LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – NOVEMBER 2003

ST-4501/STA503 – DISTRIBUTION THEORY

31.10.03                                                                                                          Max:100 marks

9.00-12.00

SECTION-A

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

 

1. Let f(x,y) = e

0          else where.

Find the marginal p.d.f of X.

2. Let the joint p.d.f of X₁ and X₂ be f(x₁,y₁) = and x₂ = 1, 2.

Find P(X₂ = 2).

3. If X ~ B (n, p), show that E
4. If X₁ andX₂ are stochastically independent, show that M (t₁, t₂) = M (t₁, 0) M (0, t₂), ” t_1, t₂.
5. Find the mode of the distribution if X ~ B .
6. If the random variable X has a Poisson distribution such that P (X = 1) = P (X = 2),

Find p (X = 4).

7. Let X ~ N (1, 4) and Y ~ N (2, 3). If X and Y are independent, find the distribution of

Z = X -2Y.

8. Find the mean of the distribution, if X is uniformly distributed over (-a, a).
9. Find the d.f of exponential distribution.
10. Define order statistics based on a random sample.

 

SECTION-B

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

 

11. Let f(x­₁, x₂) = 12

0        ;   elsewhere

 

Find P .

12. The m.g.f of a random variable X is

Show that P (= .

13. Find the mean and variance of Negative – Binomial distribution.
14. Show that the conditional mean of Y given X is E (Y÷X=x)for trinomial
15. Find the m.g.f of Normal distribution.

 

16. If X has a standard Cauchy distribution, find the distribution of X². Also identify its

distribution.

17. Let (X, Y) have a bivariate normal distribution. Show that each of the marginal

distributions is normal.

18. Let Y₁, Y₂ , Y₃ andY₄ denote the order statistics of a random sample of size 4 from a

distribution having a p.d.f.

f(x) =    2x  ;  0 < x < 1

0   ;    elsewhere   .   Find p .

 

 

SECTION-C

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

 

19. Let x (X₁, X₂) be a random vector having the joint p.d.f.

 

 

f (x₁, x₂) =         2  ;   0 < x₁ < x₂ <1

 

0  ;   elsewhere

(i) Find the correlation between x₁ and x₂                                                                        (10)

(ii) Find the conditional variance of x₁ / x₂                                                                      (10)

 

20. a) Find the mean and variance of hyper – geometric distribution. (10)
21. b) Let X and Y have a bivarite normal distribution with

 

Determine the following probabilities

1. i) P (3 < Y <8) ii) P (3 < Y< 8 ½X =7)                                                       (10)
2. i) Derive the p.d.f of ‘t’ – distribution with ‘n’ d.f (10)
3. ii) If X₁ and X₂ are two independent chi-square variate with n₁ and n₂f. respectively,

show that                                                                          (10)

22. i) Let Y₁, Y₂ and Y₃ be the order statistics of a random sample of size 3 from a

distribution having p.d.f.

1      ;      0 < x < 1

f (x) =

0     ;       elsewhere.

 

Find the distribution of sample range.                                                             (10)

ii)Derive the p.d.f of  F variate with (n₁, n₂) d.f.                                                   (10)

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 15

FOURTH SEMESTER – APRIL 2006

                                                      ST 4501 – DISTRIBUTION THEORY

(Also equivalent to STA 503)

 

 

Date & Time : 27-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION A

Answer ALL the Questions                                                            (10 ´ 2 = 20 Marks)

 

1. Define Binomial distribution.
2. Find the p.g.f. of Poisson distribution.
3. Give any two applications of Geometric distribution.
4. Write down the probability density function (p.d.f.) of Gamma distribution.
5. State the mean of Beta distribution of II kind and the condition for it to exist.
6. If X ~ N( 0, 1),  what is the value of E(X⁴)?
7. Write down the Mean deviation about median of a normal distribution.
8. State the relation between c² and F distributions.
9. State the conditions under which Binomial distribution tends to a normal distribution.
10. Define order statistics and give an example.

