Dr' K' Gururajan

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About the Author
Dr. K. Gururajan Assistant Professor Department
of Mathematics Malnad College of
Engineering Hassan 573 201 Contact Number ?
98459 84291 Email kguru.hsn_at_gmail.com
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• Introduction To
• Correlation and Regression
• Curve Fitting
• Basic Probability theory
• Random Variables
• Standard Distribution Functions of a
• Random Variable
• Hypothesis Testing

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OBJECTIVE OF THE COURSE The main purpose is to
introduce to students a new approach, namely,
probabilistic technique of solving problems for
which either the solution is un-known in advance
or is un-predictable. It is seen that problems of
this kind has received considerable attention
from engineers, researchers, and from people
working in their relevant fields.
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DISCUSSION ON CORRELATION AND REGRESSION

• It is known that in many experiments, results are
  dependent on one or more parameters. Suppose
  that the change in the numerical values of one
  variable affects the changes in the other
  variables, then we say that the variables are
  correlated. The relationship that exists is
  called correlation or co relation

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Continued . . . .

• From an analysis view point, it is necessary
  to measure the degree of relationship existing
  between the variables. This is done by using a
  measure called correlation coefficient or
  correlation index which summarizes in one figure
  the direction and degree of correlation. The
  correlation analysis refers to the techniques
  used in the measuring the closeness of the
  relationship between the variables without
  influencing or manipulation any variables.

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CURVE FITTING
Introduction-

• Scientists and engineers often want to
  represent empirical data obtained from an
  experiment, using a model based on mathematical
  equations. With the correct model and necessary
  calculus, one can determine/estimate important
  characteristics of the data, such as the rate of
  change anywhere on the curve (first derivative),
  the local minimum and maximum points of the
  function (zeros of the first derivative), and the
  area under the curve (integral) and so on.

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• Therefore, finding a best mathematical model
  to a data is an important problem from both
  theoretical and as well as from practical view
  point. With these considerations, the following
  sections deals with ways of obtaining different
  mathematical equations for a given data.

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Basic Probability Theory

• Before coming to a discussion of the topic
  probability, let us consider few examples that
  would certainly explain the importance of
  probability theory in many fields.
• Consider a person Mr. X purchasing a computer
  system from a leading firm in Bangalore.
  Naturally, Mr. X will have the following
  questions in mind before buying

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Continued . . . .

• The quality of product
• The price of the system?
• The working conditions of the components of the
  system?
• The life of the computer system?
• Is there any guarantee period for the components?
  In case, if some components get repaired, is
  there any chance of replacement etc?

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Continued . . . .

• Solutions to problems of this kind have been
  given considering probabilistic approach as the
  basis. In view of these,
• In this course, students will be introduced to
  probability, and other related topics such as
  Random Variables, Distribution Functions and so
  on. After instruction, students will be in a
  position to use theory of probability as a basis
  to solve Engineering Problems.

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Testing of Hypothesis

• In many situations, we come across the problem of
  testing some parameters about a population of
  large size. Here, usually we select a sample
  of small size using a sampling approach and study
  the sample. Based on this analysis, we try to
  predict about the characteristics of population
  parameter. This problem is called hypothesis
  testing.

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Continued . . .

• For example, consider the problem of

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What is curve fitting?

• Curve fitting means finding a best mathematical
  representation which closely match the given
  data. The procedure involves the calculation of
  parameter values of the mathematical equations.
• Most curve fitting problems is based on least
  square principle, which states that The sum of
  squares of differences between actual values and
  observed values must be least.

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• The current syllabus is restricted only to
  finding parameter values of the equations

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First we shall consider the problem of fitting a
straight line, to the following data
.
.
.

.
.
.



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Solution-

• We fit a straight line of the form
  according to Least Square Principle, i.e. the
  sum of squares of difference between actual
  values and the observed values is least. Here,
  are the actual values and the observed
  values are respectively . . .

. . . . . . .


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Considering that

• is least, We derive equations for finding
  the parameter values a and b.
• From Differential Calculus, it is known that a
  function of two variables attains an extreme
  value only at points where the first order
  partial derivatives vanishes. Now,
  differentiating S with respect a and b
  partially and equating these to zero, we obtain
  the normal equations.

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By solving these two equations, one always one
obtain the values and a and b, hence the best
straight line
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• An illustrative example
• Fit a straight line to the data given below

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Solution The two normal equations are
and
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Thus, normal equations are
Solving these two equations, we obtain Fit a
straight line to the data given below
x 1 2 3 4 5 6 y 9
8 10 12 11 13
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The two normal equations are
and
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Here, n 6. Consider
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The normal equations become
and
Solving these two, we get a and b and hence the
straight line
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