LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

## FOURTH SEMESTER – NOVEMBER 2003

**ST-4501/STA503 – DISTRIBUTION THEORY**

31.10.03 Max:100 marks

9.00-12.00

__SECTION-A__

__Answer ALL the questions.__ (10×2=20 marks)

- Let f(x,y) = e

0 else where.

Find the marginal p.d.f of X.

- Let the joint p.d.f of X
_{1}and X_{2}be f(x_{1},y_{1}) = and x_{2}= 1, 2.

Find P(X_{2} = 2).

- If X ~ B (n, p), show that E
- If X
_{1}andX_{2}are stochastically independent, show that M (t_{1}, t_{2}) = M (t_{1}, 0) M (0, t_{2}), ” t_{1, }t_{2}. - Find the mode of the distribution if X ~ B .
- If the random variable X has a Poisson distribution such that P (X = 1) = P (X = 2),

Find p (X = 4).

- Let X ~ N (1, 4) and Y ~ N (2, 3). If X and Y are independent, find the distribution of

Z = X -2Y.

- Find the mean of the distribution, if X is uniformly distributed over (-a, a).
- Find the d.f of exponential distribution.
- Define order statistics based on a random sample.

__SECTION-B__

__Answer any FIVE questions.__ (5×8=40 marks)

- Let f(x
_{1}, x_{2}) = 12

0 ; elsewhere

Find P .

- The m.g.f of a random variable X is

Show that P (= .

- Find the mean and variance of Negative – Binomial distribution.
- Show that the conditional mean of Y given X is E (Y÷X=x)for trinomial
- Find the m.g.f of Normal distribution.

- If X has a standard Cauchy distribution, find the distribution of X
^{2}. Also identify its

distribution.

- Let (X, Y) have a bivariate normal distribution. Show that each of the marginal

distributions is normal.

- Let Y
_{1}, Y_{2}, Y_{3}andY_{4}denote the order statistics of a random sample of size 4 from a

distribution having a p.d.f.

f(x) = 2x ; 0 < x < 1

0 ; elsewhere . Find p .

__SECTION-C__

__Answer any TWO questions.__ (2×20=40 marks)

- Let x (X
_{1}, X_{2}) be a random vector having the joint p.d.f.

f (x_{1}, x_{2}) = 2 ; 0 < x_{1} < x_{2} <1

0 ; elsewhere

(i) Find the correlation between x_{1} and x_{2 }(10)

(ii) Find the conditional variance of x_{1} / x_{2 }(10)

- a) Find the mean and variance of hyper – geometric distribution. (10)
- b) Let X and Y have a bivarite normal distribution with

Determine the following probabilities

- i) P (3 < Y <8) ii) P (3 < Y< 8 ½X =7) (10)
- i) Derive the p.d.f of ‘t’ – distribution with ‘n’ d.f (10)
- ii) If X
_{1}and X_{2}are two independent chi-square variate with n_{1}and n_{2}f. respectively,

show that (10)

- i) Let Y
_{1}, Y_{2}and Y_{3}be the order statistics of a random sample of size 3 from a

distribution having p.d.f.

1 ; 0 < x < 1

f (x) =

0 ; elsewhere.

Find the distribution of sample range. (10)

ii)Derive the p.d.f of F variate with (n_{1}, n_{2}) d.f. (10)