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
Answer ALL the questions . (10

2 = 20 Marks)
01.
Define an event and probability of an event.
02.
If A and B any two events, show that P (A

B
C
) = 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.
06.
Let f ( x )


 b
0
1
 a
; a
 x
 b
; elsewhere
F ind E ( X ) .
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
target is
4
; that for B is
1
2
2
and for C is .
3
Find i) the probability that
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 f ( x )



2
0 x
;
; 0
 x

1 elsewhere
Find P( ½ < X < ¾) and ii) P ( - ½ < X< ½).
14. Find the median and mode of the distribution f ( x )




3
0
( 1
 x )
2
;
; 0
 x

1 elsewhere
.
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
 respectively, then show that Z =
X
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.
Answer any TWO questions .
SECTION

C
(2

20 = 40 Marks)
19.
Let X
1 and X
2
be random variables having the joint p.d.f f ( x
1
, x
2
)



2
0
;
;
0
 x
1
 x
2 elesewhere

1
Show that the conditional means are
1

2 x
1 , 0
 x
1

1 and x
2
2
, 0
 x
2

1 .
(10+10)
20.
If f (X,Y) has a trinomial distribution, show that the correlations between
X and Y is
   p
1 p
2
( 1
 p
1
)( 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. (15+5)
22. i) Derive the p.d.f of ‘F’ variate with (n
1
,n
2
) d.f
ii) Define i) Null and alternative Hypotheses
ii) Type I and Type II errors.
and iii) critical region
* * * * *
(14)
(2)
(2)
(2)
2