07.04.2004
9.00 - 12.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - MATHEMATICS
FOURTH SEMESTER – APRIL 2004
ST-4201/STA 201- MATHEMATICAL STATISTICS
Max:100 marks
SECTION -A
(10  2 = 20 marks)
Answer ALL questions
8
1.
2.
3.
4.
1 3 
If the MGF of a random variable X is   e t  , write the mean and variance of X.
4 4 
If the random variable X has a Poisson distribution such that Pr [X = 1] = Pr [X = 2], find
Pr [X = 0].
Define the mode of a distribution.
Express the central moment  3 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
(5  8 = 40 marks)
Answer any FIVE questions
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 (  ,  2 ) , 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)
1
3
4
3
6
1
18
18
18
18
18
18
Find i) the marginal density functions and ii) E [X  Y = 0], E[Y  X = 1]
P [X=x, Y=y]
:
17. Let the random variables X and Y have the joint pdf
x+y 0<x<1, 0<y<1
f (x, y) =
0
else where,
Find the correlation coefficient between X and Y.
18. Let X1, X2 be a random sample from N (0, 1). Obtain the pdf of
X1
.
X2
1
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)
b) Show that, for a Binomial distribution

d 
 r 1  pq nr  r 1  r  .
dp 

Hence obtain  2 and  3 .
(10+2)
20. a) Discuss any five properties of Normal distribution.
(10)
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  2 in N (  ,  2 ) based on a random sample of size n. (10)
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)
b) Obtain the MGF of (X, Y) if the pdf is
n!
f(x,y) =
p 1x p 2y p3n x  y , 0  x  y  n p1  p2  p3  1
x! y!(n  x  y )!
Hence obtain E (X), Var(X) and Cov (X,Y).
(5+2+2+3)
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