LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – NOVEMBER 2003

ST-2500/STA501 – STATISTICAL MATHEMATICS – I

01.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

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

 

1. Define ‘permutation with indistinguishable objects’.  State its value in factorial notation.
2. If A, B, C are events, write the set notation for the following: (i) A or B but not C occur  (ii) None of the three events occur.
3. If A and B are independent events, show that A and B^C are independent.
4. What are the supremum and infinum of the function f(x) = x – [x], x ÎR
5. Define probability mass function (p.m.f) of a discrete random variable and state its properties.
6. “The series is divergent”  – justify.
7. ” The function f (s) = is not a probability generating function” (p.g.f)” – justify.
8. For what value of ‘a’ does the sequence {4aⁿ}define a probability distribution on the set of positive integers?
9. Find f ^‘(3) for the function f(x) = , xÎ
10. Define radius of convergence of a power series.

 

SECTION-B

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

 

11. In how many ways 3 Americans, 4 French, 4 Germans and 2 Indians be seated in a row so that those of the same nationality are seated together? Find the number  of ways, if they are seated around a circular table.
12. State and prove the Addition Theorem of Probability for two events. Extend it for three events.
13. a) Show that limit of a convergent sequence is unique.
14. b) Define monotonic sequence with an example.
15. Consider the experiment of tossing a fair coin indefinitely until a head appears. Let X = Number of tosses until first Head.  Write down the p.m.f. and c.d.f of X.
16. Discuss the convergence of the Geometric series for variations in ‘a’.
17. Investigate the extreme values of the function f(x) = 2x³-3x²-36x+10, xÎ
18. Show that the series is divergent.
19. Define Binomial distribution and obtain its moment Generating function (m.g.f). Hence find its mean and variance.

 

 

 

SECTION-C

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

 

19. a) Establish the theorem on Total probability.
20. b) Establish Baye’s theorem
21. c) Three machines produce 50%, 30% and 20% of the total products of a factory. The percentage of defectives manufactured by these machines are 3%, 4% and 5% of their total output. If an item is selected at random from the items produced in the factory, find the probability that the item is defective.  Also given that a selected item is defective. What is the probability that it was produced by the third machine?                       (6+6+8)
22. a) Show by using first principle that the function f(x) = x² is continuous at all points of R.
23. b) Identify the type of the r.v. whose c.d.f. is

0,        x < 0

f(x) =      x/3,    0 £ x < 1

2/3,    1 £ x < 2

1,       2 £  x

 

Also find P (X = 1.5), P(x < 1),  P (1£ x £ 2), P (x ³ 2) P (1 £ x < 3).                     (8+12)

21. a) Test the convergence of (i) (ii)

For each case, state the ‘test’ which you use.

1. b) Identify the probability distribution for which f (s) =is the p.g.f.  Find

the Mean and Variance of the distribution.                                                           (10+10)

22. a) Verify the applicability and validity of Mean Value theorem for the function

f(x) = x (x-1) (x-2),  xÎ[0,1/2].

1. b) Obtain the expansion of the Exponential function and hence define Poisson

distribution.                                                                                                            (10+10)

 

 

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                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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                LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 05

SECOND SEMESTER – APRIL 2007

ST 2501 – STATISTICAL MATHEMATICS – I

 

 

 

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

 

 

SECTION – A

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

 

1. Define a bounded function and give an example.

 

1. State the values of and .
2. Write down the distribution function of the number of heads in two tosses of a fair coin.
3. Investigate the nature (convergence / divergence / oscillatory) of the series

1 – 2 + 3 – 4 + 5 – ∙∙∙∙∙∙∙∙

1. State the Leibnitz test for convergence of alternating series.

 

1. Apply first principles to find f ‘(a) for the function f(x) = xⁿ.

 

1. Show the validity of Rolle’s Theorem for f(x) = , x [– 1, 1].

 

1. Define a vector space.

 

1. If M(t₁,t₂) is the joint m.g.f. of  (X,Y), express E(X) and E(Y) in terms of M(t₁,t₂).

 

1. Define an Idempotent Matrix.

SECTION – B

 

 

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

1. Show that Inf f + Inf g  ≤  Inf (f + g)  ≤  Sup (f + g) ≤ Sup f + Sup g.
2. Show by using first principles that  = 0.
3. Discuss the convergence of the following series (a) ,  (b)
4. A discrete r.v. X has p.m.f. p(x) = , x = 0, 1, 2, …… Obtain the m.g.f. and hence mean and variance of X.

 

1. Show that differentiability implies continuity. Demonstrate clearly with an example that continuity does not imply differentiability.

 

1. Obtain the coefficients in the Taylor’s series expansion of a function about ‘c’.

(P.T.O)

 

 

 

 

1. State any two properties of a bivariate distribution function. If F(x , y)  is the bivariate d.f. of (X, Y), show that

P( a < X ≤ b , c < Y ≤ d) = F( b, d) – F ( b , c) – F( a, d) + F( a, c)

 

1. Establish the ‘Reversal Laws’ for the transpose and inverse of product of two matrices.

 

SECTION – C

 

 

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

 

1. (a) Establish the uniqueness of limit of a function as x → a (where ‘a’ is any real number). Also, show that if the limit is finite, then ‘f’ is bounded in a deleted neighbourhood of ‘a’.

