LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - MATHEMATICS
THIRD SEMESTER – NOVEMBER 2004
ST 3951 - MATHEMATICAL STATISTICS - I
02.11.2004
Max:100 marks
1.00 - 4.00 p.m.
SECTION - A
(10  2 = 20 marks)
Answer ALL the questions
x
2
1. Find C such that f (x) = C   , x  1, 2, 3, .... satisfies the conditions of being a pdf.
3
2. Let a distribution function be given by
0
x<0
x 1
F(x) =
0x<1
2
1
x≥ 1
Find
1

i) Pr  3  X  
2

ii) P [X = 0].
3. Find the MGF of a random variable whose pdf is f (x) =
1
, -1 < x < 2, zero elsewhere.
3
5
1 2 
4. If the MGF of a random variable is   e t  , find Pr [X = 2].
3 3 
5. Define convergence in probability.
6. Find the mode of a distribution of a random variable with pdf
f (x) = 12x2 (1 - x), 0 < x < 1.
7. Define a measure of skewness and kurtosis using the moments.
8. If A and B are independent events, show that AC and BC are independent.
9. Show that E (X) = n  P ( X  n), for a random variable with values 0, 1, 2, 3...
10. Define partial correlation.
SECTION - B
Answer any FIVE questions.
(5  8 = 40 marks)
11. Show that the distribution function is non-decreasing and right continuous.
12. 'n' different letters are placed at random in 'n' different envelopes. Find the probability
that none of the letters occupies the envelope corresponding to it.
13. Show that correlation coefficient lies between -1 and 1. Also show that p2 = 1 is a
necessary and sufficient condition for P [Y = a + bx] = 1 to hold.
14. Derive the MGF of gamma distribution and obtain its mean and variance.
15. Let f (x, y) = 2 0 < x < y < 1 the pdf of X and Y. Obtain E [X  Y] and E [Y  X]. Also
obtain the correlation coefficient between X and Y.
16. Show that Binomial distribution tends to Poisson distribution under some conditions.
17. State Chebyshw's inequality. Prove Bernoulli's weak law of large numbers.
1
18. 4 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.
SECTION - C
(2  20 = 40 marks)
Answer any TWO questions
19. a) Let {An} be a decreasing sequence of events. Show that


P  lim An   lim P ( An ) . Deduce the result for increasing sequence.
 n 
 n 
b) A box contains M white and N - M red balls. A sample of size n is drawn from the
box. Obtain the probability distribution of the number of white balls if the sampling is
done i) with replacement ii) without replacement.
(10+10)
20. a) State any five properties of Normal distribution.
b) In a distribution exactly Normal 7% are under 35 and 89% are under 63. What are the
mean and standard deviation of the distribution?




c) If X1 and X2 are independent N  μ, σ12  and N  μ , σ 22  respectively, obtain the
2

1

distribution of a1 X 1 + a2 X2.
(5+10+5)
21. a) Show that M (t1, t2) = M (t1, 0) M (0, t2) , t1, t2 is a necessary and sufficient condition
for the independence of X1 and X2.
b) Let X1 and X2 be independent r.v's with
1  x1 m 1
f1 (x1) =
, 0 < x1 < 
e
x1
m
f2 (x2) =
1
n
e  x2 x1n 1 , 0 < x2 < 
X1
X1  X 2
Also obtain the marginal distribution of Y1 and Y2
Obtain the joint pdf of Y1 = X1 + X2 and Y2 =
c) Suppose E (XY) = E (X) E (Y). Does it imply X and Y are independent. Justify.
(6+10+4)
22. a) State and prove Lindberg-Levy central limit theorem.
b) Let Fn (x) be distribution function of the r.v Xn, n = 1,2,3... Show that the
sequence{Xn} is convergent in probability to O if and only if the sequence Fn (x)
satisfies
lim Fn ( x) = 0
x<0
1
x≥0
n 
c) Let Xn, n = 1, 2, ... be independent Poisson random variables. Let y100 = X1 + X2 + ...+
X100. Find Pr [190  Y100  210].
(8+8+4)

2