Section 1: Probability by Counting 

Experiments, Outcomes, Samples, Spaces, Events 
59:30 
 
Intro 
0:00  
 
Terminology 
0:19  
 
 Experiment 
0:26  
 
 Outcome 
0:56  
 
 Sample Space 
1:16  
 
 Event 
1:55  
 
Key Formula 
2:47  
 
 Formula for Finding the Probability of an Event 
2:48  
 
 Example: Drawing a Card 
3:36  
 
Example I 
5:01  
 
 Experiment 
5:38  
 
 Outcomes 
5:54  
 
 Probability of the Event 
8:11  
 
Example II 
12:00  
 
 Experiment 
12:17  
 
 Outcomes 
12:34  
 
 Probability of the Event 
13:49  
 
Example III 
16:33  
 
 Experiment 
17:09  
 
 Outcomes 
17:33  
 
 Probability of the Event 
18:25  
 
Example IV 
21:20  
 
 Experiment 
21:21  
 
 Outcomes 
22:00  
 
 Probability of the Event 
23:22  
 
Example V 
31:41  
 
 Experiment 
32:14  
 
 Outcomes 
32:35  
 
 Probability of the Event 
33:27  
 
 Alternate Solution 
40:16  
 
Example VI 
43:33  
 
 Experiment 
44:08  
 
 Outcomes 
44:24  
 
 Probability of the Event 
53:35  

Combining Events: Multiplication & Addition 
1:02:47 
 
Intro 
0:00  
 
Unions of Events 
0:40  
 
 Unions of Events 
0:41  
 
 Disjoint Events 
3:42  
 
Intersections of Events 
4:18  
 
 Intersections of Events 
4:19  
 
Conditional Probability 
5:47  
 
 Conditional Probability 
5:48  
 
Independence 
8:20  
 
 Independence 
8:21  
 
 Warning: Independent Does Not Mean Disjoint 
9:53  
 
 If A and B are Independent 
11:20  
 
Example I: Choosing a Number at Random 
12:41  
 
 Solving by Counting 
12:52  
 
 Solving by Probability 
17:26  
 
Example II: Combination 
22:07  
 
 Combination Deal at a Restaurant 
22:08  
 
Example III: Rolling Two Dice 
24:18  
 
 Define the Events 
24:20  
 
 Solving by Counting 
27:35  
 
 Solving by Probability 
29:32  
 
Example IV: Flipping a Coin 
35:07  
 
 Flipping a Coin Four Times 
35:08  
 
Example V: Conditional Probabilities 
41:22  
 
 Define the Events 
42:23  
 
 Calculate the Conditional Probabilities 
46:21  
 
Example VI: Independent Events 
53:42  
 
 Define the Events 
53:43  
 
 Are Events Independent? 
55:21  

Choices: Combinations & Permutations 
56:03 
 
Intro 
0:00  
 
Choices: With or Without Replacement? 
0:12  
 
 Choices: With or Without Replacement? 
0:13  
 
 Example: With Replacement 
2:17  
 
 Example: Without Replacement 
2:55  
 
Choices: Ordered or Unordered? 
4:10  
 
 Choices: Ordered or Unordered? 
4:11  
 
 Example: Unordered 
4:52  
 
 Example: Ordered 
6:08  
 
Combinations 
9:23  
 
 Definition & Equation: Combinations 
9:24  
 
 Example: Combinations 
12:12  
 
Permutations 
13:56  
 
 Definition & Equation: Permutations 
13:57  
 
 Example: Permutations 
15:00  
 
Key Formulas 
17:19  
 
 Number of Ways to Pick r Things from n Possibilities 
17:20  
 
Example I: Five Different Candy Bars 
18:31  
 
Example II: Five Identical Candy Bars 
24:53  
 
Example III: Five Identical Candy Bars 
31:56  
 
Example IV: Five Different Candy Bars 
39:21  
 
Example V: Pizza & Toppings 
45:03  

Inclusion & Exclusion 
43:40 
 
Intro 
0:00  
 
Inclusion/Exclusion: Two Events 
0:09  
 
 Inclusion/Exclusion: Two Events 
0:10  
 
Inclusion/Exclusion: Three Events 
2:30  
 
 Inclusion/Exclusion: Three Events 
2:31  
 
Example I: Inclusion & Exclusion 
6:24  
 
Example II: Inclusion & Exclusion 
11:01  
 
Example III: Inclusion & Exclusion 
18:41  
 
Example IV: Inclusion & Exclusion 
28:24  
 
Example V: Inclusion & Exclusion 
39:33  

Independence 
46:09 
 
Intro 
0:00  
 
Formula and Intuition 
0:12  
 
 Definition of Independence 
