LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

**B.Sc.** DEGREE EXAMINATION – **MATHS & PHYSICS**

FOURTH SEMESTER – **APRIL 2012**

# ST 4206/4201 – MATHEMATICAL STATISTICS

Date : 19-04-2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

__SECTION A__

**Answer all the questions. 10 X 2 = 20**

- Define random experiment.
- What are independent events?
- If Var(X) = 4, find Var(3X + 8).
- Define continuous uniform distribution and write its mean and variance.

- State any two applications of t-test.

- Write down the mean and variance of Binomial distribution.
- Define exponential distribution.
- Write any two properties of regression coefficients.
- What is null hypothesis?
- Define critical region.

__SECTION B__

**Answer any five questions. 5 X 8 = 40**

- An urn contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the probability that
- Two of the balls drawn are white.
- One ball of each colour is drawn.

- None is red.

- At least one is white
- If p
_{1}=P(A), p_{2}=P(B) and p_{3}=P(AÇB), (p_{1}, p_{2},p_{3}>0); express the following in terms of p_{1}, p_{2}and p_{3}. , P(A/B), and - A random variable X has the following probability function:

x : 0 1 2 3 4 5 6 7

p(x) : 0 k 2k 2k 3k k^{2} 2k^{2} 7k^{2} + k

Find k, evaluate P(X<6).

- State any four properties of Distribution function.
- Find mean and variance of Poisson distribution.
- Derive the r
^{th}order moments of Rectangular distribution and hence find standard deviation. - Obtain the line of regression of Y on X for the following data:

X: 65 66 67 67 68 69 70 72

Y: 67 68 65 68 72 72 69 71

- What are the steps involved in solving testing of hypothesis problem?

(PTO)

__ __

__SECTION C__

**Answer any two questions. 2 X 20 = 40**

- a) State and prove addition theorem of probability.
- b) Sixty percent of the employees of XYZ Corporation are college graduates. Of these, ten are in sales. What is the probability that
- An employee selected at random is in sales?
- An employee selected at random is neither in sales nor a college graduate? (10+5+5)
- The joint probability density function of a two-dimensional random variable (X,Y) is given by:
- Verify that whether f(x,y) is a joint p.d.f.
- Find the marginal density functions of X and of Y
- Find the conditional density function of Y given X=x and conditional density function of X given Y=y.
- Check for independence of X and Y. (5+6+6+3)
- a) A manufacturer, who produces medicine bottles, finds that 0.1 % of the bottles are defective. The bottles are packed in boxes containing 500 bottles. A drug manufacturer buys 100 boxes from the producer of bottles. Using Poisson distribution, find how many boxes will contain:
- no defective.
- atleast two defectives. (5+5)
- Derive mean and variance of Beta distribution of first kind. (10)
- Derive the p.d.f. of the F-statistic with (n
_{1}, n_{2}) degrees of freedom. (20)

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