Download Unit II - computerscience2ndyear

UNIT-II
2 Marks
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 < y <x and otherwise fin fy/x = (y/x)[A.U.model
2. State the basic properties of joint distribution of (X,Y) when X andY are random
variables.[A.U.A/M 2005]
3. If the point pdf of (X,Y) is given by
f( x,y) = e-(x+y) , x 0, Y 0 find E [XY].[A.U A/M 2005]
4.If X and Y are random variables having the joint density function f(x,y) = 1/8(6x-y) , 0< x <2, 2 < y <4, find P(X+Y<3)[A.U.A/M 2003]
5.Find the marginal density function of X, if the joint density function of two
continuous random variable X and Y is
f(x) =2(2-x-y), 0 x  y  1
0 , Otherwise
6.If the joint pdf of a 2D R V (X,Y) is given by
f(x,y) = 2, 0< x < 1; 0< y < x <1.
0, otherwise
Find the marginal density function of X and Y
7.If the joint density function of the two R V s ‘X ; and ‘Y ‘ be
f(x,y) = e-(x+y) X  0, Y0
0, otherwise
Find (i) P( X < 1) and (ii) P( X+ Y < 1) [A.U. N/D 2003]
8. Find ‘K ‘ if the joint probability density function of a bivariate random variable
(X,Y) is given by
f(x,y) = K (1-x) (1-y) if 0< x < 4 ; 1 < y < 5
0, otherwise
9.Show the Cov 2 (X,Y)  Var ( X) . Var(Y) [A.U. N/D 2004]
10. The two equations of the variables X and Y are x = 19.13 –0.87 y and y =
11.64 – 0.50 x. Find the correlation co – efficient between X and Y .[AU, May ‘99]
11.Find the acute angle between the two lines of regression [A.U A/M 2003]
12.Can Y = 5 + 2.8 x and X = 3 – 0.5 y be the estimated regression equations of Y
on X and X on Y respectively? Explain your answer.[A.U. Nov 2007]
13.The tangent of the angle between the lines of regression of y on x and x on y is
0.6 and x = ½ y , find the correlation coefficient between x and y.
14. Two random variables X and Y are related as Y = 4X+9 . Find the coefficient
between X and Y[ A.U. 2007]
15.State the equations of the two regression lines. What is the angle between
them.[ A.U N/D 2003]
16.If X and Y are linearly related find the angle between the two regression
lines.[A.U A/M 2004]
17.X and Y are independent random variables with variance 2 and 3 . Find the
variance of 3X + 4Y .[A.U.A/M 2003]
18. State the central limit theorem for independent and identify distributed random
variables.[ A.U A/M 2005]
19. Write the note on ‘ Central limit theorem’ [A.U. N/ D 2005]
20.Distinguish between correlation and regression.
16 Marks
1. The joint probability mass function (PMF) of X and Y is
P(x,y) 0 1 2
X 0 0.1 .04 .02
1 .08 .20 .06
2 .06 .14 .30
Compute the marginal PMF X and Y , P(X  1, Y  1 ) and check if X and Y are
independent . [A.U N/ D 2004]
2. If the joint density function of the two R V s ‘X ; and ‘Y ‘ be
f(x,y) = e-(x+y) X  0, Y0
0, otherwise
Find (i) P( X < 1) and (ii) P( X+ Y < 1) [A.U. N/D 2003
3.If X and Y have the joint p.d.f
¾+ xy, 0 < x < 1, 0 < y < 1
f(x,y) = 0 , otherwise
Find f(y/x ) and P( ( y > 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 <1, 0< y < 1
0, otherwise
Find P( X + Y < 1) [ A.U N/D 2005]
6. Find the marginal density function of X , if the joint density function of two
continuous random variable X and Y is
f(x,y) = 2 ( 2-x –y ), 0 x  y  1
0 , otherwise
7. If X,Y and Z are uncorrelated RV’s with zero mean and S.D 5, 12 and 9
respectively, and if U = X + Y and V = Y + Z , find the correlation coefficient
between U and V .[ A.U Model]
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 their sons Y .[A.U. N/D 2004]
X: 65 66 67 67 68 69 70 72
67 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) y = A.D . of Y if y = S.D of X = 4.
13. The random variable [X,Y ] has the following joint p.d.f
f(x,y) = ½ (x + y ) , 0 x n 2 and 0 y 2
0, otherwise
(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 5 15
10 0.2 0.4
20 0.3 0.1
15. Find the coefficient of correlation and obtain the lines of regression from the
data given below: [A.U. N/D 2003]
X 62 64 65 69 70 71 72 74
Y 126 125 139 145 165 152 180 208
16.Following table gives the data on rainfall and discharge in a certain river.
Obtain the line of regression 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 at Bombay and that at Bombay corresponding to the price 68 at Madras.
Madras Bombay
Average price 65 67
S.D of price
0.5 3.5
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 and variance 2 , find the density function of R =  X2 + Y2 and  tan1 (Y/X ). [A.U. Dec 03]
19.If X and Y are independent random variables each following N(0,2) , find the
probability density 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 variance is 400 . Using CLT , with what probability can we assert that the mean
of the sample will not differ from  = 60 by more than 4 ?(A.U. A/M 2003)