IE 207 • Midterm • Probability

Sample Space and Events

Probability

Axioms of Probability

Some Rules

Coin Example

Dice Example

Card Example 1

Card Example 2

Ball Example

Set Example

Birthday Example

Conditioning Events

Total Probability Rule

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Bayes' Rule

Bayes' Rule Example 1

Bayes' Rule Example 2

Independence

Independence Example 1

Independence Example 2

Axioms of Probability 1

Axioms of Probability 2

Axioms of Probability 3

Axioms of Probability 4

Conditional Probability and Independence 1

Conditional Probability and Independence 2

Conditional Probability and Independence 3

Conditional Probability and Independence 4

Conditional Probability and Independence 5

Conditional Probability and Independence 6

Conditional Probability and Independence 7

Conditional Probability and Independence 8

Conditional Probability and Independence 9

Bayes' Rule 1

Bayes' Rule 2

Bayes' Rule 3

Bayes' Rule 4

Bayes' Rule 5

Bayes' Rule 6

Bayes' Rule 7

Bayes' Rule 8

Bayes' Rule 9

Random Variables

Probability Mass Function (PMF) for Discrete RV

PMF Example 1

PMF Example 2

Cumulative Distribution Function for Discrete RV

CDF Example

Probability Density Function (PDF) for Continuous RV

PDF Example

Cumulative Distribution Function (CDF) for Continuous RV

CDF Example

Discrete Random Variable 1

Discrete Random Variable 2

Discrete Random Variable 3

Discrete Random Variable 4

Discrete Random Variable 5

Discrete Random Variable 6

Discrete Random Variable 7

Continuous Random Variable 1

Continuous Random Variable 2

Continuous Random Variable 3

Continuous Random Variable 4

Continuous Random Variable 5

Continuous Random Variable 6

Continuous Random Variable 7

Discrete Joint Random Variables

Discrete Joint RV Example

Marginal PMF and CDF for Discrete Joint RV

Conditional PMF and CDF for Discrete Joint RV

Continuous Joint Random Variables

Marginal PDF and CDF for Continuous Joint RV

Conditional PDF and CDF for Continuous Joint RV

Expected Value for Discrete RV

Example 1

Example 2

Variance for Discrete RV

Example 3

Example 4

Expected Value for Continuous RV

Example 5

Example 6

Variance for Continuous RV

Example 7

Expected Value for Discrete Joint RV

Variance for Discrete Joint RV

Example 8

Conditional Expectation for Discrete Joint RV

Example 9

Example 10

Expected Value and Variance for Continuous Joint RV

Conditional Expectation for Continuous Joint RV

Example 11

Example 12

Example 13

Discrete Random Variable 1- Expected Value

Discrete Random Variable 2 - Expected Value

Discrete Random Variable 3 - Variance

Discrete Random Variable 4- Expected Value

Discrete Random Variable 5 - Expected Value

Discrete Random Variable 6 - Expected Value and Variance

Continuous Random Variable 1 - Expected Value

Continuous Random Variable 2 - Expected Value

Continuous Random Variable 3 - Variance

Continuous Random Variable 4 - Expected Value

Continuous Random Variable 5 - Expected Value and Variance

Discrete Joint Distribution 1

Discrete Joint Distribution 2

Discrete Joint Distribution 3

Discrete Joint Distribution 4

Continuous Joint Distribution 1

Continuous Joint Distribution 2

Continuous Joint Distribution 3

Continuous Joint Distribution 4

Continuous Joint Distribution 5

Continuous Joint Distribution 6