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Math Help - Non technical question regarding advanced probability and statistics?

  1. #1
    Junior Member bijosn's Avatar
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    Question Non technical question regarding advanced probability and statistics?

    I would like to know from the experts which courses in their opinion are the bare minimal necessary to understand and master more advanced(graduate) level courses in statistics.

    From my research it seems that the following below would be suffiecient:
    here are some of the courses and descriptions taken from my distance learning universities math dep pamphlet.

    Linear algebra 1*
    Purpose: to obtain knowledge of systems of linear equations, Gaussian elimination and homogeneous systems; matrix algebra, partitioning of matrices,
    matrix inverses and elementary matrices; determinants, Laplace expansion and cofactor matrices; vector geometry, orthogonality, distance and the dot
    product, planes and the cross product, and least squares polynomial fi tt ing.

    Calculus A*
    Purpose: to equip students with those basic skills in diff erential and integral calculus which are essential for the physical, life and economic sciences. Some
    simple applications are covered.

    Calculus B*
    Purpose: to enable students to continue to obtain basic skills in diff erentiation and integration, and build on the knowledge provided by module MAT112.
    More advanced techniques and further basic applications are covered.

    Linear algebra 2
    Purpose: to understand and apply the following linear algebra concepts: vector spaces, rank of a matrix, eigenvalues and eigenvectors, diagonalisation
    of matrices, orthogonality in Rn, Gram-Schmidt algorithm, orthogonal diagonalisation of symmetric matrices, least squares polynomial fi tt ing, linear
    transformations, change of basis, invariant subspaces and direct sums, block triangular form.

    Calculus in higher dimensions
    Purpose: to gain clear knowledge and an understanding of vectors in n-space, functions from n-space to m-space, various types of derivatives (grad, div,
    curl, directional derivatives), higher-order partial derivatives, inverse and implicit functions, double integrals, triple integrals, line integrals and surface
    integrals, theorems of Green, Gauss and Stokes.

    Differential equations
    Purpose: to enable students to obtain knowledge of fi rst-order ordinary diff erential equations, linear diff erential equations of higher order, series solutions
    of diff erential equations (method of Frobenius), Laplace transform and partial diff erential equations (only an introduction).







    would I need any other courses like...??
    Ordinary differential equations*
    Purpose: to enable students to master the fundamental concepts and apply the methods for the solution of homogeneous and non-homogeneous systems
    of diff erential equations, as well as Gronwall’s inequality, qualitative theory, and the linearisation of nonlinear systems.

    Partial differential equations*
    Purpose: to introduce students to the following topics in partial diff erential equations; the equation of Laplace, the heat equation and the wave equation
    treated as typical examples of elliptic, parabolic and hyperbolic partial diff erential equations respectively, and methods of solution of the corresponding
    boundary value problems are also discussed

    Mechanics and the calculus of variations

    Purpose: to enable students to demonstrate a basic understanding of generalised coordinates, Hamilton’s principle, calculus of variations and the Euler-
    Lagrange equations, the problem of Lagrange and the isoperimetric problem, Hamilton-Jacobi theory and Poisson brackets, Equivalent Lagrangians,
    canonical transformations and Noether’s theorem and application of the variational principles in mechanics.



    which courses do you recommend as a minimal?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by bijosn View Post
    I would like to know from the experts which courses in their opinion are the bare minimal necessary to understand and master more advanced(graduate) level courses in statistics.

    From my research it seems that the following below would be suffiecient:
    here are some of the courses and descriptions taken from my distance learning universities math dep pamphlet.

    Linear algebra 1*
    Purpose: to obtain knowledge of systems of linear equations, Gaussian elimination and homogeneous systems; matrix algebra, partitioning of matrices,
    matrix inverses and elementary matrices; determinants, Laplace expansion and cofactor matrices; vector geometry, orthogonality, distance and the dot
    product, planes and the cross product, and least squares polynomial fi tt ing.

