LOYOLA COLLEGE (AUTONOMOUS), CHENNAI-600 034.

B.Sc. DEGREE EXAMINATION – STATISTICS

second SEMESTER – APRIL 2003

**ST 2500/ STA 501 statistical methemattics ****–**** I**

23.04.2003

9.00 – 12.00** **Max: 100 Marks

**SECTION ****–**** A **(10 ´ 2 = 20 Marks)

*Answer ALL questions. Each carries TWO marks*.

- What is ‘permutation of indistinguishable objects’? State its value in factorial notation.
- If A, B,C are events, give the set theoretic notation for the following

- Exactly one of the three events occurs
- None of the three events occur

- If P(AÈB) =7/8, P(AÇB) = 1/6, P(A
^{c}) =3/8, find P(B). - Define a bounded function and give an example.
- Define a monotonic sequence and give an example.
- “The following is not a cumulative distribution function (c.d.f)” -Justify this statement:

- “The series diverges” – Justify
- “The function is not a probability generating function (p.g.f)” –
- State the Leibnitz test for alternating series.
- Find (0) if f(x) = x |x|, xÎÂ.

**Section ****–**** B** (5 ´ 8 = 40 Marks)

# Answer any FIVE. Each carries EIGHT marks

- (a) Consider the construction of four-letter words from the word “CHEMISTRY”,

How many of these words begin and end with consonants? How many begin

with T end with a vowel?

- If there are 4 persons from Tamilnadu, 2 from Karnataka, 5 from Orissa and

3 from Kerala, find the number of ways they can be seated in a row such that

persons from the state sit together. Find the number of arrangements if they

have to be seated around a circular table. (4+4)

- State the relevant theorem and solve the following problem using it:

A box contains 10 black and 6 white marbles. Three balls are drawn at random

one by one without replacement. Find the probability that

- (i) First two draws give black and third draw gives white

(ii) First and third drawn give balls of the same colour while the second

draw gives the other colour.

- Consider the experiment of tossing a fair coin indefinitely until a Head

occurs. Write down the sample space of the experiment. If X is the number

of tosses to get the first Head, find the probability mass function (p.m.f.)

and c.d.f. of X.

- Show that the function and log x are continuous functions on (0, ¥).
- (a) Show by using first principles that

(b) Show that (4 + 4)

- Test the convergence of the following series and state the test which you use in each:

(a) (b) (4+ 4)

- Find the values of ‘s’ for which f(s) = is a p.g.f. Find the probability

distribution for which it is the p.g.f. Hence or otherwise find the mean of the distribution.

- Obtain the expansion of the exponential function and define the Poisson distribution.

**SECTION ****–**** C** (2 ´ 20 = 20 Marks)

*Answer any TWO. Each carries twenty marks*.

- (a) State the Binomial theorem for positive integer index. Find term which

contains y^{10} in the expansion of (3x^{2}y-xy^{2}/2)^{8}

(b) State and prove Baye’s theorem.

(c) Three machines produce respectively 40%, 40% and 20% of the total

production in a factory. Of their output 3%, 5% and 4% respectively are

defective items. If an item is selected at random from the entire lot, what is

the probability that it is a defective? Given that a selected item is defective,

what is the probability that it was produced by the second machine. (7+7+6)

- (a) Show that the product of two continuous functions is continuous.

(b) Consider the following c.d.f. of a r.v.X

Identify the type of the distribution. Also, find P(X =3), P(2<X £ 4),

P(0 £ X <3/2) and P(X ³ 5/2) (8+12)

- (a) Discuss the convergence of the Geometric series for all possible

variations in x. Find the value of ‘a’ for which the sequence defines

a probability distribution on the set of positive integers. Find the p.g.f and

hence the variance of the distribution.

(b) Examine the applicability and validity of Rolle’s theorem for the function

(13+7)

- (a) Investigate the extreme values of f(x) = 2x
^{5}– 10^{4 }+10x^{3 }+ 8

- Define Binomial distribution. Find the moment generating function (m.g.f.)

and hence its mean and variance. (10 +10)