0:19  
 
 Intuition 
0:49  
 
Common Misinterpretations 
1:37  
 
 Myth & Truth 1 
1:38  
 
 Myth & Truth 2 
2:23  
 
Combining Independent Events 
3:56  
 
 Recall: Formula for Conditional Probability 
3:58  
 
 Combining Independent Events 
4:10  
 
Example I: Independence 
5:36  
 
Example II: Independence 
14:14  
 
Example III: Independence 
21:10  
 
Example IV: Independence 
32:45  
 
Example V: Independence 
41:13  

Bayes' Rule 
1:02:10 
 
Intro 
0:00  
 
When to Use Bayes' Rule 
0:08  
 
 When to Use Bayes' Rule: Disjoint Union of Events 
0:09  
 
Bayes' Rule for Two Choices 
2:50  
 
 Bayes' Rule for Two Choices 
2:51  
 
Bayes' Rule for Multiple Choices 
5:03  
 
 Bayes' Rule for Multiple Choices 
5:04  
 
Example I: What is the Chance that She is Diabetic? 
6:55  
 
 Example I: Setting up the Events 
6:56  
 
 Example I: Solution 
11:33  
 
Example II: What is the chance that It Belongs to a Woman? 
19:28  
 
 Example II: Setting up the Events 
19:29  
 
 Example II: Solution 
21:45  
 
Example III: What is the Probability that She is a Democrat? 
27:31  
 
 Example III: Setting up the Events 
27:32  
 
 Example III: Solution 
32:08  
 
Example IV: What is the chance that the Fruit is an Apple? 
39:11  
 
 Example IV: Setting up the Events 
39:12  
 
 Example IV: Solution 
43:50  
 
Example V: What is the Probability that the Oldest Child is a Girl? 
51:16  
 
 Example V: Setting up the Events 
51:17  
 
 Example V: Solution 
53:07  
Section 2: Random Variables 

Random Variables & Probability Distribution 
38:21 
 
Intro 
0:00  
 
Intuition 
0:15  
 
 Intuition for Random Variable 
0:16  
 
 Example: Random Variable 
0:44  
 
Intuition, Cont. 
2:52  
 
 Example: Random Variable as Payoff 
2:57  
 
Definition 
5:11  
 
 Definition of a Random Variable 
5:13  
 
 Example: Random Variable in Baseball 
6:02  
 
Probability Distributions 
7:18  
 
 Probability Distributions 
7:19  
 
Example I: Probability Distribution for the Random Variable 
9:29  
 
Example II: Probability Distribution for the Random Variable 
14:52  
 
Example III: Probability Distribution for the Random Variable 
21:52  
 
Example IV: Probability Distribution for the Random Variable 
27:25  
 
Example V: Probability Distribution for the Random Variable 
34:12  

Expected Value (Mean) 
46:14 
 
Intro 
0:00  
 
Definition of Expected Value 
0:20  
 
 Expected Value of a (Discrete) Random Variable or Mean 
0:21  
 
Indicator Variables 
3:03  
 
 Indicator Variable 
3:04  
 
Linearity of Expectation 
4:36  
 
 Linearity of Expectation for Random Variables 
4:37  
 
Expected Value of a Function 
6:03  
 
 Expected Value of a Function 
6:04  
 
Example I: Expected Value 
7:30  
 
Example II: Expected Value 
14:14  
 
Example III: Expected Value of Flipping a Coin 
21:42  
 
 Example III: Part A 
21:43  
 
 Example III: Part B 
30:43  
 
Example IV: Semester Average 
36:39  
 
Example V: Expected Value of a Function of a Random Variable 
41:28  

Variance & Standard Deviation 
47:23 
 
Intro 
0:00  
 
Definition of Variance 
0:11  
 
 Variance of a Random Variable 
0:12  
 
 Variance is a Measure of the Variability, or Volatility 
1:06  
 
 Most Useful Way to Calculate Variance 
2:46  
 
Definition of Standard Deviation 
3:44  
 
 Standard Deviation of a Random Variable 
3:45  
 
Example I: Which of the Following Sets of Data Has the Largest Variance? 
5:34  
 
Example II: Which of the Following Would be the Least Useful in Understanding a Set of Data? 
9:02  
 
