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Concept of Moment: In statistics moment are defined as the mean values of powers of the deviation in any individual series or frequency distribution (discrete and continuous) taken about three points: (i) Origin

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Experiments: An experiment repeated under essentially homogeneous and similar conditions results in an outcome, which is unique or not unique but may be one of the several possible outcomes. When the result is unique then the experiment is called a deterministic experiment. Any experiment whose outcome cannot be predicted in advance, but is one of the set of possible outcomes, is called a random experiment. If any experiment as being performed repeatedly, each repetition is called a trial. We observe an outcome for each trial. If a coin is tossed, the result of a particular tossing, i.e., whether it will fall with head or tail can not be predicted. A die with faces marked with numbers numbers 1,2,3,4,5,6. When we throw a die the number on upper face, we can not predict which face will be on the top. Sample Space:

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Utility of Regression: In brief, the utility of regression or regression analysis can be understand in the following heads: 1. Estimation of Values: When there is a cause and effect relationship between two variables. The one can be estimated with the help of the other.

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Curve Fitting: The curve fitting process fits the equation of approximating curves to the raw field data. Nevertheless, for a given set of data, the fitting curves of a given type are generally NOT unique. Thus, a curve with a minimal deviation from all data points is desired.

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Addition Theorem of Probability: If A₁,A₂, . . . , A_k be k mutually exclusive events on a sample space associated with a random experiment, then the probability of happening one of them is the sum of their individual probabilities. In symbols, P(A1 A2 . . . Ak) = P(A1) P(A2) . . . P(Ak) or P(A1 ∪ A2 ∪. . .∪ Ak) = P(A1) P(A2) . . . P(Ak)

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Skewness: Skewness is a measure of the degree of asymmetry of a distribution. If the left tail ( Tail at small end of the distribution ) is more pronounced than the right tail ( Tail at the large end of the distribution ), the function is said to have negative skewness. If the reverse is true, it has positive skewness. If the two are equal, it has zero skewness. Types of Skewness:

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Moments about Origin in Term of Moments about any Point: Method I: Convert the moments about a in terms of moments about mean and then obtain moments about origin. Method II: Obtain by the formula.

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Random Variable: In an experiment of tossing a pair of coin we may be interested in obtaining the probabilities of getting 0,1 or 2 heads which are possible events. We consider a variable X which hold these possible values 0,1 or 2. This variable is a real valued function defined over sample space whose range is non empty set of real numbers is called random variable. i.e, A random variable is a function X(w) with domain S and range (−infinite,infinite) such that for every real number a, the event [e : X(e) ≤ a].

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Problems on Curve Fitting: Example-1 Fit straight line for the following data:

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Kurtosis: Definition 7. Kurtosis refers to the extent to which unimodal frequency curve is peaked. Definition 8. Kurtosis is a measure that refers to the peakedness of the top of the curve . Kurtosis gives the degree of flatness or peakedness in the region about the mode of a frequency distribution.

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Moment Generating Function: The function E(e^tX ) serves to generate moments of the probability distribution of the variate about the origin. and is called Moment Generating Function. The Moment Generating Function (m.g.f)of a random variable X(about origin) having the probability function f (x)

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Correlation: When for every value of a variable X we known a corresponding value of a second variable Y, i.e., the data is in the form of paired measurements, then we are interested in the relationship of these two variables. The analysis of univariate data in case of economic, social and scientific areas becomes insufficient. So, in some situations, say production price, height of father and son, marks obtained in two subjects, height of husbands and their wives, a series of expenses on advertisement and sales. If change in one variable is accompanied (or appears to be accompanied) a change in other variable and vice-versa, then the two variables are said to be correlated and this relationship is known as correlation or covariation. Correlation Analysis:

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Fitting a Curve of Different Types: Fitting a Curve of Type Y = AX BX² Let (xi,yi), i = 1,2.......,n be the n - sets of given values. Then the residual at x = xi is given by

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Methods of Determining Correlation: There are various methods of determining correlation in two variables. Here we discuss only following methods: 1. Karl Pearson’s Coefficient of correlation

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Rank Correlation: When the variables under consideration are not capable of quantitative measurement but can be arranged in serial order (ranks), we find correlation between the ranks of two series. This happens when we deal with qualitative characteristics such as honesty, beauty, etc. This method is called Spearman’s Rank Difference Method or Ranking Method and the correlation coefficient so obtained is called Rank Correlation Coefficient and is denoted by rs. This method was developed by Charles Edward Spearman, a British Psychologist in 1904. Spearman’s rank correlation coefficient is also used when the measurements are given for both the series. However, Spearman’s rank correlation coefficient rs is nothing but Karl Pearson’s correlation coefficient between the ranks, it can be interpreted in the same way as the Karl Pearson’s correlation coefficient. Case1: When ranks are different in the two series, then

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Polynomials Least-Squares Fitting: The application of the method of least square curve fitting using polynomials are briefly discussed as follows. The least square mth degree Polynomials method uses mth degree polynomials.

