LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FourTH SEMESTER – APRIL 2003

ST 4201 / sTA 201  –  MATHEMATICAL STATISTICS

 

28.04.2003

9.00 – 12.00                                                                                                     Max : 100 Marks

                                                                PART – A                                       (10´ 2=20 marks)

      Answer ALL the questions.

 

1. Two dice are thrown. What is the probability that the sum of the numbers on the two dice is eight?
2. The probability that a customer will get a plumbing contract is and the probability that he will get an electric contract is 4/9. If the probability of getting at least one is 4/5,determine the probability that he will get both.
3. Consider 2 events A and B such that and . Verify whether the given statement is true (or) false. .
4. Define i)  independent events and ii)  mutually exclusive events.
5. State any four properties of a distribution function.
6. The random variable X has the following probability function

X = x	0	1	2	3	4	5	6	7
P (X=x)	0	k	2k	2k	3k	k2	2k2	7k2+k

Find k.

 

7. Let f (x) =

0    ;   else where

Find E(X).

8. Let X ~ B (2, p) and Y~B (4, p). If P , find P.
9. Define consistent estimator.
10. State Neyman – Pearson lemma.

 

 

                                                                PART – B                                         (5´ 8=40 marks)

      Answer any FIVE questions.

 

11. A candidate is selected for three posts. For the first post three are three candidates, for the

second there are 4 and for the third there are 2. What are the chances of his getting

1. i) at least one post and  ii)  exactly one post?
2. Three boxes contain 1 white, 2 red, 3 green ; 2 white, 3 red, 1 green and 3 white, 1 red, 2 green balls. A box is chosen at random and from it 2 balls are drawn at random. The balls so drawn happen to be white and red. What is the probability that they have come from the second box?
3. Find the conditional probability of getting five heads given that there are at least four heads, if a fair coin is tossed at random five independent times.
4. Derive the mean and variance of hyper-geometric distribution.
5. Let X be a random variable having the p.d.f

 

f(x) =

Find the m.g.f. of X and hence obtain the mean and variance of X.

16. If X is B(n,p), show that E= p and E.
17. Let X be  N(m,s²).  i)  Find b so that
18. ii) If P (X < 89) =0.90 and P(X < 94) =0.95, find m and s².
19. If X and Y are independent gamma variates with parameters m and n respectively,

Show that  ~ .

 

 

 

                                                               PART – C                                         (2´20=40 marks)

Answer any TWO questions

19. If the random variables x₁ and x₂ have the joint  p.d.f

f  (x_{1 ,}x₂) =

i ) find the conditional mean of X₁ given  X₂ and  ii)  the  correlation coefficient

between  X₁ and X_2.

20. a)  Find all the odd and even order  moments of Normal distribution.
21. Let (X,Y) have a bivariate normal distribution. Show that each marginal distribution

in normal.

21. a) Derive the p.d.f of F- variate with (n₁,n₂) d.f.
22. Find the g.f of exponential distribution.

• a) Let X₁, X₂, …. X_n  be a  random sample of size n from N (q,1) . Show that the sample

mean is an unbiased estimator of the parameter q.

1. Write a short note on:
2. i) null hypothesis ii) type I and type II errors iii)    standard error
3. iv) one -sided and two -sided tests.

 

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