Example III: Calculate the Mean, Variance, & Standard Deviation 
11:48  
 
 Example III: Mean 
12:56  
 
 Example III: Variance 
14:06  
 
 Example III: Standard Deviation 
15:42  
 
Example IV: Calculate the Mean, Variance, & Standard Deviation 
17:54  
 
 Example IV: Mean 
18:47  
 
 Example IV: Variance 
20:36  
 
 Example IV: Standard Deviation 
25:34  
 
Example V: Calculate the Mean, Variance, & Standard Deviation 
29:56  
 
 Example V: Mean 
30:13  
 
 Example V: Variance 
33:28  
 
 Example V: Standard Deviation 
34:48  
 
Example VI: Calculate the Mean, Variance, & Standard Deviation 
37:29  
 
 Example VI: Possible Outcomes 
38:09  
 
 Example VI: Mean 
39:29  
 
 Example VI: Variance 
41:22  
 
 Example VI: Standard Deviation 
43:28  

Markov's Inequality 
26:45 
 
Intro 
0:00  
 
Markov's Inequality 
0:25  
 
 Markov's Inequality: Definition & Condition 
0:26  
 
 Markov's Inequality: Equation 
1:15  
 
 Markov's Inequality: Reverse Equation 
2:48  
 
Example I: Money 
4:11  
 
Example II: Rental Car 
9:23  
 
Example III: Probability of an Earthquake 
12:22  
 
Example IV: Defective Laptops 
16:52  
 
Example V: Cans of Tuna 
21:06  

Tchebysheff's Inequality 
42:11 
 
Intro 
0:00  
 
Tchebysheff's Inequality (Also Known as Chebyshev's Inequality) 
0:52  
 
 Tchebysheff's Inequality: Definition 
0:53  
 
 Tchebysheff's Inequality: Equation 
1:19  
 
 Tchebysheff's Inequality: Intuition 
3:21  
 
Tchebysheff's Inequality in Reverse 
4:09  
 
 Tchebysheff's Inequality in Reverse 
4:10  
 
 Intuition 
5:13  
 
Example I: Money 
5:55  
 
Example II: College Units 
13:20  
 
Example III: Using Tchebysheff's Inequality to Estimate Proportion 
16:40  
 
Example IV: Probability of an Earthquake 
25:21  
 
Example V: Using Tchebysheff's Inequality to Estimate Proportion 
32:57  
Section 3: Discrete Distributions 

Binomial Distribution (Bernoulli Trials) 
52:36 
 
Intro 
0:00  
 
Binomial Distribution 
0:29  
 
 Binomial Distribution (Bernoulli Trials) Overview 
0:30  
 
 Prototypical Examples: Flipping a Coin n Times 
1:36  
 
 Process with Two Outcomes: Games Between Teams 
2:12  
 
 Process with Two Outcomes: Rolling a Die to Get a 6 
2:42  
 
Formula for the Binomial Distribution 
3:45  
 
 Fixed Parameters 
3:46  
 
 Formula for the Binomial Distribution 
6:27  
 
Key Properties of the Binomial Distribution 
9:54  
 
 Mean 
9:55  
 
 Variance 
10:56  
 
 Standard Deviation 
11:13  
 
Example I: Games Between Teams 
11:36  
 
Example II: Exam Score 
17:01  
 
Example III: Expected Grade & Standard Deviation 
25:59  
 
Example IV: Pogosticking Championship, Part A 
33:25  
 
Example IV: Pogosticking Championship, Part B 
38:24  
 
Example V: Expected Championships Winning & Standard Deviation 
45:22  

Geometric Distribution 
52:50 
 
Intro 
0:00  
 
Geometric Distribution 
0:22  
 
 Geometric Distribution: Definition 
0:23  
 
 Prototypical Example: Flipping a Coin Until We Get a Head 
1:08  
 
 Geometric Distribution vs. Binomial Distribution. 
1:31  
 
Formula for the Geometric Distribution 
2:13  
 
 Fixed Parameters 
2:14  
 
 Random Variable 
2:49  
 
 Formula for the Geometric Distribution 
3:16  
 
Key Properties of the Geometric Distribution 
6:47  
 
 Mean 
6:48  
 
 Variance 
7:10  
 
 Standard Deviation 
7:25  
 
Geometric Series 
7:46  
 
 Recall from Calculus II: Sum of Infinite Series 
7:47  
 
 Application to Geometric Distribution 
10:10  
 
Example I: Drawing Cards from a Deck (With Replacement) Until You Get an Ace 
13:02  
 
 Example I: Question & Solution 
13:03  
 
Example II: Mean & Standard Deviation of Winning Pin the Tail on the Donkey 
16:32  
 
