Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
Mean and Variance :
Let X be a discrete random variable with
p.d.f. f(x). The mean or mathematical
expectation of X is defined as
all x
x f ( x)
The variance of the random variable X
with p.d.f. f(x), with mean , is defined
by
Var ( X ) ( x ) f ( x)
2
al
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial6
1. For each of the following dierential equations, locate and classify its singular points in terms
of regular and irregular
(a) x3
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial7
1. Find the general solution of the following dierential equations near the singular point x = 0
(a) x2 y x(2 x)y + (2 + x2 )y = 0,
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial8
1. Show that
(a) F(, , , x) = (1 x) ,
(b) xF(1, 1, 2, x) = ln(1 + x).
2. Verify each of the following by examining the series expan
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BITS PILANI K K BIRLA GOA CAMPUS
MATH III
1
Find the general solution of the following systems of dierential equations:
dx
= 3x + 4y
dt
dx
(b)
= 7x + 6y
dt
dx
(c)
= 5x + 2y
dt
dx
(d)
= 4x y
dt
(a)
2
Tutorial
dy
= 2x + 3y
dt
dy
= 2x + 6y
dt
dy
= 5x + 4y
d
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial10
1. Using the generating function of the Legendre polynomials prove the following:
(a) Pn (1) = 1,
(c) P2n+1 (0) = 0,
(b) Pn (1) =
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial11
1. Use Rolles theorem to show that
(a) between any two positive zeros of J0 (x), there is a zero of J1 (x).
(b) between any two po
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial12
1. Find the Laplace transform f (x) = [x] where [x] denotes the greatest integer x.
2. Show that Laplace transform of f (x) = 1/x
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial13
1. Find the Fourier series for the following functions dened by
8
> , x
>
<
2
(a) f (x) = >
> 0, < x ,
:
2
8
> , x < 0
>
<
(c) f
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial14
1. nd the eigenvalues n and eigenfunction yn (x) for the equation y + y = 0 in each of the
following cases:
(i) y(0) = 0, y(/2) =
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
1. In below we will nd the general solution to a second order dierential
equation with regular singular points, using Frobenius series. The dierential
equation
dy
d2 y
+ (4x2 + 1)y = 0
(1)
4x2 2 8x2
dx
dx
clearly has a regular singular point at x = 0. Cho
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial5
1. Show that y = c1 e x + c2 e2x is the general solution of y 3y + 2y = 0 on any interval.
2. Using the Method of Variation of Para
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial4
1. Show that y1 = ex and y2 = e2x are solutions of the dierential equation y y 2y = 0. What
is the general solution?
2. Show that y
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial3
1. Consider the equation y = y, 0 < x < , where is a real constants. Show that if is any
solution and (x) = (x)ex then (x) is a con
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
Financial Statement of the Subsidiaries F.Y. 20122013
KPIT Infosystems Limited
Share Capital
During the year, the Company increased its paid up share capital by GBP 1.82 Million.
Registered Office: Ground Floor, The Annexe, Hurst Grove, Sandford Lane, Hu
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
KPIT Fourth Quarter Results FY 2017
Q4FY17 USD Revenue grows 4.4% QoQ to $ 128.3 Million
Q4FY17 Revenues at 8,584.6 million, a QoQ growth of 3.3%
Positive on the growth outlook for FY18
Investor Release BSE: 532400  NSE: KPIT
Pune, April 26, 2017: KP
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
KPIT First Quarter Results FY 2017
KPIT Q1FY17 PAT grows 30% YoY
Q1FY17 Revenues grow 1.23% YoY in USD terms
Q1FY17 Revenues at USD 119.78 million
Investor Release BSE: 532400  NSE: KPIT
Pune, July 20, 2016: KPIT (BSE: 532400; NSE: KPIT), a global te
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
Some Special Functions of
Mathematical Physics
1
Manoj Kumar Pandey, Dept. of Mathematics
Special Functions
Legendre Polynomials
Bessel Functions and Gamma Function
2
Manoj Kumar Pandey, Dept. of Mathematics
Legendre Polynomials
The Legendre Equation is
(
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
Gausss Hypergeometric Equation
1
Anil Kumar, Department of Mathematics
Gausss Hypergeometric Equation
Consider the differential equation
x(1 x) y [c (a b 1) x] y aby,
where a, b, and c are constants.
