LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

AC 11

DEGREE EXAMINATION –STATISTICS

FOURTH SEMESTER – APRIL 2007

ST 4201 – MATHEMATICAL STATISTICS

 

 

Date & Time: 19/04/2007 / 1:00 – 4:00            Dept. No.                                                     Max. : 100 Marks

 

 

Part A

 

Answer all the questions.                                                                                  10 X 2 = 20

 

1. Define Sample space and events.
2. Let the random variable X₁ and X₂ have the joint pdf f(x₁, x₂) = 2, 0<x₁<x₂<1, zero elsewhere. Find the marginal pdf of X_1.
3. State any two properties of a distribution function.
4. State how the mean and variance are obtained from the m.g.f.
5. If X is a Poisson variable with P(X =1) = P( X = 2), find the variance of X.
6. State the p.d.f. of Exponential distribution and state its mean.
7. State the conditions under which Binomial distribution tends to Poisson distribution.
8. Define Student’s ‘t’ distribution.
9. Define a Statistic with an example.
10. What are Type I and Type II errors?

 

Part B

Answer any five questions.                                                                                 5 X 8 = 40

 

1. State and prove addition theorem of probability for two events.
2. The probabilities of X, Y and Z becoming managers are 4/9, 2/9 and 1/3 respectively. The probabilities that Bonus scheme will be introduced of X, Y and Z becomes managers are 3/10, 1/2 and 4/5 respectively.

1. a) What is the probability that Bonus scheme will be introduced, and
2. b) If the Bonus scheme has been introduced, what is the probability that the

manager appointed was x?

1. Obtain mgf of Binomial distribution.
2. Derive the mean and variance of Gamma distribution.
3. The joint probability distribution of two random variables X and Y is given by

P(x = 0, y = 1) = 1/3, P(x = 1, y = -1) = 1/3 and P(x = 1, y = 1) = 1/3.

Find i). Marginal distributions of X and of Y ii). The conditional probability

distribution of X given Y = 1.

 

 

 

 

1. Calculate mean and standard deviation for the following p.d.f.:

f(x) = (3 + 2x)/18, for 2 ≤ x ≤ 4;    0, otherwise.

 

1. Calculate the mean and variance of Beta distribution of second kind.
2. Derive t – distribution.

 

Part C

Answer any two questions.                                                                               2 X 20 = 40

 

1. a) State and prove law of total probability and hence Baye’s theorem.

b). Obtain the m.g..f of Normal distribution.                                       (12 +8)

 

1. a) Derive the recurrence relation for the moments of Poisson distribution. Obtain   beta one and beta two.

1. b) Find the mean and variance of the distribution whose p.d.f. is

f(x) = 1 / ( b – a ), a < x < b.                                                           ( 12 + 8 )

 

1. a) Variables X and Y have the joint probability density function is given by

f(x, y) = 1/3 (x +y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.

i). Find coefficient of correlation between X and Y

1. b) Let X and Y have joint pdf:

e^{-(x + y)} x³y⁴

f(x, y) =                         ,  x > 0, y > 0

G₄ G₅

Find the p.d.f. of U = X / (X + Y).                                        (10 + 10)

 

1. a)  Derive chi – square distribution.

b). Derive the m.g.f of chi-square distribution and hence establish its additive

property.                                                                                         ( 12 + 8)

 

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