SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
Assignment – II
B.Tech. - ECE
Course Category: Programme Core
Course Code/Course Title: 1151EC101 /Mathematics for E & C Max. Marks: 5
Semester: Even
Date of Submission:18-04-2018
Achievable Course Outcomes
CO3: Examine the random experiments specified by two random variables and study the Distribution of
their distributions
CO4: Determine covariance and spectral density of stationary random processes
CO5 : Derive numerical methods for various mathematical Operations and tasks, such as interpolation,
differentiation, integration, the solution of linear and nonlinear equations, and the solution of differential
equations. -K3 Level
S. Course
Assignment Topics/Questions-Written Assignment Marks Level
No. Outcome
a) The joint distribution of X and Y is given by
x+ y
f ( x, y ) = , x = 1, 2,3...; y = 1, 2,3... Find the marginal distribution.
21
1 b) Apply Gauss-seidel method to solve the equations CO3, CO5 K3
5
20x+y-2z=17: 3x+20y-z=-18: 2x-3y+20z=25.
a) Suppose the point probability density functions is given by
�6
� ( x + y ),0 �x �1, 0 �y �1
2
f ( x, y ) = �5
�
�0, otherwise
2 1 3 5 CO3, CO5 K3
Obtain the marginal pdf of X and that of Y and hence find P [ �y � ]
4 4
.
b) Find a positive root of 3x- log 10 X =6, using fixed point iteration
method.
3 a) Let X and Y have joint probability density 5 CO3, CO5 K3
f ( x, y ) = 2 0 < x < y < 1 Find the marginal distribution of f also find the
conditional density function of y given X=x
CC HOD
SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
b) Find the smallest positive root of the equation x = sin x
correct to 3 decimal places using Newton-Raphson method
a) The joint pdf of the random variable ( x , y ) is given by
2 2
f ( x, y ) = k x y e - ( x + y ) , x > 0, y > 0.
4 Find the value of k and prove that X and Y are independent. 5 CO3, CO5 K3
b) Solve by Gauss-seidel iterative procedure the system 8x-3y-2z=20:
6x+3y+12z=35: 4x+11y-z=33
�2
� (2 x + 3 y ), 0 �x �1, 0 �y �1
a) Show that the function f ( x, y ) = �5
�
�0,otherwise
5 Is a joint probability density function of X and Y . 5 CO3, CO5 K3
b) Find a real root of the equation x3+x2-1=0 by iteration method.
a) If the joint probability density function of x and Y is given by
�1
� (6 - x - y ), 0 < x < 2, 2 < y < 4
f ( x, y ) = �8 Find
�
�0 otherwise
6 (i ) P( x < 1 and y < 3)
5 CO3, CO5 K3
(ii ) P( x < 1/ y < 3)
(iii ) P ( x + y < 3).
b) Using Newton’s method, find the real root of x log 10 X=1.2 correct
to five decimal places
7 a) The joint density function of the random variable X and Y is 5 CO3, CO5 K3
given by
�8 xy ) 0 < x < 1, 0 < y < x 1 1
f ( x, y ) = � Find P ( y < / x < ) . Also find the
�0 otherwise 8 2
conditional density function f ( y / x ).
b) Apply Gauss elimination method to find the solution of the
following system :
CC HOD
SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
2x+3y-z=5: 4x+4y-3z=3: 2x-3y+2z=2.
a) Two random variables X and Y have the joint density function
(2 - x - y ), 0 < x < 1, 0 < y < 1
�
f ( x, y ) = �
0
� otherwise
-1
Show that cor ( x, y ) = .
8 144 5 CO3, CO5 K3
b) Find an iterative formula to find , where N is a positive number
and hence find
a) Suppose that the 2D random variables ( X , Y ) has the joint
probability function
�x + y, 0 < x < 1,0 < y < 1
f ( x, y ) = �
�0 otherwise
Obtain the correlation coefficient between X and Y and check
9 5 CO3, CO5 K3
whether X and Y are independent.
b) Solve the following system of equations by Gauss-Jacobi method:
27x+6y-z=85:x+y+54z=110: 6x+15y+2z=35.
a) Let X , Y and Z be uncorrelated r.v. with zero means and
standard deviation 5,12 and 9 respectively. If U = X + Y and
V = Y + Z ,find the correlation coefficient between U and V .
