BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
MATH F111 & AAOC C111
Probability and Statistics
BITS Pilani
Pilani Campus
BITS Pilani
Pilani Campus
Chapter 4
Continuous Distribution
Continuous Random Variables
Defin
BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
BITS Pilani
Pilani Campus
MATH F111 & AAOC C111
Probability and Statistics
Text Book
Introduction to Probability &Statistics
Authors : Milton & Arnold 4th Edition
Publi
BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
BITS Pilani
Pilani Campus
MATH F111 & AAOC C111
Probability and Statistics
BITS Pilani
Pilani Campus
Chapter 2
Some Probability Laws
Axioms and Definition
The definiti
Simulation
To generate a data of values of a random
variable, we need to perform the random
experiment. (several times for large data).
Such data can be used to infer more about
the random variable, like parameters.
Sometimes even though we know the densi
Question 2.4) Weightheight distribution
First we use the data on the "Data" sheet to plot the original graph in the Fig 3.1 as shown.
This graph shows the relation of weight and height for the given sample.
Sample weight distribution by height:
300
Weigh
Maximum Likelihood
Much estimation theory is presented in a rather ad hoc fashion. Minimising
squared errors seems a good idea but why not minimise the absolute error or the
cube of the absolute error?
The answer is that there is an underlying approach wh
BITS PILANI, GOA CAMPUS

Comprehensive Examination
Date: 10.05.05 (Monday)
Second Semester, 2004 2005
AAOC GCl11
Probability and Statistics
MM:120
TIME: Three Hours
Instructions:
(a)
There are two parts in this question paper, Part A and Part B. Answer
Birla Institute of Technology and Science, Piiani K K Birla Goa Campus
Course No: MATH F113
Date: 16/09/2012
First Semester 2012—2013
Test—1 (Closed Book)
Max. Mark: 75
Time: 1 Hour
Probability 8.: Statistics
Instructions: 1. Start a new question on a
BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
MATH F111 & AAOC C111
Probability and Statistics
BITS Pilani
Pilani Campus
BITS Pilani
Pilani Campus
Chapter 4
Continuous Distribution
Continuous Random Variables
Defin
Estimation of variance
Recall : S2 is an unbiased estimator for 2.
Theorem : Let X1, , Xn be the random sample of
size n from a normal population X with mean and
s.d. . Then (n1) S2/2 has chisquare distribution
with (n1) degrees of freedom.
Recall : Ch
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
Estimation of variance
Recall : S2 is an unbiased estimator for 2.
Theorem : Let X1, , Xn be the random sample of
size n from a normal population X with mean and
s.d. . Then (n1) S2/2 has chisquare distribution
with (n1) degrees of freedom.
Recall : Ch
BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
BITS Pilani
Pilani Campus
MATH F111 & AAOC C111
Probability and Statistics
Text Book
Introduction to Probability &Statistics
Authors : Milton & Arnold 4th Edition
Publi
BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
MATH F111 & AAOC C111
Probability and Statistics
BITS Pilani
Pilani Campus
BITS Pilani
Pilani Campus
Chapter 3
Discrete Distribution
Random variables
Random variables a
BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
BITS Pilani
Pilani Campus
MATH F111 & AAOC C111
Probability and Statistics
BITS Pilani
Pilani Campus
Chapter 2
Some Probability Laws
Axioms and Definition
The definiti
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
BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
MATH F111 & AAOC C111
Probability and Statistics
BITS Pilani
Pilani Campus
BITS Pilani
Pilani Campus
Chapter 3
Discrete Distribution
Random variables
Random variables a
BITS Pilani
presentation
BITS Pilani
Pilani Campus
Dr RAKHEE
Department of Mathematics
MATH F111 & AAOC C111
Probability and Statistics
BITS Pilani
Pilani Campus
BITS Pilani
Pilani Campus
Chapter 4
Continuous Distribution
Normal Approximation to
the Binom
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
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 AND SCIENCE, PILANI K. K. BIRLA GOA
CAMPUS
Second Semester 20132014
Tutorial Sheet  7
Course No. MATH F113
Course title: Probability and Statistics
Date: March 31, 2014
1. If X is uniformly distributed in (1, 1), nd g(x),
ESTIMATION AND INFERENCE
THEORY
ESTIMATION OF PARAMETER
TESTING OF HYPOTHESIS
POPULATION
Population is the collection of observations
about which conclusions are to be drawn
SAMPLE
Sample is a portion of population.
PARAMETERS
Parameters are statistical
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 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 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 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 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
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
Problem 3.45
Suppose we have 10 coins such that if the ith coin if flipped then heads will appear with
probability i/10 (i = 1, . . . , 10). One of the coins is selected at random, and then flipped.
We are told that it shows heads. What is the probability