UMA032 NUMERICAL AND STATISTICAL METHODS

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**Numerical Methods
(60% Weightage).**

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Floating-Point Numbers:** Floating-point representation, Rounding, Chopping,
Error analysis, Condition and instability.

**Non-Linear Equations**:
Bisection, Secant, Fixed-point iteration and Newton-Raphson methods, Order of
convergence.

**Linear Systems and Eigen-Values**:
Gauss-elimination method (using Pivoting strategies) and Gauss-Seidel Iteration
method. Rayleigh’s power method for eigen-values and eigen-vectors.

**Interpolation**:
Finite differences, Newton’s Forward and Stirling interpolating polynomials, Lagrange
and Newton’s divided difference interpolation formula with error analysis.

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**Numerical Integration**:
Newton-Cotes quadrature formulae (with error) and Gauss - Legendre quadrature
formulae.

**Differential
Equations:** Solution of initial value problems using Taylor Series, Euler’s
and Runge-Kutta (up to fourth order) methods.

**Statistical Methods (40% Weightage)**

**Random Variables:** Definition, Distribution Function,
Discrete and Continuous Random Variables, Probability functions, Cummulative
distributions functions, Mathematical expectation.

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**Probability Distributions:** Binomial, Poisson,
Geometric, Uniform, Normal, Exponential and Log- Normal distribution.

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**Sampling Distributions:** Sampling distribution of
Means and variance, Chi-Square distribution, t - distribution and F -
distribution.

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**Hypothesis Testing:
**General concepts, Testing a Statistical Hypothesis, one and two
tailed tests, Critical region, Confidence interval estimation. Single and two
sample tests on proportion, mean and variance.

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**Linear Regression and Correlation: **Linear Regression,
Least Square principal and the Fitted models, Karl Pearson’s Correlation
Coefficient, Rank Correlation, Lines of Regression (two variables only).

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*Laboratory Work*

Programming exercises on numerical and Statitical methods using C or C++ languages.

- To detect the interval(s) which contain(s) root of equation f(x)=0 and implement bisection Method to find root of f(x)=0 in the detected interval.
- To find the root of f(x)=0 using Newton-Raphson and fixed point iteration methods.
- To evaluate the Newton’s Forward Lagrange and divided difference interpolating polynomials of degree ≤ n, Based on (n+1) points.
- To solve linear system of equations using Gauss elimination (without pivoting) method.
- To solve linear system of equations using Gauss- seidel method.
- To find the dominant eigen-value and associated eigen-vector by Rayleigh power method.
- To integrate a function numerically using trapezoidal and Simpson’s rule.
- To solve the initial value problem using modified Euler’s and Runge-kutta methods.
- Generation of random numbers for Binomial and Poisson distributions using Linear Congruential Genrator Algorithm.
- Regression analysis using least square principle.
- Correlation analysis for bivariate distribution.

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*Text
Books*

*Conte, S.D and Carl D. Boor, Elementry Numerical Analysis: An Algorithmic approach, Tata McGraw Hill, New York (2005).**Johnson, R., Miller, I. and Freunds, J., Miller and Freund’s Probability and Statistics for Engineers, Pearson Education(2005) 7*^{th}ed.*Gerald C.F and Wheatley P.O., Applied Numerical Analysis, Pearson Education (2008) 7*^{th}ed.

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*Reference Books*

*Mathew, J.H., Numerical Methods for Mathematics, Science and Engineering, Prentice Hall Inc.J (2002).**Meyer, P.L.. Introductory Probability and Statistical Applications, Oxford (1970) 2*^{nd}ed.*Jain M.K., Iyengar, S.R.K., and Jain, R.K. Numerical Methods for Scientific and Engineering Computation, New Age International (2008) 5*^{th}ed.*Walpole, Ronald E., Myers, Raymond H., Myers, Sharon L. and, Keying Ye, Probability and Statistics for Engineers and Scientists, Pearson Education (2007) 8*^{th}ed.