 Example II: Mean 
16:33  
 
 Example II: Standard Deviation 
18:37  
 
Example III: Rolling a Die 
22:09  
 
 Example III: Setting Up 
22:10  
 
 Example III: Part A 
24:18  
 
 Example III: Part B 
26:01  
 
 Example III: Part C 
27:38  
 
 Example III: Summary 
32:02  
 
Example IV: Job Interview 
35:16  
 
 Example IV: Setting Up 
35:15  
 
 Example IV: Part A 
37:26  
 
 Example IV: Part B 
38:33  
 
 Example IV: Summary 
39:37  
 
Example V: Mean & Standard Deviation of Time to Conduct All the Interviews 
41:13  
 
 Example V: Setting Up 
42:50  
 
 Example V: Mean 
46:05  
 
 Example V: Variance 
47:37  
 
 Example V: Standard Deviation 
48:22  
 
 Example V: Summary 
49:36  

Negative Binomial Distribution 
51:39 
 
Intro 
0:00  
 
Negative Binomial Distribution 
0:11  
 
 Negative Binomial Distribution: Definition 
0:12  
 
 Prototypical Example: Flipping a Coin Until We Get r Successes 
0:46  
 
 Negative Binomial Distribution vs. Binomial Distribution 
1:04  
 
 Negative Binomial Distribution vs. Geometric Distribution 
1:33  
 
Formula for Negative Binomial Distribution 
3:39  
 
 Fixed Parameters 
3:40  
 
 Random Variable 
4:57  
 
 Formula for Negative Binomial Distribution 
5:18  
 
Key Properties of Negative Binomial 
7:44  
 
 Mean 
7:47  
 
 Variance 
8:03  
 
 Standard Deviation 
8:09  
 
Example I: Drawing Cards from a Deck (With Replacement) Until You Get Four Aces 
8:32  
 
 Example I: Question & Solution 
8:33  
 
Example II: Chinchilla Grooming 
12:37  
 
 Example II: Mean 
12:38  
 
 Example II: Variance 
15:09  
 
 Example II: Standard Deviation 
15:51  
 
 Example II: Summary 
17:10  
 
Example III: Rolling a Die Until You Get Four Sixes 
18:27  
 
 Example III: Setting Up 
19:38  
 
 Example III: Mean 
19:38  
 
 Example III: Variance 
20:31  
 
 Example III: Standard Deviation 
21:21  
 
Example IV: Job Applicants 
24:00  
 
 Example IV: Setting Up 
24:01  
 
 Example IV: Part A 
26:16  
 
 Example IV: Part B 
29:53  
 
Example V: Mean & Standard Deviation of Time to Conduct All the Interviews 
40:10  
 