    Calculus A*
    Purpose: to equip students with those basic skills in diff erential and integral calculus which are essential for the physical, life and economic sciences. Some
    simple applications are covered.

    Calculus B*
    Purpose: to enable students to continue to obtain basic skills in diff erentiation and integration, and build on the knowledge provided by module MAT112.
    More advanced techniques and further basic applications are covered.

    Linear algebra 2
    Purpose: to understand and apply the following linear algebra concepts: vector spaces, rank of a matrix, eigenvalues and eigenvectors, diagonalisation
    of matrices, orthogonality in Rn, Gram-Schmidt algorithm, orthogonal diagonalisation of symmetric matrices, least squares polynomial fi tt ing, linear
    transformations, change of basis, invariant subspaces and direct sums, block triangular form.

    Calculus in higher dimensions
    Purpose: to gain clear knowledge and an understanding of vectors in n-space, functions from n-space to m-space, various types of derivatives (grad, div,
    curl, directional derivatives), higher-order partial derivatives, inverse and implicit functions, double integrals, triple integrals, line integrals and surface
    integrals, theorems of Green, Gauss and Stokes.

    Differential equations
    Purpose: to enable students to obtain knowledge of fi rst-order ordinary diff erential equations, linear diff erential equations of higher order, series solutions
    of diff erential equations (method of Frobenius), Laplace transform and partial diff erential equations (only an introduction).







    would I need any other courses like...??
    Ordinary differential equations*
    Purpose: to enable students to master the fundamental concepts and apply the methods for the solution of homogeneous and non-homogeneous systems
    of diff erential equations, as well as Gronwall’s inequality, qualitative theory, and the linearisation of nonlinear systems.

    Partial differential equations*
    Purpose: to introduce students to the following topics in partial diff erential equations; the equation of Laplace, the heat equation and the wave equation
    treated as typical examples of elliptic, parabolic and hyperbolic partial diff erential equations respectively, and methods of solution of the corresponding
    boundary value problems are also discussed

    Mechanics and the calculus of variations

    Purpose: to enable students to demonstrate a basic understanding of generalised coordinates, Hamilton’s principle, calculus of variations and the Euler-
    Lagrange equations, the problem of Lagrange and the isoperimetric problem, Hamilton-Jacobi theory and Poisson brackets, Equivalent Lagrangians,
    canonical transformations and Noether’s theorem and application of the variational principles in mechanics.



    which courses do you recommend as a minimal?
    You will probably benefit form ODE and PDE's, the mechanics and CoV is/are probably not useful, but you may well need a few analysis courses if you are going to tackle modern theoretical probability.

    CB
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  3. #3
    Senior Member
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    All of those look good, but you are missing some important stuff. In addition to a course in Analysis (aka Advanced Calculus), and in addition to general statistics courses, you'll want some mathematical statistics courses and more advanced applied statistics courses.

    As far as just math goes, Analysis is quite important. Any higher level math will help though, just to train your mind to do well at the graduate level.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by theodds View Post
    All of those look good, but you are missing some important stuff. In addition to a course in Analysis (aka Advanced Calculus)
    Analysis is not advanced calculus, it may contain the underpinnings of calculus but it is much more.

    CB
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  5. #5
    Senior Member
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    Quote Originally Posted by CaptainBlack View Post
    Analysis is not advanced calculus, it may contain the underpinnings of calculus but it is much more.

    CB
    In general I agree, but in terms of what one is likely to see if they register for a pure math "Advanced Calculus" versus a pure math "Analysis" course, both at the introductory level, my experience suggests that they usually cover the same stuff. Given that the standard text for both of these courses is Baby Rudin, I think it is reasonable to lump them together, at least at the introductory level. At any rate, the reason I included the alias is that the courses in Real Analysis at my university are given under the title of Advanced Calculus, so if it is the same at his I wanted to make sure that he didn't think the course wasn't being offered.
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