Probability Theory and Distributions

7.

Joint Distributions and Dependence

7.1.

Joint Probability Distributions

7.1.1.

Discrete Joint Distributions

7.1.1.1.

Joint Probability Mass Function

7.1.1.2.

Properties and Validity Conditions

7.1.1.3.

Tabular Representation

7.1.2.

Continuous Joint Distributions

7.1.2.1.

Joint Probability Density Function

7.1.2.2.

Properties and Validity Conditions

7.1.2.3.

Geometric Interpretation

7.1.3.

Support of Joint Distributions

7.1.3.1.

Definition and Examples

7.1.3.2.

Rectangular vs Non-rectangular Support

7.2.

Marginal Distributions

7.2.1.

Marginal PMF from Joint PMF

7.2.1.1.

Summation Method

7.2.2.

Marginal PDF from Joint PDF

7.2.2.1.

Integration Method

7.2.3.

Relationship to Joint Distribution

7.2.3.1.

Information Loss in Marginalization

7.2.3.2.

Uniqueness Questions

7.3.

Conditional Distributions

7.3.1.

Conditional PMF

7.3.1.1.

Definition Given Discrete Condition

7.3.1.2.

Properties as Probability Distribution

7.3.2.

Conditional PDF

7.3.2.1.

Definition Given Continuous Condition

7.3.2.2.

Properties as Probability Distribution

7.3.3.

Conditional Expectation

7.3.3.1.

Definition and Calculation

7.3.3.2.

Law of Total Expectation

7.3.4.

Independence and Conditional Distributions

7.3.4.1.

Factorization Property

7.3.4.2.

Conditional Independence

7.4.

Covariance and Correlation

7.4.1.

7.4.1.1.

Definition and Interpretation

7.4.1.2.

Computational Formula

7.4.1.3.

7.4.1.3.1.

7.4.1.3.2.

Effect of Linear Transformations

7.4.1.3.3.

Covariance of Independent Variables

7.4.2.

Correlation Coefficient

7.4.2.1.

Pearson Correlation Coefficient

7.4.2.2.

Properties and Interpretation

7.4.2.3.

Range and Extremes

7.4.2.4.

Scale Invariance

7.4.3.

Linear Relationships

7.4.3.1.

Perfect Correlation Cases

7.4.3.2.

Correlation vs Causation

7.4.3.3.

Limitations of Linear Correlation

7.5.

Independence of Random Variables

7.5.1.

Definition via Joint Distribution

7.5.1.1.

Factorization Criterion

7.5.1.2.

7.5.1.3.

Continuous Case

7.5.2.

Equivalent Characterizations

7.5.2.1.

Via Conditional Distributions

7.5.2.2.

Via Moment Generating Functions

7.5.2.3.

Via Characteristic Functions

7.5.3.

Properties of Independent Variables

7.5.3.1.

Functions of Independent Variables

7.5.3.2.

Sums and Products

7.5.3.3.

Moment Properties

7.5.4.

Testing for Independence

7.5.4.1.

Graphical Methods

7.5.4.2.

Analytical Methods

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