 Example V: Setting Up 
40:11  
 
 Example V: Mean 
45:24  
 
 Example V: Variance 
46:22  
 
 Example V: Standard Deviation 
47:01  
 
 Example V: Summary 
48:16  

Hypergeometric Distribution 
36:27 
 
Intro 
0:00  
 
Hypergeometric Distribution 
0:11  
 
 Hypergeometric Distribution: Definition 
0:12  
 
 Random Variable 
1:38  
 
Formula for the Hypergeometric Distribution 
1:50  
 
 Fixed Parameters 
1:51  
 
 Formula for the Hypergeometric Distribution 
2:53  
 
Key Properties of Hypergeometric 
6:14  
 
 Mean 
6:15  
 
 Variance 
6:42  
 
 Standard Deviation 
7:16  
 
Example I: Students Committee 
7:30  
 
Example II: Expected Number of Women on the Committee in Example I 
11:08  
 
Example III: Pairs of Shoes 
13:49  
 
Example IV: What is the Expected Number of Left Shoes in Example III? 
20:46  
 
Example V: Using Indicator Variables & Linearity of Expectation 
25:40  

Poisson Distribution 
52:19 
 
Intro 
0:00  
 
Poisson Distribution 
0:18  
 
 Poisson Distribution: Definition 
0:19  
 
Formula for the Poisson Distribution 
2:16  
 
 Fixed Parameter 
2:17  
 
 Formula for the Poisson Distribution 
2:59  
 
Key Properties of the Poisson Distribution 
5:30  
 
 Mean 
5:34  
 
 Variance 
6:07  
 
 Standard Deviation 
6:27  
 
Example I: Forest Fires 
6:41  
 
Example II: Call Center, Part A 
15:56  
 
Example II: Call Center, Part B 
20:50  
 
Example III: Confirming that the Mean of the Poisson Distribution is λ 
26:53  
 
Example IV: Find E (Y²) for the Poisson Distribution 
35:24  
 
Example V: Earthquakes, Part A 
37:57  
 
Example V: Earthquakes, Part B 
44:02  
Section 4: Continuous Distributions 

Density & Cumulative Distribution Functions 
57:17 
 
Intro 
0:00  
 
Density Functions 
0:43  
 
 Density Functions 
0:44  
 
 Density Function to Calculate Probabilities 
2:41  
 
Cumulative Distribution Functions 
4:28  
 
 Cumulative Distribution Functions 
4:29  
 
 Using F to Calculate Probabilities 
5:58  
 
Properties of the CDF (Density & Cumulative Distribution Functions) 
7:27  
 
 F(∞) = 0 
7:34  
 
 F(∞) = 1 
8:30  
 
 F is Increasing 
9:14  
 
 F'(y) = f(y) 
9:21  
 
Example I: Density & Cumulative Distribution Functions, Part A 
9:43  
 
Example I: Density & Cumulative Distribution Functions, Part B 
14:16  
 
Example II: Density & Cumulative Distribution Functions, Part A 
21:41  
 
Example II: Density & Cumulative Distribution Functions, Part B 
26:16  
 
Example III: Density & Cumulative Distribution Functions, Part A 
32:17  
 
Example III: Density & Cumulative Distribution Functions, Part B 
37:08  
 
Example IV: Density & Cumulative Distribution Functions 
43:34  
 
Example V: Density & Cumulative Distribution Functions, Part A 
51:53  
 
Example V: Density & Cumulative Distribution Functions, Part B 
54:19  

Mean & Variance for Continuous Distributions 
36:18 
 
Intro 
0:00  
 
Mean 
0:32  
 
 Mean for a Continuous Random Variable 
0:33  
 
 Expectation is Linear 
2:07  
 
Variance 
2:55  
 
 Variance for Continuous random Variable 
2:56  
 
 Easier to Calculate Via the Mean 
3:26  
 
Standard Deviation 
5:03  
 
 Standard Deviation 
5:04  
 
Example I: Mean & Variance for Continuous Distributions 
5:43  
 
Example II: Mean & Variance for Continuous Distributions 
10:09  
 
Example III: Mean & Variance for Continuous Distributions 
16:05  
 
Example IV: Mean & Variance for Continuous Distributions 
26:40  
 
Example V: Mean & Variance for Continuous Distributions 
30:12  

Uniform Distribution 
32:49 
 
Intro 
0:00  
 
Uniform Distribution 
0:15  
 
 Uniform Distribution 
0:16  
 
 Each Part of the Region is Equally Probable 
1:39  
 
Key Properties of the Uniform Distribution 
2:45  
 
 Mean 
2:46  
 
 Variance 
3:27  
 
 Standard Deviation 
3:48  
 
Example I: Newspaper Delivery 
5:25  
 
Example II: Picking a Real Number from a Uniform Distribution 
8:21  
 
Example III: Dinner Date 
11:02  
 
Example IV: Proving that a Variable is Uniformly Distributed 
18:50  
 
Example V: Ice Cream Serving 
27:22  

Normal (Gaussian) Distribution 
1:03:54 
 
Intro 
0:00  
 
Normal (Gaussian) Distribution 
0:35  
 
 Normal (Gaussian) Distribution & The Bell Curve 
0:36  
 
 Fixed Parameters 
0:55  
 
Formula for the Normal Distribution 
1:32  
 
 Formula for the Normal Distribution 
1:33  
 
 Calculating on the Normal Distribution can be Tricky 
3:32  
 
Standard Normal Distribution 
5:12  
 
 Standard Normal Distribution 
5:13  
 
 Graphing the Standard Normal Distribution 
6:13  
 
Standard Normal Distribution, Cont. 
8:30  
 
 Standard Normal Distribution Chart 
8:31  
 
Nonstandard Normal Distribution 
14:44  
 
 Nonstandard Normal Variable & Associated Standard Normal 
14:45  
 
 Finding Probabilities for Z 
15:39  
 
Example I: Chance that Standard Normal Variable Will Land Between 1 and 2? 
16:46  
 
 Example I: Setting Up the Equation & Graph 
16:47  
 
 Example I: Solving for z Using the Standard Normal Chart 
19:05  
 
Example II: What Proportion of the Data Lies within Two Standard Deviations of the Mean? 
20:41  
 
