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)

 

Go To Main page

 

 

 

 

Latest Govt Job & Exam Updates:

View Full List ...