 

SECTION B

Answer any FIVE Questions                                                            (5 ´ 8 = 40 Marks)

 

1. Let X have the distribution with p.m.f.

X :   -1        0       1

Pr:   0.3      0.4     0.3

Find the distributions of (i) X²   (ii) 2X +3

1. Find the mode of Binomial distribution.
2. Derive the conditional distributions associated with a Trinomial distribution.
3. Derive the mean and variance of Uniform distribution on (a,b).
4. For N(m,s²) distribution, show that the even order central moments are given by m_2n = 1^.3^…..(2n – 1) s²ⁿ ” n ³ 1.
5. Show that the limiting form of Gamma distribution G(1, p ) as p ®µ, is normal distribution.
6. Define Student’s t in terms of Normal and Chi-Squared variates and derive its p.d.f.

 

18. Let

e^{– ( x – q )}, x > q

f(x ; q) =

0, otherwise,

where q Î R. Suppose X₁,X₂,…,X_n denote a random sample of size n from the

above distribution find E(X₍₁₎).

 

 

SECTION C

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

 

19. (a)Let f(x, y) = 2, 0 < x < y < 1 be the joint p.d.f of (X,Y). Find the marginal and conditional distributions. Examine whether X and Y are independent.

(b) Derive the m.g.f of Trinomial distribution.                                     (12 + 8)

20. (a) Show that for a normal distribution Mean = Median = Mode.

(b) Show that, under certain conditions (to be stated), the limiting form of

Poisson distribution is Normal distribution.                                         (15 +5)

 

21. (a) Let the joint p.d.f of (X₁, X₂) be

f(, ) = exp(–),  , > 0

Obtain the joint p.d.f. of Y₁ = X₁ + X₂ and Y₂ = X₁ / (X₁ + X₂).

(b)Show that mean does not exist for Cauchy distribution.                   (12 +8)

 

22. (a) Find the joint density function of i ^th and j ^th order statistics.

(b) Let X₁,X₂,…,X_n be  a random sample of size n from a standard uniform

distribution. Find the covariance between X ₍₁₎ and X ₍₂₎.                      (6 + 14)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.  DEGREE EXAMINATION – STATISTICS

AC 15

FOURTH SEMESTER – APRIL 2007

ST 4501  – DISTRIBUTION THEORY

 

 

Date & Time: 24/04/2007 / 9:00 – 12:00         Dept. No.                                                     Max. : 100 Marks

 

 

Section A

Answer ALL the questions                                                                                  (10×2=20)

1. State the properties of a distribution function.
2. Define conditional  and marginal distributions.
3. What is meant by ‘Pair-wise Independence’ for a set of ‘n’ random variables.
4. Define Conditional Variance of a random variable  X given a  r.v. Y=y.
5. Define Moment Generating Function(M.G.F.) of a  random variable X.
6. Examine the validity of the statement “ X is a Binomial variate with mean 10 and standard deviation 4”.
7. If X is binomially distributed with parameters n and p, what is the distribution of

Y=(n-x)?

1. Define ‘Order Statistics’ and give an example.
2. State any two properties of Bivariate Normal distribution.
3.  State central limit theorem.

Section B

Answer any FIVE questions                                                                                                     (5×8 =40)

11. Let (X,Y) have the joint p.d.f. described as follows:

(X,Y) : (1,1)   (1,2)   (1,3)   (2,1)   (2,2)   (2,3)

f(X,Y) :  2/15   4/15    3/15    1/15   1/15   4/15. Examine if X and Y are independent.

1. The joint p.d.f. of  X₁ and X₂ is : f(x₁, x₂) =.

[a] Find the conditional p.d.f. of X₁ given X₂=x_2.

[b] conditional mean and variance of X₁ given X₂=x₂

13. Derive the recurrence relation for the probabilities of Poisson distribution
14. Obtain Mode of Binomial distribution
15. Obtain Mean deviation about mean of  Laplace distribution
16. The random variable X follows Uniform distribution over the interval(0,1). Find the

distribution of Y = -2 log X.

