1. Let X be a random variable with probability density function

f(x) =   1, 0 < x < 1

0, otherwise

Find the moment generating function and hence the mean and variance of X.

1. Show that   (a > 0)   , converges for p > 1 and diverges for p ≤ 1.
2. Discuss the convergence of the following improper integrals:

(i)                                        (ii)

 

1. Show that  (i)      and      (ii) b (m, n) = b (n, m)
2. Show that L _{(f + g)}(s) = L_f (s) + L_g (s) and L_{c f} (s) = c L_f (s), where L_f (s) is the Laplace transform of the function f.
3. Evaluate   over the region between the line x = y and the parabola y = x ².

 

SECTION C

 

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

 

1. (a) Let f, g Є R [a, b], then show that f + g Є R [a, b] and

(b) Let f (x) =      5 x ⁴, 0 ≤ x < 1

0, otherwise

be the probability density function of the random variable X.

 

Find (i) P (1/2 < X < 1) (ii) P (-2 < X < 1/2) (iii) P (0 < X < 3/4) (iv) P (1/4 < X < 3)

 

1. (i) State and prove the first fundamental theorem of Integral calculus.

 

(ii) Derive the differential difference equations for a Poisson process.

21. (i) Show that b (m, n) =

      (ii) If X has the probability density function f (x) = c x ² e^-x, 0 < x < . Find

            Expectation of X and variance of X.

 

22. (a) If  converges absolutely, then show thatconverges.

      (b) Discuss the convergence of the following integrals.

(i)    (ii)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2010

ST 3503/ST 3501/ST 3500 – STATISTICAL MATHEMATICS – II

 

 

 

Date : 30-10-10                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL the Questions                                                                                                 (10 x 2 = 20 marks)

 

1. Define upper and lower sums of a function on [a, b] corresponding to partition.
2. State the linearity property of integrals.
3. Define Gamma integral.
4. Evaluate
5. State the -test for an improper integral I kind.
6. State how the mean and variance are found from the m.g.f.
7. State the relationship between the characteristic roots and trace and determinant of a matrix.
8. Solve: .
9. Verify whether the following system of equations is consistent:

x – y + z – 2 = 0, 3x – y + 2z = 0,  3x + y + z +18 = 0.

1. Find the characteristics roots of .

 

PART – B

Answer any FIVE Questions                                                                                                             (5 x 8 = 40 marks)

1. Evaluate  from first principles.
2. Show that if  is a monotonically increasing  function on [ a, b], then .
3. Discuss the convergence of for various value of p.
4. Solve .
5. Find the m.g.f and hence the mean and variance of a distribution with p.d.f.
6. Discuss the convergence of : (a)   (b)  (c) .
7. Evaluate the integralover the upper half of the circle.
8. Find any non trivial solution which may exist:

.

(P.T.O.)

 

 

 

 

 

PART – C

Answer any TWO Questions                                                                                        (2 x 20 = 40 Marks)

1. (a) If and , show that and that .

(b) State and prove the First Fundamental Theorem of Integral Calculus. Deduce the Second

Fundamental theorem.                                                                                                       (10 + 10)

 

1. (a) Show that .                                                                                                     (10)

(b) Show that mean does not exist for the distribution with p.d.f. .   (10)

1. a) If have jont pdf , find the p.d.f. of U=X₁/X₂.

(15)

1. b) Discuss the convergence of the integral for various values of a. (5)

1.  (a) State and prove Cayley-Hamilton Theorem.                                                                              (10)

 

(b) Find the inverse of the following matrix by using Cayley-Hamilton theorem.

.                                                                                                                       (10)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – STATISTICSTHIRD SEMESTER – NOVEMBER 2012ST 3503/3501/3500 – STATISTICAL MATHEMATICS – II
Date : 02/11/2012 Dept. No.         Max. : 100 Marks                 Time : 9:00 – 12:00
PART-A  Answer ALL the questions: [10×2 =20]
1. Define upper sum and lower sums.2. Give an example of a function that is not Riemann integrable.    3. Define improper integrals.   4. State the comparison test for the improper integrals    5. Define variance-covariance matrix.     6. When do we say that a system of equations is homogenous?7. State the order and degree of the differential equation:  8. Obtain the Laplace transform of  t > 0. 9. Define characteristic equation and characteristic roots. 10. Write down the importance of Caley-Hamilton Theorem.

PART – BAnswer any FIVE questions:      [5×8 =40]11. Evaluate   from first principles.12. Show that every continuous function defined on a closed interval of the real line is       Riemann integrable.
13. Find the mean and variance of Beta distribution of I kind.
14. Discuss the convergence of gamma integral.
15. Find the covariance between X and Y whose joint p.d.f.  is  .
16. Evaluate  over the upper half of the circle with centre (0, 0) and radius 1.

 

17. a) Solve the differential equation   .    (b) Obtain the inverse Laplace transform of  .18. Find the characteristic roots and corresponding vectors of the matrix          .

PART – C Answer any TWO questions: [2×20 =40]
19. (a) State and Prove the first fundamental theorem on integral calculus.       (b) Derive the MGF of normal distribution. Hence find its mean and variance.  20. (a) Establish the relation between the Beta and Gamma integrals. Hence find the            value of β (3, 4).
(b) Solve the differential equation:   .
21. (a) Solve the following initial value problem using Laplace transform,            where y(0) = -2 and y'(0) = 5 :  .     (b) Let X and Y be two independent one parameter Gamma random variables with            parameters 1 and 2 respectively. Use ‘transformation of variables’ method to            obtain the distribution  of  .
22. (a) Show that if λ  is a characteristic root of A, then  λn is a characteristic root of An.
(b) Solve the system of equations using matrix inverse method.       .

 

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