10 5 CO3, CO5 K3
b) Apply Gauss-Jacobi method to find the solution of the following
system: 10x+y+z=12: 2x+10y+z=13: x+y+5z=7
a) The two regression lines are 4 x - 5 y + 33 = 0 and 20 x - 9 y = 107 ,
variance of x = 25 .Find the means of x and y .
b) Find the negative root of x2 + 4sinx = 0 by Newton’s
11 Raphson method 5 CO3, CO5 K3
CC HOD
SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
a) The two regression lines are 8 x - 10 y + 66 = 0 and
40 x - 18 y - 214 = 0 , variance of x = 9 .Find the means of x and y
,correlation coefficient between x and y .
b) Solve the following system of equation by Gauss - seidel
12 5 CO3, CO5 K3
method
a) Consider the random process
where is uniformly Distributed in the interval
to check whether is stationary or not? Find
13 5 CO3, CO5 K3
the first and second moments of the process
b) Solve the following system of equations by Gauss-Seidel
method:
14 5 CO3, CO5 K3
a) Show that if the process
is SSS, where ‘a’ and ‘b’ are independent random
variable then they are normal.
b) Solve by Gauss elimination method:
CC HOD
SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
a) The process whose probability distribution
under certain condition is
given by
15 5 CO3, CO4 K3
Show that it is “not stationary”.
b) Use Newton’s method to find a real root of the equation +
x -1=0
CC HOD
SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
a) Consider the random process
where A and are
Independent variables. A is a random variable with
mean 0 and variance 1. is uniformly distributed in
16
Find mean and auto correlation and Hence 5 K3
show that is WSS. CO3, CO5
b) Using Lagrange’s interpolation formula, find from the
following table.
3 7 9 10
168 120 72 63
a) Show that the poisson process is a Markov process.
b) Solve the following system of equations by Gauss-Seidel
method:
17 5 CO3, CO5 K3
a) A customer arrive at a certain in accordance with a
poisson process with a mean rate of 2 per min.
b) Find the probability that the interval between 2
consecutive arrivals is a)more than 1 min b)between
18 1 min and 2 min c) 4 min or less 5 CO3, CO5 K3
c) Use Newton’s method to find a real root of the equation
+ x - 1 = 0.
CC HOD
SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
a) If the tpm of a markov chain is find the
19 steady state distribution of the chain. 5 CO3, CO4 K3
b) Solve by Gauss elimination method:
a) The transition probability matrix of the markov
chain } n=1,2,.. having 3 states 1,2 and 3 is
and the initial distribution is
Find
20 5 CO3, CO4 K3
(i) =3 and (ii)
b) Using Lagrange’s interpolation formula, find from the
following table.
3 7 9 10
168 120 72 63
21 a) Consider the random process 5 CO3, CO4 K3
where A and are
Independent variables. A is a random variable with
mean 0 and variance 1. is uniformly distributed in
CC HOD
SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
Find mean and auto correlation and Hence
show that is WSS.
b) Using Lagrange’s interpolation formula, find from the
following table.
3 7 9 10
168 120 72 63
a) The two regression lines are 8 x - 10 y + 66 = 0 and
40 x - 18 y - 214 = 0 , variance of x = 9 .Find the means of x and y
,correlation coefficient between x and y .
b) Solve the following system of equation by Gauss - seidel
22 5 CO3, CO5 K3
method
c) The joint distribution of X and Y is given by
x+ y
f ( x, y ) = , x = 1, 2,3...; y = 1, 2,3...
21 Find the marginal distribution.
23 5 CO3, CO5 K3
d) Apply Gauss-seidel method to solve the equations
20x+y-2z=17: 3x+20y-z=-18: 2x-3y+20z=25.
a) Suppose the point probability density functions is given by
�6
� ( x + y ), 0 �x �1, 0 �y �1
2
f ( x, y ) = �5
�
�0, otherwise
Obtain the marginal pdf of X and that of Y and hence find
24 5 CO3, CO5 K3
1 3
P [ �y � ]
4 4 .
b) Find a positive root of 3x- log 10 X =6, using fixed point
iteration method.
CC HOD
SCHOOL OF ELECTRICAL & COMMUNICATION
DEPARTMENT OF ECE
VTU R 2015
a) Let X and Y have joint probability density
f ( x, y ) = 2 0 < x < y < 1 Find the marginal distribution of f also find the
conditional density function of y given X=x
25 5 CO3, CO4 K3
b) Find the smallest positive root of the equation x = sin x
correct to 3 decimal places using Newton-Raphson method
CC HOD
