B.Sc. DEGREE EXAMINATION –STATISTICS
SUPPLEMENTARY EXAMINATION – JUNE 2007
ST 4201 - MATHEMATICAL STATISTICS
Date & Time: 25/06/2007 / 9:00 - 12:00 Dept. No. Max. : 100 Marks
PART A
Answer all the questions. 10 X 2 = 20
1.
Define mutually exclusive events
2.
A box contains 3 black and 2 blue balls. If 2 balls are drawn at random, find the probability that 2 of them are black.
3.
Define Axiomatic probability.
4.
What is distribution function?
5.
Define the covariance of any two random variables X and Y. What happens when they are independent?
6.
State any two properties of expectation.
7.
Define Rectangular distribution.
8.
State any two application of t- distribution.
9.
State any two properties of chi-square distribution.
10.
What is consistent estimator?
PART B
Answer any five questions. 5 X 8 = 40
11.
An urn contains 6 white, 4 red and 3 green balls. If 3 balls are drawn at random, find the probability that: a). two of the balls drawn are white. b). one is of each colour. c). none is red. d). atleast one is white.
12.
State and prove Boole’s inequality.
13.
The contents of Urns, I, II and III are as follows i). 1 white, 2 black and 3 red balls ii). 2 white, 1 black and 1 red balls iii). 4 white, 5 black and 3 red balls
One urn is chosen at random and 2 balls are drawn from it. They happen to be white and red. What is the probability that they come from urns I, II and III.
14.
Establish the recurrence relation for the central moments of Binomial distribution.
15.
Derive the mgf of N(
 , σ 2
)
16.
If random variable X and Y have the following joint p.d.f f(x,y) = x + y, 0 < x <1, 0 < y < 1, zero elsewhere then show that the correlation coefficient between X and Y is – 1/11
17.
Obtain mean and variance of first kind of Beta distribution.
18.
Write short notes on i) Statistic ii). Unbiased estimator iii). Maximum likehood estimation iv). Type I and Type II errors

PART C
Answer any two questions.
19.
a). State and prove Baye’s theorem. b). A random variable has the following probability function:
Value of x: -2 -1 0 1 2 3
P(x) : 0.1 k 0.2 2k 0.3 k
2 X 20 = 40 i). Find the value of k and find mean and variance. ii). Construct cumulative distribution function of X. iii). What is the smallest value of x when the probability value is 0.7. (8+6+4+2)
20.
If two random variables X and Y have the following p.d.f f(x,y) = 2-x-y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, zero elsewhere, find, a). Marginal probability density function of X and Y. b). Conditional density function of Y given X = x and X given Y = y. c). Variances of X and Y. d). Covariance between X and Y. (4+4+8+4)
21.
a). Show that Binomial distribution tends to Poisson distribution under some conditions.
(10) b). X is normally distributed and mean of X is 12 and SD is 4. Find out the probability of the following: i). X ≥ 20 , X ≤ 20 and 0 ≤ X ≤ 12. ii). Find x
1
, when p(X>x
1
) = 0.24 iii). Find x
0
'
and x
1
'
, when p(x
0
'
< X < x
1
'
) = 0.50 and p(X > x
1
'
) = 0.25. (4+2+4)
22.
a). Obtain mean and variance of exponential distribution. b). If X and Y are independent Gamma variates with parameters

and v respectively, show that the variables U = X + Y, Z = X / (X+Y) are independent and that U is a

(

+v) variate and z is a

1
(

,v) variate. (8+12)
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