The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics many processes are described probabilistically, from the kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

As a more specific example of an application, the cache language models and other statistical language models used in natural language processing to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.

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Advantages of Linear Least Squares     

Linear least squares regression has earned its place as the primary tool for process modeling because of its effectiveness and completeness.
    Though there are types of data that are better described by functions that are nonlinear in the parameters, many processes in science and engineering are well-described by linear models. This is because either the processes are inherently linear or because, over short ranges, any process can be well-approximated by a linear model.
    The estimates of the unknown parameters obtained from linear least squares regression are the optimal estimates from a broad class of possible parameter estimates under the usual assumptions used for process modeling. Practically speaking, linear least squares regression makes very efficient use of the data. Good results can be obtained with relatively small data sets.
    Finally, the theory associated with linear regression is well-understood and allows for construction of different types of easily-interpretable statistical intervals for predictions, calibrations, and optimizations. These statistical intervals can then be used to give clear answers to scientific and engineering questions.

Disadvantages of Linear Least Squares     

The main disadvantages of linear least squares are limitations in the shapes that linear models can assume over long ranges, possibly poor extrapolation properties, and sensitivity to outliers.
    Linear models with nonlinear terms in the predictor variables curve relatively slowly, so for inherently nonlinear processes it becomes increasingly difficult to find a linear model that fits the data well as the range of the data increases. As the explanatory variables become extreme, the output of the linear model will also always more extreme. This means that linear models may not be effective for extrapolating the results of a process for which data cannot be collected in the region of interest. Of course extrapolation is potentially dangerous regardless of the model type.
    Finally, while the method of least squares often gives optimal estimates of the unknown parameters, it is very sensitive to the presence of unusual data points in the data used to fit a model. One or two outliers can sometimes seriously skew the results of a least squares analysis. This makes model validation, especially with respect to outliers, critical to obtaining sound answers to the questions motivating the construction of the model.

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The reason why eigenvalues are so important in mathematics are too many. Here is a very short and extremely incomplete list of the main applications I encountered in my path and that are coming now in mind to me:

Theoretical applications: 

• The eigenvalues of the Jacobian of a vector field at a given point determines the local geometry of the flow and the stability of that point; 
• An iterative method yk+1=Ayk is convergent if the spectral radius ρ(A) (the maximum absolute value of the eigenvalues of A) is < 1. 

Practical applications:

• Google Page Rank: The order in which your search results appear in Google is determined by computing an eigen vector. 
• Face Recognition: You can automatically recognize faces by computing eigenvectors of images. 
• In mechanics, the eigen vectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.
• Geology and glaciology: In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. 
• Molecular orbitals: In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. 
• Vibration analysis: Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom.

And there are also other type of eigenvalue problems, more difficult to solve, e.g. Generalized and Nonlinear Eigenvalue Problems, with even more interesting applications.

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Direct Methods

We use direct methods in absence of round off errors. Such method would yield the exact soluton within finite number of steps.

Iterative Methods

It is useful for problems involving special, very large matrices.

Drawbacks of Direct Methods

• In time dependent problems, direct method can not make good use of such information. 
• Sometime only matrix vector products are given. In other words, the matrix is not available explicitly or is very expensive to computer for example in DSP (Digital Signal Processing)

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Following are the advantages of iterative methods over direct methods:

• Each cycle is O(N2) operations for full storage mode.
• Round off errors only occurs during O(N2) operations which accure in O(N3) in direct method. 
• Since each cycle only produces an approximation for the next cycle, an error in a guess will be handled by the next cyle.
• We can consider round off error to accrue only during the last iterations. 

when to use iterative methods?

1. When very large matrices are there since they reduce the round off problems. 
2. Sparse but not bounded matrices since they can reduce computational efforet not oprationg on zeros.
3. Very large sparse banded matrices due to effeciency. 

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1. How can you solve the system of nonlinear equations? Explain the importance of Jacobian Matrix. Write down its algorithm and explain the steps.
2. If the date is not evenly spaced, which interpolation method you apply? Explain it with suitable example and algorithm.
3. What are the advantages of Least Square Method? Derive the equation for quadratic curve fitting. Write down its algorithm and explain it.
4. Why Numerical Integration is required? Explain it with suitable example. Derive the general quadrature formula.  Write down its algorithm and explain it.
5. Differentiate between direct and iterative method for these solution of liner system of equations. Derive the equation for Gauss Seideal Method, explain it with suitable example and write its algorithm. 
6. Write down the general form of second order partial differential equation and derive the Poission's equation. Explain the equation and write down its algorithm. 
7. Justify the Measure of Central Tendency, Dispersion, and Direction are important in the statistics. Explain with suitable equations and examples. 
8. Why distribution is important in the probability? Explain Poisson's distribution with suitable example and write down its algorithm.
9. Why do you require optimization? Explain the steps for Simplex method with suitable example and write down its algorithm. Compare it with dynamic programming. 
10. Why we require Discrete Fourier Transformation? Explain with suitable example. Compare it with Fast Fourier Transformation and write its algorithm. 

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An initial value problem is how to aim my gun. A boundary value problem is how to aim my gun so that the bullet hits the target.

Qualitatively the methods of solution are sometimes different, because Taylor series approximate a function at a single point, i.e. at 0.

Here is an example:
For an ordinary 2nd order linear differential equation
y" + f(t) y' + g(t)y = 0,

- the initial value problem is to find the partial integral (solution) of the equation which satisfies the initial conditions
y(t0) = a, y'(t0) = b (t0, a and b are arbitrary constants);

- the boundary value problem is to find the partial integral which satisfies the boundary conditions
y(t0) = c, y(t1) = d (t0, t1, c and d are arbitrary constants, t0 < t1).

In other words: the initial value problem involves conditions in one point (the initial point of an interval), the boundary value problem involves conditions in various points (boundary points of the interval). 

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Probability sampling is a method for selecting choices on a completely random basis. Commonly, probability sampling is used to ensure that the selected sample is totally random, and not subject to any controls or rigging. This system works well, as long as all rules are followed, and the system is not violated in any way.

• Advantages
The principal advantage of probability sampling is fairness, as in contests where names are selected from a box full of entry forms. A selector will reach into the box, without looking, and pluck out the winner's name. This sort of selection process guarantees that every entrant has an equal chance at winning a prize.

• Disadvantages

The disadvantage of probability sampling is the possibility of flaws to the randomness model - in other words, people may cheat the system or interfere with the innate fairness of the probability sampling system. For example, in a contest, illegal or duplicate ballots may be added to a box, weighting the odds in someone's favor.

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