SSN COLLEGE OF ENGINEERING, KALAVAKKAM
DEPARTMENT OF MATHEMATICS
ASSIGNMENT ON RANDOM VARIABLES
PART A
1. If two random variables X and Y have probability density function (PDF)
f ( x, y)  ke( 2 x  y ) for x, y>0, evaluate ‘k’
2. Let X and Y be integer valued random variables with
P X  m, Y  n  q 2 p m n2 ; n, m  1, 2, . . . . . and p  q  1. Are X and Y
independent?
3. The p.d.f. of a random variable X is f(x) =2 x , 0 < x <1, find the p.d.f. of
Y=3X+1.
4. If X and Y are random variables having the joint density function
f x , y  
1
6  x  y , 0  x  2, 2  y  4, find P  Y  3
8
5. Can the joint distributions of two random variables X and Y be got if their
marginal distributions are known?
6. State the basic properties of joint distribution of (X, Y) where X and Y are
random variables
 x  y, 0  x  1, 0  y  1
check whether X
otherwise,
 0
7. If X and Y have joint pdf f ( x, y)  
and Y are independent.
8. Is the function defined as follows a density function?
0

1
f ( x )   ( 3  2x )
18
0

for 0  x  2
for 2  x  4
for
x4
9. Find the marginal density function of X and Y if f ( x, y )  2 x  5 y ,
2
5
0  x 1
0  y 1
.
10.A continuous random variable X has the p.d.f. f(x) given by
f(x)= {C e  x    x  . Find the value of C and C.D.F. of X.
1
11.If the joint p.d.f. of (X, Y) is f(x, y) =  4 , 0  x, y  2 find P( x  y 1).
 0 otherwise.
12.The following table gives the joint probability distribution of X and Y. Find
the marginal density functions of X and Y.
X
1
2
3
1
0.1
0.1
0.2
2
0.2
0.3
0.1
Y
13.If the joint pdf of the random variables (X, Y) is given by
f ( x, y)  kxye( x
2
 y2 )
, x  0, y  0, find the value of k.
14.Find k if the joint probability density; function of a bivariate random variable
k 1  x  (1  y) if 0  x, y  1
otherwise.
0

(X, Y) is given by 
15.Two random variable X and Y has joint probability density function
 xy if

f ( x, y )   96

0
0  x  4,
1 y  5
otherwise
Find E (XY).
16.The joint probability mass function of (X, Y) is given by P (x, y) = K (2x +
3y), x = 0, 1, 2; y= 1, 2, 3. Find the marginal probability distribution of X :
{i, Pi* }
Y
X
1
2
3
0
3K
6K
9K
1
5K
8K
11K
2
7K
10K
13K
17.A continuous random variable X that can assume any value between x=2 and
x=5 has a density function given by; f(x) = k (1+x). Find P [X<4].
18.A random variable X has to p.d.f. f(x) given by
Cx e  x , if x  0
Find the value of C and C.D.F. of X.
f ( x)  
if x  0.
 0,
19.If the joint p.d.f. of (X, Y) is given by f ( x, y)  e ( x y ) , x  0, y  0 Find
E ( XY).
20.Find the value of (a) C and (b) mean of the following distribution given
C ( x  x 2 ) for 0  x  1
f ( x)  
otherwise
 0
PART B
1. The joint probability mass function of random variables X and Y
is
p( x, y )
e  x p x q x y
; y  0,1 2, ....x and x  0,1, 2 . . . .where  0, 0  p  1, p  q  1
y! ( x  y )!
are constants. Find the marginal and conditional distributions.
2. Let the number X be selected from among the set of integers {1, 2, 3,
4}and the number Y be chosen from among those at least as X. Obtain
the joint PMF of (X, Y). Hence prove that covariance (X, Y) =5/8.
3. Two dimensional random variable (X, Y) has the joint PDF
f ( x, y )  8 xy, 0  x  y  1; 0 otherwise Find (1) marginal and conditional
distributions, and (2) Test whether X and Y are independent.
4. If X has the probability density f ( x)  ke3 x forx  0; find
k , P0.5  X  1and the mean of X.
5. If the joint density of X 1 and X 2 is given by
6 e 3 x1 2 x2
for x1  0, x2  0
find the probability density of
f x1 x2   
0
otherwise