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Fitting an Exponential Curve: Consider the equation y = ae^bx. Taking on both sides, we get log y = log a bx log e

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Control Charts for Variables: Control charts for variables monitor characteristics that can be measured and have a continuous scale, such as height, weight, volume, or width. When an item is inspected, the variable being monitored is measured and recorded. For example, if we were producing candles, height might be an important variable.We could take samples of candles and measure their heights. Two of the most commonly used control charts for variables monitor both the central tendency of the data (the mean) and the variability of the data (either the standard deviation or the range). Note that each chart monitors a different type of information. When observed values go outside the control limits, the process is assumed not to be in control. Production is stopped, and employees attempt to identify the cause of the problem and correct it.

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Normal Distribution: The perfectly smooth and symmetrical curve, resulting from the expansion of the binomial (q p) ⁿ when n approaches infinitely is known as the normal curve. Thus, the normal curve may be considered as the limit toward which the binomial distribution approaches as n increases to infinity. Alternatively we can say that the normal curve represents a continues and infinite binomial distribution when neither p nor q is very small. Even though p and q are not equal, for very large n we get perfectly smooth and symmetrical curve. Such curves are also called Normal Probability curve or Normal curve of Error or Normal Frequency curve. It was extensively developed and utilized by German Mathematician and Artsonomer Karl Gauss. This is why it called Gaussian curve in his honor. The normal distribution was first discovered by the English Mathematician Dr. A. De Moivre. Later on French Mathematician Pierre S. Laplace developed this principle. This principle was also used and developed by Quetlet, Galton and Fisher. Assumptions of Normal Distribution:

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Techniques of One-way Analysis of Variance: 1. In One-way analysis of variance there are k samples, one from each of k normal populations with common variance σ ² and means μ₁,μ₂, ...μ_n. The number of observations n_i in samples may be equal or unequal i.e. n₁ n₂ ....... n_k = N

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Control Charts for Attributes: Control charts for attributes are used to measure quality characteristics that are counted rather than measured. Attributes are discrete in nature and entail simple yes-or-no decisions. For example, this could be the number of nonfunctioning light-bulbs, the number of rotten apples, or the number of complaints issued. Two of the most common types of control charts for attributes are p-charts and c-charts. The qualitative characteristic of the manufactured product can not be expressed numerically; but simply we consider its presence and absence. Here we consider the number of defects per item. The control charts for the such situations are called control charts for attributes. There are three types of control charts for attributes :

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Analysis of Variance: Analysis of variance basically is an arithmetical method by which we split up the total variability into component variations ascribable to different sources of causes. In the words of Yule and Kendall, The analysis of variance is essentially a procedure for testing the difference between different groups of data for homogeneity. While defining analysis of variance, Ronal A. Fisher wrote, The separation of the variance ascribable to one group of causes from the variance ascribable to other groups. In simple words, Analysis of Variance is a statistical technique, with help of which total variation, is partitioned into variation caused by each set of independent factors and homogeneity of several means is tested.

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Binomial Distribution: The binomial distribution¹ is a discrete probability distribution which is used when there are exactly two mutually exclusive outcomes of trials. These outcomes are appropriately labeled success and failure. This is used to obtain the probability of observing x successes in n trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials. Consider a random experiment which has two possible outcomes success and failure. The probability of success in each trial is p and remains same for all n trials. The trials are independent. Let X is a random variable which represents the number of successes in n trial. Then X has n values named as 0 success, 1 success, 2 successes, ..., n successes., i.e.