17. Obtain raw moments of Student’s ‘t’ distribution, Hence fine the Mean and the Variance.
18. Let Y1, Y2, Y3, and Y4 denote the order statistics of a random sample of size 4 from a

distribution having p.d.f.   f(x) =.       Find P (Y3 > ½ ).

 

 

Section C

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

 

19.[a]  The random variables X and Y have the joint p.d.f. f(x, y) =.

Compute Correlation co-efficient between X and Y.                                                       [15]

[b] Let X and Y are two r.v.s with the p.d.f. f(x, y) =.

Examine whether X and Y are stochastically independent.                                            [5]

 

20 [a] Derive M.G.F. of Binomial distribution and hence find its mean and variance.

[b] Show that  E(Y| X=x) = (n-x) p₂/ (1-p₁),if (X,Y) has a Trinomial distribution with

parameters n, p₁ and p_2.

21. [a] Prove that Poisson distribution is a limiting case of Binomial distribution, stating the

assumptions involved.

[b] Let X and Y follow Bivariate Normal distribution with;m₁=3  m₂=1  s₁² =16  s₂² =25 and

r = 3/5.

Determine the following probabilities: (i) P[(3<Y<8)|X=7]   (ii) P[ (-3<X<3) | Y=4]

22. [a[ If X and Y are two independent Gamma variates with parameters m and n respectively.

Let U=X+Y and V= X / (X+Y)

[i] Find the joint p.d.f. of  U and V

[ii] Find the Marginal p.d.f.s’ of U and V

[iii] Show that the variables U and V are independent.                                                [12]

[b] Derive the p.d.f of F-variate with (n1, n2) d.f.                                                            [8]

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

NO 19

FOURTH SEMESTER – APRIL 2008

ST 4501 – DISTRIBUTION THEORY

 

 

 

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

Time : 9:00 – 12:00

 

SECTION – A

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

 

1. Explain the joint p.d.f of two continuous random variables X and Y.
2. Define conditional probability mass function.
3. Let . Find.
4. Write down any two properties of negative Binomial distribution.
5. Define Laplace distribution and find its mean.
6. Define Beta distribution of first kind.
7. Define students-t statistic and write down its probability density function.
8. State the additive property of Chi-square distribution.
9. Define order statistic and give an example.
10. Define conditional expectation and conditional variance of a random variable X given Y= y.

 

SECTION – B

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

 

1. The joint p.d.f of random variables X and Y is given by

• Find the value of k
• Verify whether X and Y are independent.

1. Derive Poisson distribution as limiting form of Binomial distribution.
2. Define multinomial distribution and find the marginal distributions.
3. Explain joint distribution function of two dimensional random variable (X,Y) and establish any two of its properties..
4. Show that for normal distribution mean, median and mode coincide.
5. Find the MGF of Bivariate normal distribution.
6. State and prove central limit theorem.
7. Derive the p.d.f of F-distribution with n₁ and n₂ degrees of freedom.

 

SECTION – C

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

 

1. a) Obtain mean deviation about mean of Laplace distribution.

1. b) Show that exponential distribution satisfies lack of memory property.

1. a) Derive MGF of negative Binomial distribution and show that its mean is less than its variance.

1. b) Find the factorial moments of hyper-geometric distribution.

1. a) If X and Y are independent Chi-square variates with n₁ and n₂ d.f, find the p.d.f of x/x+y.

1. b) Obtain the MGF. of Binomial distribution with n=7 and p=0.6 and hence find .

1. a) Let X and Y follow Bivariate normal distribution with and P=0.4. Find the following probabilities.

