Transformations of Random Variables

Overview

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Explore a comprehensive video series on transformations of random variables in statistics. Learn techniques for handling discrete and continuous, univariate and multivariate random variables. Master methods like CDF technique, change of variable, and multivariate transformations. Dive into specific transformations involving various distributions such as Uniform, Normal, Binomial, Gamma, Beta, Cauchy, and Chi-square. Study advanced topics including order statistics, linear transformations of random vectors, orthogonal transformations, and the independence of sample mean and variance in normal distributions. Gain practical insights through examples like comparing adverse event rates and proving statistical theorems.

Syllabus

Transformations: Discrete Univariate Random Variables.
Transformations: CDF Technique.
Transformations: Univariate Change of Variable.
Transformations: Multivariate to Univariate CDF Technique.
Transformations: Discrete Multivariate Sum of Binomial Random Variables.
Transformations: Multivariate Change of Variable.
Transformations: Product of Uniform Random Variables.
Transformations: Ratio of Standard Normal Random Variables to a Cauchy Random Variable.
Transformations: Uniform Random Variable to a Cauchy Random Variable.
Transformations: t Random Variable to a Beta Random Variable.
Transformations: Sums and Ratios of Gamma Random Variables.
Transformations: Derive the Joint pdf of Order Statistics.
Transformations: F Random Variable to a Beta Random Variable.
Transformations: Sums and Ratios of Exponential Random Variables.
Transformations: Sums of Chi square Random Variables.
Transformations: Xj has a Continuous and Strictly Increasing CDF.
Transformations: X~Unif[0,2] to Y=(X/(1-X))**2.
Transformations: Sums of Binomial Random Variables with Unequal Proportions.
Linear Transformations of Random Vectors.
Orthogonal Transformation of Independent Normal Random Variables.
Sample Mean and Variance are Independent if Data are Normal.
Ratio of Independent Gamma rv's to a Beta Prime Distribution.
Beta Prime Random Variable to a Beta Distribution and Vice Versa.
Multivariate Normal Random Variable transformed to a Multivariate Uniform Random Variable.
Distribution for the Sum of Negative Binomial Random Variables Using the MGF.
Derivation of the Dirichlet Distribution.
Distribution of the Mid Range Statistic.
Example using Vandermonde's Identity: Sum of 2 Binomial Random Variables w/ Equal p..
Crazy Transformation: y=sin(x)^2 where x~unif(0,2Pi).
Transform 2 Normal RVs to a Rayleigh RV (part 1/2).
Transform 2 Normal RVs to a Rayleigh RV (part 2/2).
Comparing Adverse Event Rates Between Treatment Arms.
Joint Distribution of the Mid-Range and the Range.
Prove Pearson's Goodness of Fit Test Statistic limits to a Chi-sq Distribution.
Show that R & Theta are Independent in Polar Coordinates.