Assignment 2
1. Prove the following relations
i.
P A B 1 P A B
ii.
P A B 1 P A P B P A B
iii.
P A B 1 P A P A B
iv.
P A B P B P A B
2. Show that the probability of occurrence of only one of the events A and B is
P A P B 2 P A B .
3. Two urns contain 3
Assignment 1
1. A student wishes to take either a mathematics course or a biology course, but not
both. If there are 4 mathematics courses and 3 biology courses for which the student
has the necessary prerequisites, then the student can choose a course to
Assignment 1
1. Prove the following relations
i.
P A B 1 P A B
ii.
P A B 1 P A P B P A B
iii.
P A B 1 P A P A B
iv.
P A B P B P A B
2. Show that the probability of occurrence of only one of the events A and B is
P A P B 2P A B .
3. Two urns contain 3 w
6.4
Markov chains
Markov Chain: If, for all ,
=

=
,
=
,
=
=
=

=
, then
, ,
the process
, = 0,1, , is called a Markov chain and , ,
are called the states of the Markov chain. The conditional probability
= 
=
is called the onestep transition probabili
6.3
Processes Depending on Stationary Gaussian Process
Square law detector process
If
is a zero mean stationary Gaussian process and if
is called a square law detector process.
,
[Since
=
=
and
=
=
=
=
are jointly normal,
=
=
0 +2
+2
[since
Since the RH
6.2
Gaussian Process
Many random phenomena in physical problems including noise are well
approximated by a special class of random process, namely Gaussian random
process. A number of processes such as the Wiener process and the shot noise
process can be
Unit6
Special Stochastic Processes
6.1
Poisson Process
There are many practical situations where the random times of occurrences of
some specific events are of primary interest. For example, we may want to study
the times at which components fail in a la
4.5
Central Limit Theorem
n
be a sequence of independent random variables. Let Sn = X i . In laws
Let
i =1
of large numbers we considered convergence of
to
which is a constant
either in probability (in case of WLLN) or almost surely (in case of SLLN). Her
4.4
Strong Law of Large Numbers
is said to satisfy the strong law of large
Definition: A sequence of r.vs
numbers (SLLN) if
.
0 as
We state the following theorems without proof which are useful in checking
whether a given sequence satisfies SLLN or
4.3
Weak Law of Large Numbers
Let
be a sequence of r.vs and let X n
1 n
X i be the mean of first
n
i 1
r.vs. The
weak laws deal with limits of probabilities involving X n . The strong laws deal with
probabilities involving limits of X n .
Definition of
4.2
Convergence of Sequence of Random Variables
In this module we investigate convergence properties of sequences of random
variables. Throughout this module we assume that
or
is a
sequence of r.vs and is a r.v. We consider four different modes of converg
UnitIV
Order Statistics and Limit Theorems
4.1
Order Statistics
Independent and identically distributed random variables:
We say that
are independent and identically distributed random
variables (i.i.d.r.vs) if
and
where
(identically distributed)
is the
3.5
Probability Generating Function
Let
be a nonnegative integer valued random variable with p.m.f.
. Then the probability generating function (p.g.f.) of
defined by
where
is a dummy variable.
Advantages:
1.
2.
3.
4.
It is easy to compute.
Moments and so
3.4
Cumulant Generating Function
Just as the moment generating function (m.g.f.)
or characteristic function
(ch.f.)
of a r.v.X generates its moments, the logarithm of
or
generates a sequence of numbers called the Cumulants of . Cumulants are of
interest f
3.3
Characteristic Function
In some cases m.g.f. does not exist. For example, consider the p.m.f. given by
Its m.g.f. is given by
etx
M X t e p x 2 2 ,
x1 x
x 1
tx
6
which is divergent. Thus,
does not exist. A more serviceable function that
the m.g.f. is
3.2
Moment Generating Function
Certain derivations presented in modules 2.4, 2.5 and 2.6 have been somewhat
heavy on algebra. For example, determining the mean and variance of the
Binomial distribution turned out to be fairly tiresome. Another example of
Unit 3
Probability Inequalities and Generating Functions
3.1
Probability Inequalities
Inequalities are useful for bounding quanties that might otherwise be hard to
compute. They will also be used in the theory of convergence and limit
theorems.
Chebychevs
Numerical Solution of Ordinary and Partial Differential Equations
MATHEMATIC 206

Fall 2014
Numerical solution of ordinary and
partial dierential equations
Module 1: Introduction
Dr.rer.nat. Narni Nageswara Rao
August, 2011
Many problems in science and engineering can be formulated in terms of
dierential equations. Most of these realistic mathem
2.2. General properties of congruences
In this module we now discuss some general properties of congruences.
Common nonzero factors cannot always be cancelled from both members of the
congruence as they can be done in equations.
For example, both members
2.4. Simultaneous Linear Congruences and the
Chinese Remainder Theorem
Having considered a single linear congruence, it is natural to discuss the problem
of solving a system
(
)
(
)
)
.
(
of simultaneous linear congruences.
It may be noted that a sy
1.8. The Euclidean Algorithm
The gcd of two integers can be found by listing all their positive divisors and
taking out the largest common divisor. Clearly, it is a cumbersome for large
numbers. A more efficient process, involving repeated application of
2.4. Simultaneous Linear Congruences and the
Chinese Remainder Theorem
Exercise:
1. Solve the following system of linear congruences:
3 , 2
5 , 3
7
Ans:
a) 1
b) 5
11 , 14
29 , 15
31
Ans:
c) 5
6 , 4
11 , 3
17 Ans:
d) 2 1
5 , 3
2.2. General properties of congruences
Exercise
1. Prove that
2. Find the remainders when
is divisible by .
and
are divided by . (Ans: and )
3. What is the remainder when the sum
is divided by . (Ans: )
4. Using the congruence theory prove the following:
1.9. The Diophantine Equation
An equation in one or more unknowns with integer coefficients admitting
integer solutions is called a Diophantine equation and the study of such
equations is known as Diophantine analysis.
The equation
for Pythagorean triples
2.3. Linear Congruences
We consider polynomials
with integer coefficients, so that these polynomials
will be integers when is an integer.
Let
be a given positive integer. Let
=
+
+
+ +
0
be a polynomial with integer coefficients of degree , i.e.,
1.6. Fundamental Theorem of Arithmetic
Theorem 1: Fundamental Theorem of Arithmetic (or unique Factorization
Theorem)
Every integer
can be represented as a product of prime factors in only one
way, apart from the order of the factors.
Proof: We prove the
UNIT2
2.1. The Theory of Congruences
Another approach to divisibility related questions is through the arithmetic
of remainders or the theory of congruences, as it is now commonly known.
The concept and notation was first introduced by the German
mathema
1.5
Prime and Composite Numbers
We start with prime numbers. Prime numbers are crucial for understanding of
natural numbers. Infact they are the building blocks of natural numbers.
Prime number:
An integer is called a prime number if
are and . That is, a