Y  X 1  X 2 and its mean.
6. Random variables X and Y have the joint distribution

x
0  p  1 and   0 : px , y   e  .
y!
p y . q x y
, y  0,1, 2, . . . .x; x  1, 2, 3, . . .
 x  y !
Find the marginal and conditional distributions and evaluate P{X<1}.
7. Suppose the point probability density function (PDF) is given by


6
f  x, y    x  y 2 ; 0  x  1, 0  y  1
. Obtain the marginal PDF of X and
5
0
otherwise
1
3
that of Y. Hence or otherwise find p   y  .
4
4
8. The joint probability mass function (PMF) of X and Y is
P(x, y)
X
0
1
2
0
0.1
.04
.02
1
.08
.20
.06
.06
.14
.30
2
Compute the marginal PMF X and of Y, PX  1, Y  1 and check if X and Y are
independent.
9. The joint density function of random variables X and Y is
f ( x, y )  2, 0  x  y  1, find the marginal and conditional probability
density functions. Are X and Y independent?
10.The joint p.d . f . of X and Y is given by f x, y   e  x  y  , x  0, y  0, find the
probability density function of U 
X Y
2
11.If the joint pdf of a two dimensional random variable (X, Y) is given by
xy
; 0  x  1, 0  y  2
1
Find (i) P X   (ii) PY  X and
3
2

 0, elsewhere
f ( x, y)  x 2 

1
1
(iii) P Y  X   .
2
2

12.If X is the proportion of persons who will respond to one kind of mail
order solicitation, Y is the proportion of persons who will respond to
another kind of mail-order solicitation and the joint probability density of
2
X and Y is given by f ( x, y)   5 ( x  4 y ) for 0  x  1, 0  y  1, Find the


0
elsewhere
Probabilities that (i) at least 30% will respond to the first kind of mailorder solicitation.(ii) atmost 50% will respond to the second kind of
mail-order solicitation given that there has been 20% response to the first
kind of mail-order solicitation.
13.Suppose the probability density function ( pdf ) f ( x, y ) of ( x, y ) is given by
6

