01.11.2003
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
SECOND SEMESTER – NOVEMBER 2003
ST-2500/STA501 - STATISTICAL MATHEMATICS - I
Max:100 marks
SECTION-A
Answer ALL the questions.
(10x2=20 marks)
1. Define 'permutation with indistinguishable objects'. State its value in factorial notation.
2. If A, B, C are events, write the set notation for the following: (i) A or B but not C occur
(ii) None of the three events occur.
3. If A and B are independent events, show that A and BC are independent.
4. What are the supremum and infinum of the function f(x) = x - [x], x R
5. Define probability mass function (p.m.f) of a discrete random variable and state its
properties.

n
6. "The series 
is divergent" - justify.
n 1 2 n  1
s2
is not a probability generating function" (p.g.f)" - justify.
3 s
8. For what value of 'a' does the sequence {4an} n 1 define a probability distribution on the
set of positive integers?
9. Find f '(3) for the function f(x) = x , xR.
10. Define radius of convergence of a power series.
7. " The function  (s) =
SECTION-B
Answer any FIVE the questions.
(5x8=40 marks)
11. In how many ways 3 Americans, 4 French, 4 Germans and 2 Indians be seated in a row so
that those of the same nationality are seated together? Find the number of ways, if they
are seated around a circular table.
12. State and prove the Addition Theorem of Probability for two events. Extend it for three
events.
13. a) Show that limit of a convergent sequence is unique.
b) Define monotonic sequence with an example.
14. Consider the experiment of tossing a fair coin indefinitely until a head appears. Let X =
Number of tosses until first Head. Write down the p.m.f. and c.d.f of X.

15. Discuss the convergence of the Geometric series
a
n
for variations in 'a'.
n 1
16. Investigate the extreme values of the function f(x) = 2x3-3x2-36x+10, xR.

1
17. Show that the series  is divergent.
n 1 n
18. Define Binomial distribution and obtain its moment Generating function (m.g.f). Hence
find its mean and variance.
SECTION-C
Answer any TWO the questions.
(2x20=40 marks)
19. a) Establish the theorem on Total probability.
b) Establish Baye's theorem
c) Three machines produce 50%, 30% and 20% of the total products of a factory. The
percentage of defectives manufactured by these machines are 3%, 4% and 5% of their
total output. If an item is selected at random from the items produced in the factory, find
the probability that the item is defective. Also given that a selected item is defective.
What is the probability that it was produced by the third machine?
(6+6+8)
2
20. a) Show by using first principle that the function f(x) = x is continuous at all points of R.
b) Identify the type of the r.v. whose c.d.f. is
0,
x<0
f(x) = x/3, 0  x < 1
2/3, 1  x < 2
1,
2 x
Also find P (X = 1.5), P(x < 1), P (1 x  2), P (x  2) P (1  x < 3).
(8+12)
2n
 1
1 

 
12  2 2  .....  n 2
n

21. a) Test the convergence of (i) 
(ii)

n4 1
en
n 1
n 1
For each case, state the 'test' which you use.
s2
3
b) Identify the probability distribution for which  (s) =
, s  is the p.g.f. Find
3  2s
2
the Mean and Variance of the distribution.
(10+10)
22. a) Verify the applicability and validity of Mean Value theorem for the function
f(x) = x (x-1) (x-2), x[0,1/2].
b) Obtain the expansion of the Exponential function and hence define Poisson
distribution.
(10+10)
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