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Exploring High-Dimensional Spaces with Mathematical Insights

Discover the intriguing world of N-dimensional Euclidean space, Stirling's approximation, binomial bounds, and Chebyshev's inequality. Dive into fascinating facts about volume, angles, variance, and the Law of Large Numbers. Unveil the secrets of mathematical preliminaries and their applications in high-dimensional realms.

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Exploring High-Dimensional Spaces with Mathematical Insights

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  1. Chapter 9 Mathematical Preliminaries

  2. Stirling’s Approximation Fig. 9.2-1 by trapezoid rule take antilogs . . . . Fig. 9.2-2 by midpoint formula take antilogs  9.2

  3. Binomial Bounds Show the volume of a sphere of radius λn in the n-dimensional unit hypercube is: Assuming 0  λ  ½ (since the terms are reflected about n/2) the terms grow monotonically, and bounding the last by Stirling gives: 9.3

  4. N.b. 9.3

  5. by convention The Gamma Function For n > 1, integrate by parts: dg = e−xdxf = xn−1 Idea: extend the factorial to non-integral arguments. 9.4

  6. dr rdθ r dx dy area = rdrdθ area = dxdy 9.4

  7. N – Dimensional Euclidean Space Use Pythagorean distance to define spheres: Consequently, their volume depends proportionally on rn converting to polar coordinates 9.5

  8. r2 t just verify by substitution 9.5

  9. Interesting Facts aboutN-dimensional Euclidean Space Cn→ 0 as n → ∞  Vn(r) → 0 as n → ∞ for a fixed r Volume approaches 0 as the dimension increases! Almost all the volume is near the surface (as n→∞) end of 9.5

  10. Angle between the vector (1, 1, …, 1) and each coordinate axis: As n→ ∞: cos θ → 0,  θ → π/2. length of projection along axis What about the angle between random vectors, x and y, of the form (±1, ±1, …, ±1)? length of entire vector For large n, the diagonal line is almost perpendicular to each axis! By definition: Hence, for large n, there are almost 2n random diagonal lines which are almost perpendicular to each other! end of 9.8

  11. Chebyshev’s Inequality x2× p(x) ≥ 0 Let X be a discrete or continuous random variable with p(xi) = the probability that X = xi. The mean square is Chebyshev’s inequality 9.7

  12. Variance The variance of X is the mean square about the mean value of X, So variance, (via linearity) is: Note: V{1} = 0 → V{c} = 0 & V{cX} = c²V{X} Also: V{X − b} = V{X} 9.7

  13. The Law of Large Numbers Let X and Y be independent random variables, with E{X} = a, E{Y} = b, V{X} = σ2, V{Y} = τ2. because of independence Then E{(X− a) ∙ (Y − b)} = E{X − a} ∙ E{Y − b} = 0 ∙ 0 = 0And V{X + Y} = V{X − a + Y − b} = E{(X− a + Y − b)2} = E{(X − a)2} + 2E{(X − a)(Y − b)} + E{(Y − b)2} = V{X} + V{Y} = σ2 + τ2 Consider n independent trials for X; called X1, …, Xn. The expectation of their average is (as expected!): 9.8

  14. The variance of their average is (using independence): So, what is the probability that their average A is not close to the mean E{X} = a? Use Chebyshev’s inequality: Let n→ ∞ Weak Law of Large Numbers: The average of a large enough number of independent trials comes arbitrarily close to the mean with arbitrarily high probability. 9.8

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