(i)

(ii)

1. b) Let X₁, X₂, …. X_n be a random sample with common p.d.f

Find p.d.f, mean and variance of X₍₁₎, the first order statistic.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 19

B.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – April 2009

ST 4501 – DISTRIBUTION THEORY

 

 

 

Date & Time: 24/04/2009 / 9:00 – 12:00        Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer ALL Questions                                                                                                   (10 x 2 =20)

1. Show that the function  is a probability density function.
2. Define : Conditional Variance with reference to a bivariate distribution.
3. Write the density function of discrete uniform distribution and obtain its mean.
4. Find the mean of binomial distribution using its moment generating function.
5. Find the maximum value of normal distribution with mean and variance.
6. Write down the density function of bivariate normal distribution.
7. Define : t Statistic.
8. Find the density function of the random variable where is the distribution function of the continuous random variable .
9. State central limit theorem for iid random variables.
10. Obtain the density function of first order statistic in.

 

PART – B

 

Answer any FIVE Questions                                                                                         (5 x 8 =40)

 

1. Let be iid RVs with common PDF . Write . Show that and are pairwise independent but not independent.
2. Let be independent RVs with common density given by

Find the distribution of .

1. Derive the characteristic function of Poisson distribution. Using the same find the first three central moments of Poisson distribution
2. Establish the lack of memory property of geometric distribution
3. Obtain the marginal distribution of X if (X,Y) follows bivariate normal distribution
4. Obtain the moment generating function of chi-square distribution and hence establish its additive property
5. Show that if , then
6. Derive the formula for the density function of rth order statistic

 

PART –  C

 

Answer any TWO Questions                                                                              (2 x 20 =40)

 

1. Let (X,Y) be jointly distributed with density

 

Find

 

1. (a) Obtain the moment generating function of Normal distribution and hence find its

mean and variance

 

(b) Show that, if X and Y are independent poisson variates with parameters and

,  then the conditional distribution of X given X+Y is binomial.

 

1. Derive the density function of t distribution and obtain its mean.

 

1. Write descriptive notes on the following :

• Stochastic independence
• Multinomial distribution
• Transformation of variables
• Stochastic Convergence

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2011

ST 4502/ST 4501 – DISTRIBUTION THEORY

 

 

Date : 07-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL Questions                                                                                                   10 x 2 =20

 

1. Define : Independence of random variables
2. Show that under usual notations,  .
3. State the additive property of Binomial distribution..
4. Write down the density function of Negative Binomial Distribution.
5. Write E[X|Y=y] when (X,Y) has bivariate normal distribution.
6. What is meant by Lack of Memory Property?.
7. Write the density function of t-statistic with n degrees of freedom
8. Define : F Statistic.
9. Write down the general formula for the density function of the first order statistic.
10. Mention the use of Central limit theorem.

 

PART – B

 

Answer any FIVE Questions                                                                                         5 x 8 =40

 

1. Find E[X|Y=3], if the joint probability density function of and Y is given by

.

1. Lethave the joint probability density function

 

 

Compute the correlation coefficient of X and Y.

 

1. Establish the additive property of independent Poisson variates.
2. If the moment generating function of a random variable is compute P(X=2 or X=3).
3. Obtain the mean and variance of beta distribution of first kind with parameters m and n.
4. Show that if X has uniform distribution defined over [0,1] then -2logx has chi-square distribution with 2 degrees of freedom.
5. Let and be independent standard normal variates. Derive the distribution of using the moment generating function method.
6. Find the limiting distribution of sample mean based on a sample of size n drawn from normal distribution with given mean and variance.

 

PART –  C

Answer any TWO Questions                                                                                          2 x 20 =40

 

1. (a) Let and  be  jointly distributed with density

 

 

Find .

 

(b) Derive the moment generating function of Negative binomial distribution.

 

1. (a) Show that if X and Y are independent Poisson variates with means and then the

conditional distribution of X given X+Y is binomial.

 

(b) Obtain the distribution of if X and Y are independent exponential variates with

parameter .

 

1. (a) Derive the density function of F- distribution.

 

(b) Derive the moment generating function of chi-square distribution with n degrees of freedom

and hence find its mean and variance.