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Variance and Standard Deviation of Binomial Distribution: Let the Binomial distribution be

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Time Series and Forecasting: Time series analysis is the basis for understanding past behaviour, evaluating current accomplishment, planning future operations. It is also used for comparing the components of different time series. It is used to determine the patterns in the data of the past over a period of time and extrapolate the data into the future. Previous performances are studied to forecast future activity. Prof. W.Z. Hirsch writes, A main objective in analyzing a time series is to understand, interpret and evaluate changes in economic phenomena in the hope of more correctly anticipating the course of the events. In short, following are the use of Time Series Analysis:

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Poisson Distribution: Let X be a discrete random variable where probability of X = x is given by

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Constants of Binomial Distribution: Let the Binomial Distribution be

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Test for the Difference of Two Sample Means: Suppose we draw two samples from two population (or some population). Then to test the hypothesis, ‘The difference of the means of two samples is not significant or the two samples have been taken from the same population. For this we consider the difference between two sample means and the standard error of this difference. As usual, if the difference is greater than three times standard error, its considered significant otherwise not significant. 1. When population standard deviation (σ_pop) is known:

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Chi-Square Test: Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. For example, if, according to Mendel’s laws, you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8 males, then you might want to know about the “goodness to fit” between the observed and expected. Were the deviations (differences between observed and expected) the result of chance, or were they due to other factors. How much deviation can occur before you, the investigator, must conclude that something other than chance is at work, causing the observed to differ from the expected. The chi-square test is always testing what null hypothesis, which states that there is no significant difference between the expected and observed result.

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Universe of Population: In any statistical investigation often called statistical survey, observations are made on a group of objects or individuals called elementary units as they are without any interference. The aggregate of individuals under study in any statistical survey is called a population. According to A.C. Rosander: A Population is the totality of objects under consideration. Types of Population:

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Statistical Quality Control Method: Statistical quality control (SQC) is the term used to describe the set of statistical tools used by quality professionals. Statistical quality control can be divided into two broad categories: 1. Statistical process control (SPC) involves inspecting a random sample of the output from a process and deciding whether the process is producing products with characteristics that fall within a predetermined range. SPC answers the question of whether the process is functioning properly or not.

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Measurement of Secular Trend: Trends are measured by the following methods: 1. Free hand graphic method

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Difference Operators: So far we have studied the operators Δ, and δ. Here we will discuss some more operators which play a vital role in numerical analysis. 1. Shifting Operator:

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Backward Differences: The differences y₁ − y₀, y₂ − y₁, ..., y_n − y_n−1 when denoted by y₁, y₂, ..., y_n respectively, are called the first backward differences. Here, (read as Nabla) is backward differences operator. Thus we have;

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Regular Falsi Method: This method is also called as false position method. Consider the equation Bisection Method. Let a and b, such that a < b, be two values of x such that f (a) and f (b) are of opposite signs. Then the graph of y = f (x) crosses the x-axis at some point between a and b. The equation of the chord joining the two point (a, f (a)) and (b, f (b)) is,

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Newton Forward Interpolation: Let the function y = f (x) take the values y₀, y₁, y₂, ...,y_n corresponding to the values x₀, x₁, x₂, ...,x_n. Where x_i = x₀ ih, i = 0,1,2, ...,n. Suppose it is required to evaluate f (x) for the x = x₀ ph, where p is any real number. We have for any real number p, we have defined E such that

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Problems on Interpolation Method: Example-1 Prove that Lagrange’s formula can be put in the for

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Newton Raphson Method: When an approximate value of a root of an equation is given, this method is used to obtain better and closer approximation to the root. Let x₀ be an approximation of a root of the given equation f (x) = 0, Let x₁ = x₀ h be the exact approximation of the root. Then f (x₀ h) = 0. By Taylor’s theorem, we have

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Lagranges Interpolation: Let f (x₀), f (x₁), ..., f (x_n) be n 1 entries of a function y = f (x), where f (x) is assumed too be polynomial corresponding to the arguments x₀, x₁, ...,x_n. The polynomial f (x) may be written as;

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Bisection Method: Suppose f (x) is continuous on an interval [a,b], such that f (a). f (b) < 0;

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Rate of Convergence: Here, the speed at which a convergent sequence approaches its limit is called the rate of convergence. This concept is of practical importance if we deal with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. Convergence of A Sequence:

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Newton’s Divided Difference Interpolation Formula: The Lagrange’s formula has the drawback that if another interpolation values were inserted then the interpolation coefficients are required to be recalculated. This labour of recomputing is saved by using Newton’s general interpolation formula which is based on divided differences. Divided Difference:

