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SRI VENKATESWARA COLLEGE ENGINEERING
DEPARTMENT OF APPLIED MATHEMATICS
MA 2262 PROBABILITY AND QUEUEING THEORY
II- TWO DIMENSIONAL RANDOM VARIABLES
PART-A
1) Define joint probability density function of two random variables X and Y and state its properties.
2) The joint p.m.f of (X,Y) is given by P(x,y)= k(2x+3y) , x = 0,1,2;y = 1,2,3. Find all marginal probability distributions.
Also find the probability distribution of (X+Y).
3) If the joint pdf of (X,Y) is given by f ( x, y)  e  ( x y ) , x  0, y  0 . Find E(XY).
4) Find the value of k if f(x,y) = k(1-x)(1-y) for 0<x,y<1is to be density function.
5) Find the value of A if the joint pdf f ( x, y)  A y e  x , x  0, 0  y  2 .
(1  x 2 )(1  y 2 ) if x, y  0
6) If F ( x, y)  
, then find the joint pdf.
 0 otherwise
7) Prove that COV 2 ( X , Y ) Var ( X )  Var (Y )
8) If X 1 has mean 4 and variance 9. While X 2 has mean -2 and variance 5 and the two independent. Find their Var(2 X 1 + X 2 -5).
9) Find the acute angle between the two lines of regression.
10) Distinguish between Correlation and Regression.
11) Let X and Y joint distributed with the correlation coefficient r(x,y) = ½ ,  x  2,  y  3 . Find Var(2X- 4Y+3).
12) Find the mean of x and y given two regression lines are x+6y = 4 and 2x+3y =1
PART-B
13) The joint probability distribution of a pair (X,Y) is random variable is
Y
1
2
3
X
1
0.1
0.1 0.2
2
0.2
0.3
0.1
Find the following (i) The conditional distribution of X given Y=yj (ii) the conditional distribution of Y given X = xi.
(iii) P(X  2) (iv) P(Y  3) (v) P(X+Y<4).
14) Three balls are drawn at random with out replacement from a box containing 2 white,3 re, 4 black balls. If X denotes
The number of white balls drawn and Y denotes the number of red balls drawn, find the joint probability distribution of
(X,Y).Also verify X and Y are independent or not.
 2 xy
 x  , 0  x  1,0  y  1
15) Let X and Y be continuous random variables with joint pdf f ( x, y )  
. Find the conditional
3
 0 otherwise
density functions and show that X and Y are not independent.
c x ( x  y ) , 0  x  2,  x  y  x
16) If the joint density function of X and Y is f ( x, y )  
otherwise
0
 y
(i)Find c (ii) find the marginal density functions of X and Y(iii) find f Y   .
x
X 
2, 0  y  x  1
17) Two random variable (X,Y) has the joint pdf f ( x, y )  
. (i) Find the Marginal& Conditional density
0 otherwise
functions (ii) Find the distribution function F(x,y) (iii) Find P[X<1/2 /Y<1/4] (iv) Are X and Y independent?
 2 x2
,0  x  2, 0  y  1
 xy 
18) The joint pdf of a two random variables (X,Y) is given by f ( x, y )  
8
 0
otherwise

Compute (i) P(X>1),(ii)P(Y<1/2 ),(iii)P(X>1/Y<1/2),(iv)P(Y<1/2/X>1)(v)P(X<Y)(vi)P(X+Y≥1).
c (6  x  y), 0  x  2, 2  y  4
19) If X and Y are two random variables having joint density function f ( x, y)  
otherwise
 0
Find (i)c (ii) P(X<1∩Y<3), (iii)P(X<1/Y<3), (iv)P(X+Y<3), (v) P(5/2< Y<7/2).
2
2
k xy e  ( x  y ) , x, y  0
20) The joint pdf of a two random variables (X,Y) is given by f ( x, y )  
 0
otherwise
Find the value of (i) k(ii) find the conditional density functions(iii) Are X and Y independent?
8
 xy ,0  x  y  2
21) The joint pdf of a two random variables (X,Y) is given by f ( x, y )   9
.(i) find the conditional