 Example II: Setting Up the Equation & Graph 
20:42  
 
 Example II: Solving for z Using the Standard Normal Chart 
24:38  
 
Example III: Scores on an Exam 
27:34  
 
 Example III: Setting Up the Equation & Graph, Part A 
27:35  
 
 Example III: Setting Up the Equation & Graph, Part B 
33:48  
 
 Example III: Solving for z Using the Standard Normal Chart, Part A 
38:23  
 
 Example III: Solving for z Using the Standard Normal Chart, Part B 
40:49  
 
Example IV: Temperatures 
42:54  
 
 Example IV: Setting Up the Equation & Graph 
42:55  
 
 Example IV: Solving for z Using the Standard Normal Chart 
47:03  
 
Example V: Scores on an Exam 
48:41  
 
 Example V: Setting Up the Equation & Graph, Part A 
48:42  
 
 Example V: Setting Up the Equation & Graph, Part B 
53:20  
 
 Example V: Solving for z Using the Standard Normal Chart, Part A 
57:45  
 
 Example V: Solving for z Using the Standard Normal Chart, Part B 
59:17  

Gamma Distribution (with Exponential & Chisquare) 
1:08:27 
 
Intro 
0:00  
 
Gamma Function 
0:49  
 
 The Gamma Function 
0:50  
 
 Properties of the Gamma Function 
2:07  
 
Formula for the Gamma Distribution 
3:50  
 
 Fixed Parameters 
3:51  
 
 Density Function for Gamma Distribution 
4:07  
 
Key Properties of the Gamma Distribution 
7:13  
 
 Mean 
7:14  
 
 Variance 
7:25  
 
 Standard Deviation 
7:30  
 
Exponential Distribution 
8:03  
 
 Definition of Exponential Distribution 
8:04  
 
 Density 
11:23  
 
 Mean 
13:26  
 
 Variance 
13:48  
 
 Standard Deviation 
13:55  
 
Chisquare Distribution 
14:34  
 
 Chisquare Distribution: Overview 
14:35  
 
 Chisquare Distribution: Mean 
16:27  
 
 Chisquare Distribution: Variance 
16:37  
 
 Chisquare Distribution: Standard Deviation 
16:55  
 
Example I: Graphing Gamma Distribution 
17:30  
 
 Example I: Graphing Gamma Distribution 
17:31  
 
 Example I: Describe the Effects of Changing α and β on the Shape of the Graph 
23:33  
 
Example II: Exponential Distribution 
27:11  
 
 Example II: Using the Exponential Distribution 
27:12  
 
 Example II: Summary 
35:34  
 
Example III: Earthquake 
37:05  
 
 Example III: Estimate Using Markov's Inequality 
37:06  
 
 Example III: Estimate Using Tchebysheff's Inequality 
40:13  
 
 Example III: Summary 
44:13  
 
Example IV: Finding Exact Probability of Earthquakes 
46:45  
 
 Example IV: Finding Exact Probability of Earthquakes 
46:46  
 
 Example IV: Summary 
51:44  
 
Example V: Prove and Interpret Why the Exponential Distribution is Called 'Memoryless' 
52:51  
 
 Example V: Prove 
52:52  
 
 Example V: Interpretation 
57:44  
 
 Example V: Summary 
63:54  

Beta Distribution 
52:45 
 
Intro 
0:00  
 
Beta Function 
0:29  
 
 Fixed parameters 
0:30  
 
 Defining the Beta Function 
1:19  
 
 Relationship between the Gamma & Beta Functions 
2:02  
 
Beta Distribution 
3:31  
 
 Density Function for the Beta Distribution 
3:32  
 
Key Properties of the Beta Distribution 
6:56  
 
 Mean 
6:57  
 
 Variance 
7:16  
 
 Standard Deviation 
7:37  
 
Example I: Calculate B(3,4) 
8:10  
 
Example II: Graphing the Density Functions for the Beta Distribution 
12:25  
 
Example III: Show that the Uniform Distribution is a Special Case of the Beta Distribution 
24:57  
 
Example IV: Show that this Triangular Distribution is a Special Case of the Beta Distribution 
31:20  
 