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Convergence of Newton-Raphson Method: In this section, we will see the condition for the convergence of Newton-Raphson method. We have;

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Newton Backward Interpolation: Let the function y = f (x) take the values y₀, y₁, y₂, ...,y_n corresponding to the values x₀, x₁, x₂, ...,x_n. Where x_i = x₀ ih, i = 0,1,2, ...,n. Suppose it is required to evaluate f (x) for the x = x₀ ph, where p is any real number. We have for any real number p, we have defined E such that

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Analytical Functions: In this chapter we consider the linear space A(Ω) of all analytic functions on an open set Ω and introduce a metric d on A(Ω) with the property that convergence in the d-metric is uniform convergence on compact subsets of Ω. We will characterize the compact subsets of the metric space (A(Ω), d) and prove several useful results on convergence of sequences of analytic functions. After these preliminaries we will present a fairly standard proof of the Riemann mapping theorem and then consider the problem of extending the mapping function to the boundary. Also included in this chapter are Runge’s theorem on rational approximations and the homotopic version of Cauchy’s theorem. The Spaces A(Ω) and C(Ω):

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Complexification of the Integrand: We evaluate

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Isolated Singularities: It is often important to consider a function that is holomorphic on a punctured open set U \ { P }  C. Refer to Figure below. In this chapter we shall obtain a new kind of infinite series expansion which generalizes the idea of the power series expansion of a holomorphic function about a (nonsingular) point. We shall in the process completely classify the behavior of holomorphic functions near an isolated singular point.

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Laurent Series: A Laurent series on D(P, r) is a (formal) expression of the form

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Relation with Harmonic Functions: Analytic Functions are intimately related to harmonic functions. We say that a real-valued function h(x, y) on the plane is harmonic if it obeys Laplace's equation:

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Annulus of Convergence: The set of convergence of a Laurent series is either an open set of the form {z : 0 ≤ r₁ ≤ |z−P| ≤ r₂}, together with perhaps some or all of the boundary points of the set, or a set of the form {z : 0 ≤ r₁ ≤ |z −P| ≤ }, together with perhaps some or all of the boundary points of the set. Such an open set is called an (generalized) annulus centered at P. We shall let,

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Hurwitz Theorem: Suppose that { fn } is sequence in A (Ω) that coverges to f uniformly on compacts subsets of Ω. Let D (Zo,r) Ω and assume that f(z) ≠ 0 for [z - zo] = r. Then there is a positive integer N such that for n ≥ N, fn and f have same numbers of zeros in D (zo,r). Proof : Let ε = Then for sufficiently large n, . By Rouche theorem fn and f have the same numbers of zero in D (zo,r).

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Complex Arithmetic: The Real Numbers: We assume the reader to be familiar with the real number system R. We let R² = {(x, y) : ∈ R , y ∈ R} . These are ordered pairs of realnumbers. As we shall see, the complex numbers are nothing other than R² equipped with a special algebraic structure.

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Taylor Series: If function f (z) is analytic at all points inside a circle c, with its centre at the point a and radius R, then at each point z inside c,

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Cauchy Integral Theorem: If f is a holomorphic function on an open disc U in the complex plane, andif : [a, b] → U is a C¹ curve in U with (a) = (b), then

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The Exponential Function: We define the complex exponential as follows: 1. If z = x is real, then

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Polar Form of a Complex Number: A consequence of our first definition of the complex exponential— is that if ζ C, |ζ | = 1, then there is a unique number  θ, 0 ≤ 2, such that ζ = e^iθ (see below Figure). Here θ is the (signed) angle between the positive x axis and the ray Now, if z is any non-zero complex number, then

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Roots of Complex Numbers: The properties of the exponential operation can be used to find the n^th roots of a complex number. Example-1: To find all sixth roots of 2, we let re^iθ be an arbitrary sixth root of 2 and solve for r and θ. If

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Formula for the Derivative: Let U  C be an open set and let f be holomorphic on U. Then f C(U). Moreover, if D(P, r)  U and z D(P, r), then

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Topology of the Complex Plane:

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Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra: Let p(z) be a nonconstant (holomorphic) polynomial. Then p has a root. That is, there exists an α C such that p( α) = 0. Proof: Suppose not. Then g(z) = 1/p(z) is entire. Also when |z| → , then |p(z)| → . Thus 1/ |p(z)| → 0 as |z| → ; hence g is bounded. By Liouville’s Theorem, g is constant, hence p is constant. Contradiction: The polynomial p has degree k ≥ 1, then let α₁ denote the root provided by the Fundamental Theorem. By the Euclidean algorithm (see [HUN]), we may divide z − α₁ into p with no remainder to obtain;

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Evaluation of Definite Integrals: One of the most classical and fascinating applications of the calculus of residues is the calculation of definite (usually improper) real integrals. It is an over-simplification to call these calculations, taken together, a “technique”: it is more like a collection of techniques. We present several instances of the method.