0
otherwise
density function of Y given X=x (ii) find the conditional density function of X given Y = y.
22) Calculate the correlation coefficient of the following heights (in inches) of fathers(X) and their sons(Y) .
X : 65 66 67 67 68 69 70 72
Y : 67 68 65 68 72 72 69 71
x  2y
, x  1,2, y  1,2

23) Let X and Y be two discrete random variables with joint p.m.f P( X  x, Y  y )   18
 0
otherwise
Find the correlation coefficient rXY .
2  x  y, 0  x, y  1
24) Two random variables X and Y have the joint pdf f ( x, y )  
elsewhere
0
1
Show that correlation coefficient  XY 
.
11
 x  y, 0  x  1, 0  y  1
25) Let the random variable X and Y have joint pdf f ( x, y )  
elsewhere
 0
Find the correlation coefficient between X and Y.
26) If X ,Yand Z are uncorreleated R.V’s with zero mean and standard deviations 5,12,9 respectively and if U=X+Y, V=Y+Z.
Find the correlation coefficient between U and V.
27) LetX1and X2 be two independent random variables with means 5 and 10 and standard deviations 2 and 3 respectively.
Obtain the correlation coefficient between U = 3X1+ 4X2 and V = 3X1-X2.
28) If X and Y have a bivariate normal distributions and U = X+Y and V = X-Y, find an expression for the correlation coefficient of
U and V.
 xy
 , 0  x  4,1  y  5
29) Two random variables X and Y have the joint pdf f ( x, y )   96
. Find (i) E(X), (ii) E(Y) ,(iii)Var(X),
 0 otherwise
(iv) Var(Y) (v) E(XY), (vi)E(2X+3Y) ,(vii) COV(X,Y). What can infer from COV(X,Y).
30) From the following data
Marks in Mathematics 25 28 35 32 31 36 29 38 34 32
Marks in Statistics
43 46 49 41 36 32 31 30 33 39
find (i) the two regression equations(ii) the coefficient of correlation between the marks in mathematics and statistics
(iii) the most likely marks un statistics when marks in mathematics are 30.
31) Two regression lines are 4x-5y + 33 = 0 and 20x -9y = 107 and variance of x = 25. Find the means of x and y. Also find
the value of rXY .
x  y
, 0  x  1 ,0  y  2

32) If the joint pdf of f ( x. y )   3
. Obtain the regression of Y on X and X on Y.
 0
otherwise
3 2
2
 ( x  y ) , 0  x, y  1
33) Let X and Y be continuous random variables with pdf f ( x, y )   2
. Find the regression curves.
 0 otherwise
34) The regression equations of two random variables X and Y is x 
5
33
20
107
y
 0 and y 
x
 0 . The S.D of X is 3.
4
5
9
9
Find the S.D of Y.
4 xy , 0  x, y  1
35) Let X and Y be continuous random variables with joint pdf f ( x, y )  
.Find the joint probability
 0 otherwise
distributions of X 2 and XY .
36) The joint pdf of X and Y is given by f ( x, y)  e ( x  y ) , x  0, y  0 . Find the probability density function
X Y
of U 
.Are X and Y independent?.
2
37) Let X and Y be independent (strictly positive) exponential random variables with parameter . Are the random
X
variables X  Y and
independent?
Y
38) If X and Y are independent uniform variables over (0,1).Find the pdf of Z = X + Y.
39) If X and Y are independent random variables each normally distributed with mean zero and variance  2 .Find the
density functions of r  x 2  y 2 and   tan 1 ( y / x) .
 x  y, 0  x  1, 0  y  1
40) Let the random variable X and Y have joint pdf f ( x, y )  
.Find the pdf of XY.
elsewhere
 0
41) State and prove Central limit theorem.
42)A random sample of size 100is taken from a population whose is 60 and variance is 400.Using central limit theorem,
with what probability can we assert that the mean of the sample will not differ from μ = 60 by more than 4?.
43)The life time of a certain brand to tube light may be considered as a random variable with mean 1200 hrs and S.D 250 hrs.
Find the probability, using central limit theorem , that life time of 60 lights exceeds 1250 hrs.
44) A distribution with unknown mean  has variance equal to 1.5. Use central limit theorem to fine how large a sample should be
taken from the distribution in order that the probability will be with in   0.5.
45) If 20 dice are thrown, then Find the approximate probability that the sum obtained is between 65 and 75 by central limit
theorem.
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