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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI– 600 034
B.Sc. DEGREE EXAMINATION  MATHEMATICS
FOURTH SEMESTER  NOVEMBER 2003
ST 4201/STA 201 MATHEMATICAL STATISTICS
14.11.2003
Max: 100 Marks
9.00  12.00
SECTION  A
(10  2 = 20 Marks)
Answer ALL the questions.
01. Define an event and probability of an event.
02. If A and B any two events, show that P (ABC) = P(A)  P(AB).
03. State Baye’s theorem.
04. Define Random variable and p.d.f of a random variable.
05. State the properties of distribution function.
 1
;a  x  b

Find E ( X ).
06. Let f ( x)   b  a
0
;
elsewhere

07. Define marginal and conditional p.d.fs.
08. Examine the validity of the given Statement “X is a Binomial variate with
mean 10 and S.D 4”.
09. Find the d.f of exponential distribution.
10. Define consistent estimator.
SECTION  B
(5  8 = 40 Marks)
Answer any FIVE questions.
11. An urn contains 6 red, 4 white and 5 black balls. 4 balls are drawn at random.
Find the probability that the sample contains at least one ball of each colour.
12. Three persons A,B and C are simultaneously shooting. Probability of A hit the
1
1
2
target is
; that for B is
and for C is . Find i) the probability that
4
2
3
exactly one of them will hit the target ii) the probability that at least one of them
will hit the target.
13. Let the random variable X have the p.d.f
2 x ; 0  x  1
f ( x)  
0 ; elsewhere
Find P( ½ < X < ¾) and
ii) P ( - ½ < X< ½).
14. Find the median and mode of the distribution
3 (1  x) 2 ; 0  x  1
.
f ( x)  
; elsewhere
0
1
15.
Find the m.g.f of Poisson distribution and hence obtain its mean and variance.
16. If X and Y are two independent Gamma variates with parameters  and 
X
respectively, then show that Z =
~  (,).
X Y
17. Find the m.g.f of Normal distribution.
18. Show that the conditional mean of Y given X is linear in X in the case of
bivariate normal distribution.
SECTION  C
Answer any TWO questions.
(2  20 = 40 Marks)
19. Let X1and X2 be random variables having the joint p.d.f
2 ; 0  x1  x2  1
f ( x1 , x2 )  
 0 ; elesewhere
Show that the conditional means are
1  x1
, 0  x1  1 and x2 ,0  x2  1.
2
2
(10+10)
20. If f (X,Y) has a trinomial distribution, show that the correlations between
p1 p 2
X and Y is   
.
(1  p1 )(1  p 2 )
21. i) Derive the p.d.f of ‘t’ distribution with ‘n’ d.f
ii) Find all odd order moments of Normal distribution.
22.
(15+5)
i) Derive the p.d.f of ‘F’ variate with (n1,n2) d.f
(14)
ii) Define i) Null and alternative Hypotheses
ii) Type I and Type II errors.
and
iii) critical region
(2)
(2)
(2)
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