EC 221 Probability and Random Processes Assignment-I
1. Given S cfw_1, 2,3 findtwofieldscontainingthesubsetcfw_2.
2. Suppose isafieldofsubsetsofasetSand B .Showthat C cfw_ A B | A isa
fieldofthesubsetsofB.
3. Suppose S (thesetofnaturalnumbers)and C cfw_
Assignments for practice.
1. Study the application of Jensens, Triangular and Cauchy Schwartz inequalities for random variables.
2.
are i.i.d random variables with an unknown parameter .Find
for the following
X 1 , X 2 ,., X N
i.
X i : Poi ( )
ii.
1 ( x )
EE664 Assignment-1
1. Write a program to solve Ax=b using Gaussian Elimination, where A is an nxn dense matrix.
2. Write a program to solve Ax=b using sparse LU decomposition, where A is an nxn sparse matrix stored
in Harwell-Boeing Format.
3. A tube hold
Assignment Problems are given below.
S. No.
Problem Statement
1
Basic Programs on Matrix into Vector Computations using Pthreads, OpenMP and
MPI
2
Matrix-Matrix multiplication simple, Cannons, DNS algorithm OpenMP and MPI
3
Gaussian elimination simple, pa
Pattern Recognition and Machine Learning :
Assignment 1
The assignment is due on February 12.
The assignments are meant to enhance your learning, so that you master the basics
by the close of this semester. You may work in groups for discussion, but it
Maximum Likelihood Estimator (MLE)
Suppose X 1, X 2 ,., X n are random samples with
function f X , X
1
2 ,., X n /
( x1 , x2 ,., xn )
the joint probability density
which depends on an unknown nonrandom
parameter .
f X / ( x 1,x2 , .,xn / )
is called the l
Probability Review
Definition: Let S be a sample space and a sigma field defined over it. Let P : be a mapping from the sigma-algebra into the real
line such that for each A , there exists a unique P( A) . Clearly P is a set function and is called probabi
Computational
Complexity
Gaurav Trivedi
EEE Dept.
IIT Guwahati
Well talk about this first
one today.
http:/www.claymath.org/millennium/
In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CM
ESTIMATION THEORY
4.1. Introduction:
When we fit the random data by an AR model, we have to determine the process
parameters observed data.
In RADAR signal processing, we have to determine the location and the velocity of a
target by observing the receive
MA 201 Complex Analysis
Lecture 18: Evaluation of integrals
Lecture 18
Evaluation of integrals
Evaluation of certain contour integrals: Type IV
Type IV Integrals of the form
Z
0
sin x
dx
x
can be evaluated using Cauchy residue theorem.
Before we discuss i
MA 201 Complex Analysis
Lecture 17: Evaluation of integrals
Lecture 17
Evaluation of integrals
Evaluation of certain contour integrals: Type I
Type I: Integrals of the form
Z
2
F (cos , sin ) d
0
dz
.
iz
Substituting for sin , cos and d the definite integ
MA 201 Complex Analysis
Lecture 16: Residues Theorem
Lecture 16
Residues theorem and its Applications
Residues
Question: Let is a simple closed contour in a simply connected domain D
and let z0 doesnt
lie on . If f has singularity only at z0 then what cou
Tutorial9Date16/3/11
1. Suppose X and Y denote the independent random variables with the identical density
function f. Find the expressions for the PDF of (i) Z = min( X , Y ) and (ii)
Z = max( X , Y )
2. Suppose X ~ exp(1) and Y ~ exp(1) are two RVs and
Tutorial 11 30/3/11
1. The Poisson process
cfw_ X (t ), t 0 is
a random
process where
the increments
X (t j ) X (ti ), t j > ti are independent Poisson random variables with the rate parameter
(t2 t1 ) .
(a) Find p X ( t1 ), X ( t2 ) (k1 , k2 ) for t1 >
Tutorial 4
1. A player has equal probability of win in each game against his opponent. Which is more
porobable?
(a) Three wins from 5 games or 4 wins from 7 games?
(b) At least three wins from from 5 games or at least 4 wins from 7 games?
2. A player toss
EC 221:Probability and Random Processes
Assignment-2
1. Consider the sequence cfw_ An given by
An ( a
1
n 1
, b n1 ), a, b
Show that lim supcfw_ An lim infcfw_ An ( a, b] and hence lim An exists.
n
n
n
2.Define the Borel sigma algebra as the minim
Tutorial 8
1.
9/3/11
(a) ForrandomvariablesXandY,thejointPDFisgivenby
1
, x 1, y 1
f X ,Y ( x, y ) 4
0 otherwise
Show that XandYareindependentrandomvariables.
(b) Suppose XandYaredistributedas
1
2
2
, x y 1
f X ,Y ( x, y )
0 otherwise
ExamineifXandYare
! " #$%
"#$
#
D( f | g ) =
&!
f ( x)log ( f ( x) / g ( x)dx
'!
+
,- !
( '!
(.
I A ( s) =
( /
&! * (
.
1 ( (
.
.
1 if s A
0 otherwise
&
2
,
2+
*
( /
&
X
/
( a>03
P(cfw_ X + a)
4
&
I A B = I A I B = min( I A , I B )
!+
0
D ( f | g ) 0
) * +
2
+ a2
2
Consi
EC 221:Probability and Random Processes
Assignment-3
1. A well shuffled deck of 52 cards is distributed among 4 players North, East, South and
West. What are the probabilities that
(a) North has 2 kings,
(b) West has 4 kings, 4 aces and 4 queens.
(c) Sout
Tutorial 5
1. Suppose X is a binomial random variable with the PMF
()
p X ( k ) = n p k (1 p ) n k
k
0 p 1, k = 0,1,., n
(a) Find the most likely value(s) of X.
(b) Find a condition on n and p so that
(c) p X (k ) = p X (n k ) k = 0,1,., n
(d) If p=0.3, f
Tutorial 6
1. (a) Suppose X is a discrete random variable with RX = cfw_0,1,. show that
EX = P (cfw_ X n)
n =0
(b) Suppose X is a non-negative continuous random variable. Show that
EX = (1 FX ( x) dx
0
2. (a) Let X be a continuous random variable with
1
f
EE664: Introduction to Parallel
Computing
Dr. Gaurav Trivedi
Lectures 5-14
1
Some Important Points
Automatic vs. Manual Parallelization
Understand the Problem and the Program
Partitioning
Communications
Synchronization
Data Dependencies
Load Balancing
Gra