Example V: Morning Commute 
37:39  
 
 Example V: Identify the Density Function 
38:45  
 
 Example V: Morning Commute, Part A 
42:22  
 
 Example V: Morning Commute, Part B 
44:19  
 
 Example V: Summary 
49:13  

MomentGenerating Functions 
51:58 
 
Intro 
0:00  
 
Moments 
0:30  
 
 Definition of Moments 
0:31  
 
MomentGenerating Functions (MGFs) 
3:53  
 
 MomentGenerating Functions 
3:54  
 
 Using the MGF to Calculate the Moments 
5:21  
 
MomentGenerating Functions for the Discrete Distributions 
8:22  
 
 MomentGenerating Functions for Binomial Distribution 
8:36  
 
 MomentGenerating Functions for Geometric Distribution 
9:06  
 
 MomentGenerating Functions for Negative Binomial Distribution 
9:28  
 
 MomentGenerating Functions for Hypergeometric Distribution 
9:43  
 
 MomentGenerating Functions for Poisson Distribution 
9:57  
 
MomentGenerating Functions for the Continuous Distributions 
11:34  
 
 MomentGenerating Functions for the Uniform Distributions 
11:43  
 
 MomentGenerating Functions for the Normal Distributions 
12:24  
 
 MomentGenerating Functions for the Gamma Distributions 
12:36  
 
 MomentGenerating Functions for the Exponential Distributions 
12:44  
 
 MomentGenerating Functions for the Chisquare Distributions 
13:11  
 
 MomentGenerating Functions for the Beta Distributions 
13:48  
 
Useful Formulas with MomentGenerating Functions 
15:02  
 
 Useful Formulas with MomentGenerating Functions 1 
15:03  
 
 Useful Formulas with MomentGenerating Functions 2 
16:21  
 
Example I: MomentGenerating Function for the Binomial Distribution 
17:33  
 
Example II: Use the MGF for the Binomial Distribution to Find the Mean of the Distribution 
24:40  
 
Example III: Find the Moment Generating Function for the Poisson Distribution 
29:28  
 
Example IV: Use the MGF for Poisson Distribution to Find the Mean and Variance of the Distribution 
36:27  
 
Example V: Find the Momentgenerating Function for the Uniform Distribution 
44:47  
Section 5: Multivariate Distributions 

Bivariate Density & Distribution Functions 
50:52 
 
Intro 
0:00  
 
Bivariate Density Functions 
0:21  
 
 Two Variables 
0:23  
 
 Bivariate Density Function 
0:52  
 
Properties of the Density Function 
1:57  
 
 Properties of the Density Function 1 
1:59  
 
 Properties of the Density Function 2 
2:20  
 
 We Can Calculate Probabilities 
2:53  
 
 If You Have a Discrete Distribution 
4:36  
 
Bivariate Distribution Functions 
5:25  
 
 Bivariate Distribution Functions 
5:26  
 
 Properties of the Bivariate Distribution Functions 1 
7:19  
 
 Properties of the Bivariate Distribution Functions 2 
7:36  
 
Example I: Bivariate Density & Distribution Functions 
8:08  
 
Example II: Bivariate Density & Distribution Functions 
14:40  
 
Example III: Bivariate Density & Distribution Functions 
24:33  
 
Example IV: Bivariate Density & Distribution Functions 
32:04  
 
Example V: Bivariate Density & Distribution Functions 
40:26  

Marginal Probability 
42:38 
 
Intro 
0:00  
 
Discrete Case 
0:48  
 
 Marginal Probability Functions 
0:49  
 
Continuous Case 
3:07  
 
 Marginal Density Functions 
3:08  
 
Example I: Compute the Marginal Probability Function 
5:58  
 
Example II: Compute the Marginal Probability Function 
14:07  
 
Example III: Marginal Density Function 
24:01  
 
Example IV: Marginal Density Function 
30:47  
 
Example V: Marginal Density Function 
36:05  

Conditional Probability & Conditional Expectation 
1:02:24 
 
Intro 
0:00  
 
Review of Marginal Probability 
0:46  
 
 Recall the Marginal Probability Functions & Marginal Density Functions 
0:47  
 
Conditional Probability, Discrete Case 
3:14  
 
 Conditional Probability, Discrete Case 
3:15  
 
Conditional Probability, Continuous Case 
4:15  
 
 Conditional Density of Y₁ given that Y₂ = y₂ 
4:16  
 
 Interpret This as a Density on Y₁ & Calculate Conditional Probability 
5:03  
 
Conditional Expectation 
6:44  
 
 Conditional Expectation: Continuous 
6:45  
 
 Conditional Expectation: Discrete 
8:03  
 
Example I: Conditional Probability 
8:29  
 
Example II: Conditional Probability 
23:59  
 
Example III: Conditional Probability 
34:28  
 
Example IV: Conditional Expectation 
43:16  
 
Example V: Conditional Expectation 
48:28  

Independent Random Variables 
51:39 
 
Intro 
0:00  
 
Intuition 
0:55  
 
 Experiment with Two Random Variables 
0:56  
 
 Intuition Formula 
2:17  
 
Definition and Formulas 
4:43  
 
 Definition 
4:44  
 
 Short Version: Discrete 
5:10  
 
 Short Version: Continuous 
5:48  
 
Theorem 
9:33  
 
 For Continuous Random Variables, Y₁ & Y₂ are Independent If & Only If: Condition 1 
9:34  
 
 For Continuous Random Variables, Y₁ & Y₂ are Independent If & Only If: Condition 2 
11:22  
 