This famous equation is called Gausss Hypergeometric
Eq
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
MATH F211
Mathematics III
Dr. Jajati Keshari Sahoo
Department of Mathematics
1
Power Series Solutions and Special
functions
2
Finding the general solution of a linear differential equation
depends on determining a fundamental set of solutions of the
homo
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI  KK BIRLA
GOA CAMPUS
FIRST SEMESTER 20142015
MATHEMATICS  III
Tutorial2
1. Determine which of the following equations are exact, and solve the ones that are:
1
x
x
(ii) dx + 2 sin
dy = 0,
y
y
y
(iv) 2x s
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
Lecture9
MATHEMATICSIII (MATH F211)
Dr. P. DHANUMJAYA
Department of Mathematics
BITSPilani K K Birla Goa Campus
Solution Techniques for Second Order ODEs
Dr. P. Dhanumjaya
Solution Techniques for Second Order ODEs
Solution Techniques for Second Order O
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
SERIES SOLUTION OF LINEAR
EQUATIONS
at
Regular Singular Points
Solutions about Singular Points
If we attempt to use previous methods to solve the
differential equation in a neighborhood of a singular
point x0, we will find that these methods fail.
This is
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2013
Chapter 4: Continuous distributions
Continuous Random
Variables
The Uniform Distribution
The Gamma Distribution
The Normal Distribution
The Normal Approximation
to
Binomial Distribution
Exponential Distribution
Chi square Distribution
Rectangular or Unifo
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2016
Chapter # 5
Joint Distributions
Single Random Variables: (Univariate)
Discrete
Continuous
Two Dimensional Random variables (Bivariate)
Continuous
Discrete
Bivariate distribution occurs when we observe 2
nondeterministic quantities, one followed by another
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2016
Chapter 9
Inferences on Proportions
This
chapter
will
deal
inferences
on
proportions
hypothesis tests on them.
with
and
We will see how to employ the
standard normal distribution to
construct confidence intervals on p
and test hypotheses concerning its
va
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Spring 2016
Chapter 6
Descriptive Statistics
Statistics
In Statistics, we want to study
properties of a (large) group of objects,
generally termed as population.
Methods of statistics study small
subsets of population. This is called
sample. The science developed for
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
Probability and Statistics
Welcome to AAOC C111
Text book : Johnson and Gupta : Miller and
Freunds Probability and Statistics for
Engineers.
Hand outs : To be distributed in tutorials.
Evaluation : 2 tests (20% each)
Tutorials (20%)
Compre. (40%)
Problems
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
Joint Distributions
Discrete Case
Recall that if X is a discrete
random variable taking the
values a,b,c, then the
probability P(X=a) is denoted as
f(a).
Consider now two discrete
random variables X1 and X2
defined on the same sample
space. The probabilit
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
7.5: Hypothesis Concerning One Mean:
Statistic for test concerningmean ( known) :
Z
X 0
n
In order to test the Hypothesis, we need
To know:
1. Null Hypothesis (= 0 (say)
2. Alternative Hypothesis ( 0 (say)
3. Level of Significance :
Critical Regions: The
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
Sampling
Parameters are numerical descriptive
measures for populations.
For the normal distribution, the
location and shape are described by
and
For a binomial distribution
consisting of n trials, the shape is
determined by p.
Often the values of parame
Birla Institute of Technology & Science, Pilani  Hyderabad
Probability and statistics
MATH 113

Winter 2012
Random Variables
Many a times it is advantageous to
associate a numerical measure to
outcomes.
Most of the times these numerical
values represent an important
characteristic of the outcome.
For example, rather than keeping track
of the sequence of head