(b) Identify the type of the r.v. X whose distribution function is

F ( x) =

Also, find P( X ≥ 4 / 3 ) and P(X ≤ 1).                                                          (12 + 8)

 

1. (a) Investigate the extreme values of the function f(x) = ( x + 5)²(x³ – 10)

(b) State the Generalized Mean Value Theorem. Examine its validity for the functions f(x) = x², g(x) = x⁴ for x[1, 2].                                                   (12 + 8)

 

1. (a) For the following function, show that the double limit at the origin does not exist but the repeated limits exist:

f (x ¸y) =

(b) Show that the mixed partial derivatives of the following function at the origin

are unequal:

f (x , y) =              (8 + 12)

 

1. (a) Show that if two (non-zero) vectors are orthogonal to each other, they are

linearly independent.

(b) Find the inverse of the following matrix by ‘partitioning’ method:

(5 + 15)

 

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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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

NO 5

SECOND SEMESTER – APRIL 2008

ST 2501 – 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 a bounded function and give an example.
2. Write the formula for a_n for the sequence 1, –3, 5, –7, . . . .
3. State any two properties of a distribution function.
4. Give reason (state the relevant result) as to why the series  is divergent.
5. State the Limit Form of Comparison Test.
6. For the function f(x) = | x |, x  R, find f ‘(0 +) and f ‘(0 –).
7. Define ‘Stationary Point’ of a function.
8. Is the space V   = { (x₁, x₂, 2x₁ – x₂) | x₁, x₂  R }a vector subspace of R³ ? Justify your answer.
9. Define symmetric matrix
10. Define rank of a matrix.

 

SECTION – B

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

 

1. Show that the function f(x) = xⁿ is continuous at every point of R
2. Show that a convergent sequence is bounded. Give an example to show that the converse is not true.
3. Show that the series 1 + + + + · · · · · ·    is convergent
4. Find the m.g.f. of the random variable with p.m.f. p(x) = p q^x, x = 0, 1, 2, ….

Hence find the mean.

1. Verify the Mean Value theorem for the function f(x) = x² +3x – 4   on the interval [1, 3].
2. Examine the continuity of the following function at the origin (by using first principles):

f(x) =

1. Verify whether the vectors [2, –1, 1]’, [1, 2, –1]’, [1, 1,–2]’ are linearly independent or dependent.
2. Find the rank of the matrix

(P.T.O)

 

 

SECTION – C

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

 

1. (a) If f(x) = ℓ₁ and g (x) = ℓ₂ ≠ 0, then show that  = ℓ₁ / ℓ₂

(b) If p(x) = x / 15,  x = 1, 2, 3, 4, 5 be the probability mass function of a random variable X, obtain the distribution function of X.                                          (12+8)

 

1. (a) State the Cauchy’s condensation Test and using it discuss the convergence of the series for variations in ‘p’.

(b) Check whether the following series are conditionally convergent / absolutely convergent / divergent:

(i)   (ii)                                           (10+10)

 

1. (a) Obtain the Maclaurin’s Series expansion for the function f(x) = log (1 + x). Show that the expansion indeed converges to the function for –1 < x < 1 by analyzing the behaviour of the remainder term(s).

(b) Discuss the extreme values of the function f(x) = 2 x³ – 15 x² + 36 x + 1

(14+6)

 

1. (a) Establish the uniqueness of the inverse of a non-singular matrix. Also, establish the ‘Reversal Law’ for the inverse of product of two matrices.

(b) Find the inverse of the following matrix by using any method:

(8+12)

 

 

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       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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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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2012

ST 2502/ST 2501/ST 2500 – STATISTICAL MATHEMATICS – I

 

Date : 16-04-2012                   Dept. No.                                          Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

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

1. Define monotonically decreasing sequences.
2. Define random variable.
3. Define divergence sequences.
4. What is meant by linear dependence?
5. 5. Find the trace of the matrix A = 
6. State Rolle’s Theorem.
7. The probability distribution of a random variable X is: Determine       the constant k.
8. Define symmetric matrix. Give an example.
9. Find the determinant of the matrix
10. Define stochastic matrix.

PART – B

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

11. The diameter, say X, of an electric cable, is assumed to be continuous random variable with p.d.f
12. i) Check that the above is a p.d.f. ;   ii) Obtain an expression for the c.d.f of x ;

iii)  Compute ;  iv) Determine the number K such that P(X < k) = P(X > k)

12. Prove that a convergent sequence is also bounded.
13. By using first principles, show that the sequences , where, n = 1, 2, . . . ,

converges to   .

14. Show that differentiability of a function at a point implies continuity. What can you say about the

converse? Justify your answer.

15. State and prove Lagrange’s Mean Value Theorem. (P.T.O.)
16. Obtain the Maclaurin’s Series expansion for log(1+x), for – 1 < x < 1 .
17. If the joint distribution function of X and Y is given by
18. a) Find the marginal densities of X and of Y ;     b) Are X and Y independent?
19. c) Find P(X  1 Y ;
20. Find inverse of the matrix

PART – C

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

19. Examine the validity of the hypothesis and the conclusion of Rolle’s theorem for the          function f defined in  in each of the following cases:
20. i) , a = 0, b = 2
21. ii) , a = -3, b = 0
22. Two fair dice are thrown. Let X₁ be the score on the first die and X₂ the score on the second die. Let Y denote the maximum of X₁ and X₂ i.e. max(X₁, X₂).
23. a) Write down the joint distribution of Y and X₁.
24. b) Find E (Y), Var (y) and Cov (Y, X₁).
25. Suppose that two-dimensional continuous random variable (X, Y) has joint probability density        function given by
26.  i) Verify that
27.   ii) Find P (0 < X <,  P(X+Y < 1),  P(X > Y),  P(X < 1 | Y < 2)
28. (a) Find the rank of .

(b) Verify whether the vectors (2, 5, 3), (1, 1, 1) and (4,–2, 0) are linearly independent.       (10 +10)

 

 

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