 x  y 2 0  x  1, 0  y  1 
f ( x, y )   5
 . Obtain the marginal pdf of X, the

0
otherwise


conditional pdf of Y given X = 0.8 and then E(Y/x=0.8).
14.Given is the joint distribution X and Y:
X
0
Y
1
2
0
0.02
0.08
0.10
1
0.05
0.20
0.25
2
0.03
0.12
0.15
Obtain Marginal Distributions and the conditional distribution of X given
Y = 0.
8 xy, 0  x  y 1
 0, otherwise.
15.Given the joint pdf of (X, Y) as f ( x, y)  
16.Find the marginal and conditional pdfs of X and Y. Are X and Y
independent?
17.The joint pdf of random variable X and Y is given by
  x 2  y 2 
 x  0 . Find the value of K and prove also that X
f ( x, y )  Kxye 
y0
and Y are independent.
18.The joint p.d.f. of a bivariate R.V. (X, Y) is given by:
k xy; 0  x 1, 0  y 1
Where k is a constant (1) Find the value of
otherwise,
 0,
f(x, y) = 
K (2) Find P(X+Y<1) (3) Are X and Y independent random variables.
Explain.
19.Let X and Y be independent standard normal random variables. Find the
p.d.f. of Z 
X
.
Y
20.If the Joint p.d.f. of random variables X and Y is
 x e  x (1 y ) , x  0 y  0
f(x, y) = 
0
otherwise
find f(y/x) and E(Y/X=x).
21.Let X and Y be independent uniform random variables over (0, 1).Find
the p.d.f. of Z=X+Y
22.If the joint density function of the two random variables X and Y be
f ( x, y)  e  ( x  y ) x  0, y  0
0
otherwise.
Find : (1) P( x  1) (2) P ( X  Y 1).
 cx( x  y ), 0  x  2,  x  y  x
Find (1)C, (2) The marginal
elsewhere.
 0,
23.Given f ( x, y)  
distributions f(x) and f(y) and (3)The conditional density of Y given X
f(y/x)
24.If X and Y each follow an exponential distribution with parameter 1 and
are independent, find the pdf of U=X-Y.
25.The diameter of an electric cable X is a continuous random variable with
pdf f(x)=kx(1-x), 0  x  1. Find (A) the value of k (B) the cumulative
distribution function of X (C) P (X  1/ 21/ 3  X  2 / 3).
26.If X and Y are independent random variable with pdf e x , x  0
and e  y , y  0 find the density function of U 
X
andV  X  Y . Are they
X Y
independent?
27.If the joint pdf of a random variable (X, Y) is given by
f ( x, y )  x 2 
xy
, 0  x  1, 0  y  2, find the conditional densities of X given
3
Y and Y given X.
28.The pdf of X and Y is given by f ( x, y)  kxye( x  y ) , x  0, y  0.
29.Find k and prove that X and Y are independent
2
2
30.X and Y are two random variables having joint density function
1
f ( x )  (6  x  y ) 0  x  2 , 2  y  4
8
0
otherwise
Find (1) P ( X  1  Y  3) (2) ( X  Y  3) (3) P ( X  1 / Y  3).
31.Two random variables X and Y have the following joint probability
2  x  y, 0  x  1, 0  y  1
Find (1)
0,
otherwise.

density function f(x, y) = 
Marginal probability density functions of X and Y (2) Conditional
density functions (3) var (X) and var (Y).
32.Let (X, Y) be a two-dimensional non-negative continuous random
 4 xy
variable having the joint density. f ( x, y )  
0,
e ( x
2
 y2 )
, x  0, y  0
elsewhere ,
Find the density function of U  X 2  Y 2 .
33.The joint p.d.f. of R.V.s X and Yis given by
3 ( x  y), 0  x  1, 0  y  1, x  y  1
Find the marginal p.d.f. of X ,
f ( x, y)  
0,
othersise

P(X+Y < ½),Cov (x, Y).
34.The joint p.d.f. of R. vs X and Y is given by
1
f(x) =  y
e  x / y e  y ; x  0, y  0

 0,
FindP( X  1 / Y  y ).
otherwise
35.The random variables X and Y have joint p.d.f.


f ( x, y )  


x2 
0,
xy
,
0  x  1, 0  y  2 Are X and Y
3
otherwise
independent? Find the conditional p.d.f. of X given Y.
36.Suppose X and Y are two random variables having the joint p.d.f.
4 xye ( x  y ) , x, y  0
f ( x, y )  
. Find the p.d.f of

0 otherwise
2
2
z
X 2 Y 2
37.In producing gallium – arsenide microchips, it is known that the ratio
between gallium and arsenide is independent of producing a high
percentage of workable wafer, which are main components of
microchips. Let X denote the ratio of gallium to arsenide and Y denote
the percentage of workable mierowafers retrieved during a 1-hour
period. X and Y are independent random variables with the joint density;
 x(1  3 y 2 )
, 0  x  2, 0  y  1 Show that E
being known as f ( x, y)  
4

0
, otherwise

(XY)=E(X). E(Y).
38.If the joint density of X 1and X 2 given by
6.e 3 x1 2 x2 forx1  0, x2  0
find the probability density of
f ( x1 , x2  
otherwise
 0,
Y  X 1  x2
39.Two random variables X and Y have joint density function
f XY ( x, y )  x 2 
xy
; 0  x 1 y  y  2
Find the conditional density functions.
3
Check whether the conditional density functions are valid
40.If the joint probability density of X 1 and X 2 is given by
 e (  x1  x2 ); for x1  0, x2  0
X1
find the probability of Y 
f ( x1 , x2 )  
.
X1  X 2
otherwise
0,