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
(10 2=20 marks)
PART  A
Answer ALL the questions.
01. Two dice are thrown. What is the probability that the sum of the numbers on the two dice
is eight?
2
02. The probability that a customer will get a plumbing contract is
and the probability that
3
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.
03. Consider 2 events A and B such that P A  1 PB / A  1 and P A / B   1 .
4
4,
2
Verify whether the given statement is true (or) false. P A c / B c  3 .
4
04. Define i) independent events and ii) mutually exclusive events.
05. State any four properties of a distribution function.
06. The random variable X has the following probability function
X=x
0
1
2
3
4
5
6
7
2
2
P (X=x)
0
k
2k
2k
3k
k
2k
7k2+k
Find k.
1
; a xb
ba
07. Let f (x) =
0 ; else where
Find E(X).
08. Let X ~ B (2, p) and Y~B (4, p). If P  X  1  5 , find P Y  1 .
9
09. 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
i) at least one post and ii) exactly one post?
12. 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?
13. 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.
14. Derive the mean and variance of hypergeometric distribution.
15. Let X be a random variable having the p.d.f
 xe x ; 0  x  
f(x) = 
; elsewhere
0
Find the m.g.f. of X and hence obtain the mean and variance of X.
2
 X
X
  p1  p 
16. If X is B(n,p), show that E   = p and E   p   
.
n
n
 
 n
X 


 b   .90
17. Let X be N(,2). i) Find b so that P  b 



ii) If P (X < 89) =0.90 and P(X < 94) =0.95, find  and 2.
18. If X and Y are independent gamma variates with parameters  and  respectively,
X
Show that
~ 1 ( , ) .
X Y
(220=40 marks)
PART  C
Answer any TWO questions
19. If the random variables X1 and X2 have the joint p.d.f
 x x

2 e 1 2 ; o  x1  x2  
f (x1 ,x2) = 
; else where

0
i ) find the conditional mean of X1 given X2 and ii) the correlation coefficient
between X1 and X2.
20. a) Find all the odd and even order moments of Normal distribution.
b) 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.
b) Find the m.g.f of exponential distribution.
22) a) Let X1, X2, …. Xn be a random sample of size n from N (,1) . Show that the sample
mean is an unbiased estimator of the parameter .
b) Write a short note on:
i) null hypothesis ii) type I and type II errors iii)
standard error
iv) one sided and two sided tests.
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