LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2004

ST-4201/STA 201- MATHEMATICAL STATISTICS

07.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

SECTION -A

 

Answer ALL questions                                                                               (10 ´ 2 = 20 marks)

1. If the MGF of a random variable X is , write the mean and variance of X.
2. If the random variable X has a Poisson distribution such that P_r [X = 1] = P_r [X = 2], find P_r [X = 0].
3. Define the mode of a distribution.
4. Express the central moment in terms of the raw moments.
5. The MGF of a chi-square distribution with n degrees of freedom is ___________ and its variance is ____________.
6. Write any two properties of a distribution function.
7. There are 2 persons in a room. What is the probability that they have different birth days assuming 365 days in the year?
8. Define an unbiased estimator.
9. Explain Type I error.
10. If the MGF of a random variable X is M (t), express the MGF of Y = aX + b in terms of M(t).

 

SECTION – B

 

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

 

11. State and prove Baye’s theorem.
12. State and prove Chebyshev’s inequality.
13. Obtain the mode of Poisson distribution.
14. Derive the pdf of t – distribution.
15. If the random variable X is N , obtain the MGF of X. Derive the mean and variance.
16. Let X and Y have the joint pdf

(X, Y)             :      (0, 0)    (0, 1)  (1, 0)    (1, 1)    (2, 0)    (2, 1)

 

P [X=x, Y=y]       :

Find i) the marginal density functions and ii) E [X ½ Y = 0], E[Y ½ X = 1]

 

17. Let the random variables X and Y have the joint pdf

x+y    0<x<1,  0<y<1

f (x, y) =

• else where,

Find the correlation coefficient between X and Y.

18. Let X_1,  X₂ be a random sample from N (0, 1).  Obtain the pdf of  .

SECTION – C

Answer TWO questions                                                                               (2 ´ 20 = 40 marks)

 

19. a) Show that Binomial distribution tends to Poisson distribution under certain conditions (to be stated).                                                                                                       (8)
20. b) Show that, for a Binomial distribution

.

Hence obtain .                                                                                          (10+2)

 

20. a) Discuss any five properties of Normal distribution.            (10)
21. b) Of a large group of men , 5% are under 60 inches in height and 40% are between 60 and 65 inches. Assuming Normal distribution find the mean and variance. (10)

 

21. a) Obtain the MLE of and  in N (,) based on a random sample of size n.   (10)
22. b) State and prove Neyman- Pearson theorem. (10)

 

22. a) Four distinct integers are chosen at random and without replacement from the first 10

positive integers.  Let the random variable X be the next to the smallest of these 4

numbers.  Find the pdf of X.                                                                                        (8)

1. b) Obtain the MGF of (X, Y) if the pdf is

f(x,y) =  p, 0

 

Hence obtain E (X), Var(X) and Cov (X,Y).                                                    (5+2+2+3)

 

Go To Main page

 

 

 

 

Latest Govt Job & Exam Updates:

View Full List ...