This course covers advanced topics in integration, including finding integrals of complex functions, Riemann Stieltjes Integration, Fourier Series integration, and more. Students will learn techniques such as LOTUS, Gaussian Integrals, Spherical Coordinates, and solving integrals using the Beta Function. The course is designed for statisticians and individuals with a strong background in calculus and mathematical analysis. The teaching method includes theoretical explanations, proofs, and practical illustrations using both univariate and multivariate examples.
Overview
Syllabus
Find the Integral of (1+x^2)^-n.
2nd Mean Value Theorem for Integrals.
Show that f(t)=sin(t/2)^-1 - (t/2)^-1 is integrable in (0,d).
Integration of a Fourier Series (F9).
Riemann Stieltjes Integration for Statisticians.
Fourier Coefficients: Riemann Lebesgue Theorem (F1).
LOTUS - Law of the Unconcious Statistician.
Probability of 1st Quadrant for a Standard Bivariate Normal Random Variable.
Integrating a Bivariate Normal Distribution.
Find the integral of z^(k+l+1) * (w/z - 1)^((2k+1)/2.
Illustration using univariate LOTUS: Derive the MGF for a 1 df noncentral Chi square Distribution.
Illustration using multivariate LOTUS: Derive the MGF or a k df noncentral Chi square Distribution.
Probability of 4th Quadrant for a Standard Bivariate Normal Random Variable.
Integrating over a Rectangle in Polar Coordinates.
Probability of 1st Quadrant for a Scaled Bivariate Normal Random Variable.
Using R: Calculating Probability for a Bivariate Normal Random Variable.
Gaussian Integrals.
Spherical Coordinates.
Useful Trig Integral: Sin & Cos power reduction.
Proof of the Hermite - Hadamard Inequality.
Solving Integrals using the Beta Function.
Useful Trig Integral: Secant power reduction.
Viral Calculus Problem - ATM Card Pin Number: Solution using Trig functions.
Taught by
statisticsmatt