 

22. Derive the distribution of sample mean and sample variance based on a sample drawn from

normal distribution. Also prove they are independently distributed.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2012

ST 4502/ST 4501 – DISTRIBUTION THEORY

 

 

 

Date : 21-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL Questions:                                                                                                       (10 x 2 =20)

1. 1. Find the value of for which the function is a probability density function.
2. Define: Correlation coefficient.
3. Write down the density function of hyper geometric distribution.
4. Obtain the mean of geometric distribution.
5. Write down the density of bivariate normal distribution.
6. Define: Chi-square statistic
7. State any two properties of t-distribution
8. Write down the distribution function of if .
9. Obtain the density function of the nth(largest) order statistic when a sample of size n is drawn from a

population with pdf

10. Define: Stochastic convergence.

PART – B

 

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

 

11. Let and have the joint pdf  described as follows:

	(0,0)	(0,1)	(1,0)	(1,1)	(2,0)	(2,1)
	1/18	3/18	4/18	3/18	6/18	1/18

 

Obtain the marginal probability density functions and the conditional expectations

12. Let and have the joint pdf

Examine whether the random variables are independent.

13. Establish the lack of memory property of geometric distribution.
14. State and prove the additive property of Binomial distribution.
15. Find the median of Cauchy distribution with location parameter and scale paramter  .
16. Obtain the moment generating function of standard normal distribution.
17. Show that ratio of two independent standard normal variates has Cauchy distribution.
18. State and prove Central limit theorem for iid random variables.

PART – C

 

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

 

19. (a) Derive the mean and variance of Poisson distribution.

(b) Let and have a bivariate binomial distribution with  and

. Obtain

 

20. (a) Write down the density function of two parameter gamma distribution. Derive its moment

generating function and hence the mean and variance of the distribution.

(b) Let  . Find the density function of

 

21. (a) Derive the distribution of t-statistic.

(b) Derive the sampling distribution of sample mean from a normal population.

 

22. (a) Find where  is the largest order statistic based on a sample of size four from a

population with  pdf

(b) Obtain the limiting distribution of nth order statistic based on a sample of size n drawn from

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – NOVEMBER 2012

ST 4502/4501 – DISTRIBUTION THEORY

 

 

 

Date : 7/11/2012               Dept. No.                                        Max. : 100 Marks

Time : 1.00 – 4.00

 

PART – A

 

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

 

1. Suppose that two dimensional continuous random variable (X, Y) has joint p.d.f. given by

Find E (xy).

2. Prove that sum of squares of deviations is minimum when the deviations taken from mean.
3. If X₁ and X₂ are independent Poisson variates with parameters l₁ and l₂ find the distribution

of X₁ + X₂.

4. Under what conditions Binomial tends to poisson distribution?
5. Define MGF of a random variable.
6. State the properties of normal distribution.
7. Identify the distribution of sum of n independent exponential variates.
8. Write the pdf of the Laplace distribution.
9. Obtain the distribution of when X has F(n₁, n₂).
10. Define Stochastic convergence.

PART – B

 

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

 

11. The two dimensional random variable (x,y) has the joint density function,

Find marginal density function of x, y and mean of x, y.

 

12. Find the recurrence relation for the moments of binomial distribution with parameters n and p.
13. Explain memory less property. Prove that Geometric distribution has this property.
14. Derive the distribution of k th order statistic.
15. Find the moment generating function of Gammma distribution. Hence find the mean and variance.
16. Derive the mean and variance of Beta distribution.
17. State and prove central limit theorem for for iid random variables.
18. Define chi-square variate. Find its probability density function using moment generating function.

PART – C

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

 

19. a) Find the marginal distribution of X and conditional distribution X given Y=y in a bivariate

normal distribution.

1. b) State and prove the additive property of poisson distribution.
2. a) Prove that for a Normal distribution all odd order central moments vanish and find the

expression for even order moments.

 

1. b) Derive the pdf of t-distribution.

 

21. a) Define the Hyper – geometric distribution. Find its mean and variance.

 

1. b) Show that t-distribution tends to standard Normal distribution as

 

22. Identify the distribution of sample mean and sample variance. Also prove that they are

independently distributed.  Assume the parent population is Normal.

 

 

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