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TWO DIMENSIONAL RANDOM VARIABLES

PART = A

1.The joint pdf of two random variables X and Y is given by fxy(x,y) = 1/8x(x-y) ; 0 < x < 2; -x 12 / X = 1/2 ) A.U 2000,

4. X and Y are two random variable having joint function

f(x,y) = 1/8 (6-x y ) 0 < x < 2, 2< y < 4

0 , otherwise

Find the (i) P( X < 1 Y < 3 ) (ii) P ( X + Y < 3 ) (iii) P ( X < 1/Y < 3 ) [ A.U A/M 2003

5. The joint p.d.f of a R .V (X,Y ) is given by,

f(x,y) = 4xy , 0 < x

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8. Find the covariance of the two random variables whose p.d.f is given by [A.U. May 2000]

f(x,y)= 2 for x > 0 , y> 0 , x+ y < 1

0, otherwise

9.Calculate the correlation co-efficient for the following heights ( in inches) of fathers X theirsons Y .[A.U. N/D 2004]

X: 65 66 67 67 68 69 70 7267 68 65 68 75 72 69 71

10.Suppose that the 2D RVs ( X,Y ) has the joint p.d.f

f(x,y) = x+y, 0 < x < 1, 0 < y < 1

0 , otherwise

Obtain the correlation co-efficient between X and Y .

11. Two independent random variables X and Y are defind by

f( x) = 4ax , 0 x 1

0 , otherwise

Show that U = X + Y and V = X Y are uncorrelated. [A.U. A/M 2003]

12.A statistical investigator obtains the following regression equations in a survey:

X Y 6 = 0 and 0.64 X + 0.48 = 0 .

Here X = are of husband and Y = age of wife. Find

(i) Mean of X and Y

(ii) Correlation coefficient between X and Y and

(iii) sy = A.D . of Y if sy = S.D of X = 4.

13. The random variable [X,Y ] has the following joint p.d.f

f(x,y) = (x + y ) , 0 x n 2 and 0 y 2

0, otherwise

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(1) Obtain the marginal distribution of X.

(2) E(X) and E ( X2)

(3) Compute co-variance (X,Y ) [ A.U A/M 2005]

14 . Find the Cor ( x,y ) for the following discrete bi variate distribution

X 515

10 0.20.4

20 0.30.1

15. Find the coefficient of correlation and obtain the lines of regression from the data givenbelow: [A.U. N/D 2003]

X62

64 65 69 70 71 72 74

Y126

125 139 145 165 152 180 208

16.Following table gives the data on rainfall and discharge in a certain river. Obtain the line ofregression of y on x . [A.U. May,99]

Rain fall( inches) (X) 1.53 1.78 2.60 2.95 3.42

Discharge ( 1000 C.C ) (Y) 33.5 36.3 40.0 45.8 53.5

17. For the following data find the most likely price at Madras corresponding to the price 70 atBombay and that at Bombay corresponding to the price 68 at Madras.

city

Madras Bombay

Average price65 67

S.D of price 0.5 3.5

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S.D of the difference between the price at Madras and Bombay is 3.1?

18.If X and Y are independent random variables each normally distributed with mean zero andvariance s2 , find the density function of R = X 2 + Y2 and j tan-1 (Y/X ). [A.U. Dec 03]

19.If X and Y are independent random variables each following N(0,2) , find the probabilitydensity function of Z = 2X + 3Y . [A.U A/M 2003]

20. A random sample of size 100 is taken from a population whose mean is 60 and the varianceis 400 . Using CLT , with what probability can we assert that the mean of the sample will notdiffer from m = 60 by more than 4 ?(A.U. A/M 2003)