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Evaluation of Real Integrals by Contour Integration: For such questions consider a unit radius circle with center at origin, as contour

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Calculus of Residues: Functions with Multiple Singularities: It turns out to be useful, especially in evaluating various types of integrals, to consider functions that have more than one “singularity.” We want to consider the following general question: Suppose that f : U \ { P₁, P₂, . . . , P_n } → C is a holomorphic function on an open set U  C with finitely many distinct points P₁, P₂, . . . , P_n removed. Suppose further that

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Sequences of Holomorphic Functions and their Derivatives: A sequence of functions gj defined on a common domain E is said to converge uniformly to a limit function g if, for each ε > 0, there is a number N > 0 such that for all j > N it holds that | gj (x)− g (x) | <  for every x E . The key point is that the degree of closeness of gj (x) to g (x) is independent of x E. Let fj : U → C, j = 1, 2, 3 . . . , be a sequence of holomorphic functions on an open set U in C. Suppose that there is a function f : U → C such that, for each compact subset E (a compact set is one that is closed and bounded of U, the restricted sequence f _j|E converges uniformly to f |E. Then f is holomorphic on U. [In particular, f C(U).] If fj , f, U are as in the preceding paragraph, then, for any k {0, 1, 2, . . . }, we have,

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Solution of System of Linear Equations: This chapter covers extremely important problem how to solve a large system of equations? The algebraic system of linear equations occur abundantly in various fields of science and engineering, like elasticity, electrical engineering ,fluid dynamics, heat transfer, structural analysis, statistics, biomedical engineering, vibration engineering and in control theory. A number of problems in numerical analysis can be reduced to, or approximated by, a system of linear equations. One of the most common source of such systems is through discretization of ordinary or partial differential equations or integral equations. The system of n-linear equations in n-unknowns can be written as:

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Gauss-Seidel Method: To understand the GaussSeidel method, we consider a system of 3 linear equations in 3 variables:

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Problems on Solving Linear System: Example-1 Solve the system of linear equations: x y z = 1, 3x y − 3z = 5, x − 2y − 5z = 10 by Crout’s method. Solution: Here,

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Simpson’s 3/8 Rule: Setting n = 3 in Newton’s Cotes Quadrature Formula, we obtain

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Simpson’s 1/3 Rule: Setting n = 2 in Newton’s Cotes Quadrature Formula, we obtain

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Problems on Numerical Technique - II: Example-1 find from the following table, the area bounded by the curve and x-axis from x = 7.47 to x = 7.52

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Picard's Method: First Order Differential Equations: Consider the first order differential equation;

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Runge-Kutta Method: C Runge and W. Kutta, two German mathematicians, developed this method and as such the method was named after them. It was developed to avoid the computation of higher order derivations which Taylor’s method involves. Runge-Kutta formulas posses the advantage of requiring only the functional values at some selected points. This method agrees with Taylor’s series solutions up to the term h^r where r differs from method to method and is called the order of that method. Here, we will discuss Runge-Kutta method of order four. Fourth Order Runge-Kutta Method:

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Euler's Method: First Order Differential Equations: Consider the differential equation;

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Crout Method: In this method , the coefficient matrix A is decomposed into matrices L and U. Here the matrix L is lower triangular matrix and U is the unit upper triangular matrix. Let us consider the system of linear equations:

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Numerical Differentiation: The general method for deriving the numerical differentiation formulae is to differentiate the interpolating polynomial. Here we illustrate the derivation with Newton’s forward difference formula only. The method of derivation being the same with regard to the other formulae. Consider Newton’ forward difference formula:

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Newton’s Cotes Quadrature Formula: Let where y = f (x) takes the values y₀, y₁, y₂, ...,y_n for x₀, x₁, x₂, ...,x_n. Let us divide the interval (a,b) into n equal parts of width h, so that;