Explore advanced integration techniques and applications in statistics through a comprehensive 3.5-hour tutorial. Delve into complex topics such as the 2nd Mean Value Theorem, Riemann-Stieltjes Integration, Fourier Series, and the Law of the Unconscious Statistician (LOTUS). Master the intricacies of bivariate normal distributions, including probability calculations and integration methods. Learn to apply these concepts to real-world problems, such as deriving moment-generating functions for non-central chi-square distributions. Gain practical skills in using R for bivariate normal probability calculations. Cover additional mathematical tools like Gaussian integrals, spherical coordinates, and the Beta function. Conclude with solving challenging calculus problems, including a viral ATM card PIN number question using trigonometric functions.
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
