1: Basic Definitions and Results

• 1.1Basic Definitions and Results Is Probability Intuitive or Mathematical?
• Basic Definitions---Borel Sets
• Axioms of Probability
• Immediate Deductions from Axioms of Probability
• From Mathematical to Intuitive Definition of Probability
• 1.2Bayes Theorem: Conditional Probabilities---1) Frequency Approach
• Bayes Theorem: Conditional Probabilities---2) Frequency and Probability
• Bayes Theorem: Conditional Probabilities---3) Bayes Law
• Estimation of Image Scanning Efficiency---I
• Example: Estimation of Image Scanning Efficiency---II
• Example: Analysis of Unreliable Test
• The Bayes Controversy Naive Application for Estimation---I
• The Bayes Controversy Naive Application for Estimation---II
• The Bayes Controversy Naive Application for Estimation---III
• 1.3Continuous Distributions General (Borel) Formalism
• Gaussian Distribution---I
• Properties of Gaussian Distribution
• 1.4Functions of a Random Variable
• Joint Probability Distributions
• Example of Joint Probability Distributions: Bayes Law for Densities---I
• Example of Joint Probability Distributions: Bayes Law for Densities---II
• 1.5Properties of Joint Distribution Functions
• Means---Moments, etc. for One-dimensional Distributions
• 1.6Means--Moments--Correlations for Multidimensional Distributions I
• Means---Moments---Correlations for Multidimensional Distributions II
• Errors and Moment Matrices
• Density Matrix Element Example---I
• Density Matrix Element Example---II
• Examples of Correlations 2D Track Finding---I
• Examples of Correlations 2D Track Finding---II
• Examples of Correlations Exponential Fits---I
• Examples of Correlations Exponential Fits---II
• Errors in Linear Combinations of Uncorrelated Variables
• Error in Lack of Correlation Assumption
• Parameters of a Gaussian Distribution---I
• Parameters of a Gaussian Distribution---II
• Addition of Independent Random Variables
• 1.7Leading Up to the Central Limit Theorem Moment Generating Function---I
• Moment Generating Function of a Sum of Two Random Variables
• Moment Generating Function III
• Central Limit Theorem Statement Laplace
• Central Limit Theorem Statement---Liapounoff
• Proof of Central Limit Theorem Laplace---I
• Proof of Central Limit Theorem Laplace---II
• Monte-Carlo Integration and the Central Limit Theorem---I
• Monte-Carlo Integration and the Central Limit Theorem---II
• Monte-Carlo Integration and the Central Limit Theorem---III
• Extension of Central Limit Theorem to Correlated Random Variables---I
• Extension of Central Limit Theorem to Correlated Random Variables---II
• Central Limit Theorem for Quotients
• (i) Example from Histogramming (Physics) Events---I
• (i) Example from Histogramming (Physics) Events---II
• (i) Example from Histogramming (Physics) Events---III
• (ii) Analysis of Scattering Experiments---I
• (ii) Analysis of Scattering Experiments---II
• (iii) Example of Weighted Histograms