Example I: Use the Definition to Determine if Y₁ and Y₂ are Independent 
12:49  
 
Example II: Use the Definition to Determine if Y₁ and Y₂ are Independent 
21:33  
 
Example III: Are Y₁ and Y₂ Independent? 
27:01  
 
Example IV: Are Y₁ and Y₂ Independent? 
34:51  
 
Example V: Are Y₁ and Y₂ Independent? 
43:44  

Expected Value of a Function of Random Variables 
37:07 
 
Intro 
0:00  
 
Review of Single Variable Case 
0:29  
 
 Expected Value of a Single Variable 
0:30  
 
 Expected Value of a Function g(Y) 
1:12  
 
Bivariate Case 
2:11  
 
 Expected Value of a Function g(Y₁, Y₂) 
2:12  
 
Linearity of Expectation 
3:24  
 
 Linearity of Expectation 1 
3:25  
 
 Linearity of Expectation 2 
3:38  
 
 Linearity of Expectation 3: Additivity 
4:03  
 
Example I: Calculate E (Y₁ + Y₂) 
4:39  
 
Example II: Calculate E (Y₁Y₂) 
14:47  
 
Example III: Calculate E (U₁) and E(U₂) 
19:33  
 
Example IV: Calculate E (Y₁) and E(Y₂) 
22:50  
 
Example V: Calculate E (2Y₁ + 3Y₂) 
33:05  

Covariance, Correlation & Linear Functions 
59:50 
 
Intro 
0:00  
 
Definition and Formulas for Covariance 
0:38  
 
 Definition of Covariance 
0:39  
 
 Formulas to Calculate Covariance 
1:36  
 
Intuition for Covariance 
3:54  
 
 Covariance is a Measure of Dependence 
3:55  
 
 Dependence Doesn't Necessarily Mean that the Variables Do the Same Thing 
4:12  
 
 If Variables Move Together 
4:47  
 
 If Variables Move Against Each Other 
5:04  
 
 Both Cases Show Dependence! 
5:30  
 
Independence Theorem 
8:10  
 
 Independence Theorem 
8:11  
 
 The Converse is Not True 
8:32  
 
Correlation Coefficient 
9:33  
 
 Correlation Coefficient 
9:34  
 
Linear Functions of Random Variables 
11:57  
 
 Linear Functions of Random Variables: Expected Value 
11:58  
 
 Linear Functions of Random Variables: Variance 
12:58  
 
Linear Functions of Random Variables, Cont. 
14:30  
 
 Linear Functions of Random Variables: Covariance 
14:35  
 
Example I: Calculate E (Y₁), E (Y₂), and E (Y₁Y₂) 
15:31  
 
Example II: Are Y₁ and Y₂ Independent? 
29:16  
 
Example III: Calculate V (U₁) and V (U₂) 
36:14  
 
Example IV: Calculate the Covariance Correlation Coefficient 
42:12  
 
Example V: Find the Mean and Variance of the Average 
52:19  
Section 6: Distributions of Functions of Random Variables 

Distribution Functions 
1:07:35 
 
Intro 
0:00  
 
Premise 
0:44  
 
 Premise 
0:45  
 
Goal 
1:38  
 
 Goal Number 1: Find the Full Distribution Function 
1:39  
 
 Goal Number 2: Find the Density Function 
1:55  
 
 Goal Number 3: Calculate Probabilities 
2:17  
 
Three Methods 
3:05  
 
 Method 1: Distribution Functions 
3:06  
 
 Method 2: Transformations 
3:38  
 
 Method 3: Momentgenerating Functions 
3:47  
 
Distribution Functions 
4:03  
 
 Distribution Functions 
4:04  
 
Example I: Find the Density Function 
6:41  
 
 Step 1: Find the Distribution Function 
6:42  
 
 Step 2: Find the Density Function 
10:20  
 
 Summary 
11:51  
 
Example II: Find the Density Function 
14:36  
 
 Step 1: Find the Distribution Function 
14:37  
 
 Step 2: Find the Density Function 
18:19  
 
 Summary 
19:22  
 
Example III: Find the Cumulative Distribution & Density Functions 
20:39  
 
 Step 1: Find the Cumulative Distribution 
20:40  
 
 Step 2: Find the Density Function 
28:58  
 
 Summary 
30:20  
 
Example IV: Find the Density Function 
33:01  
 
 Step 1: Setting Up the Equation & Graph 
33:02  
 
 Step 2: If u ≤ 1 
38:32  
 
 Step 3: If u ≥ 1 
41:02  
 
 Step 4: Find the Distribution Function 
42:40  
 
 Step 5: Find the Density Function 
43:11  
 
 Summary 
45:03  
 
Example V: Find the Density Function 
48:32  
 
 Step 1: Exponential 
48:33  
 
 Step 2: Independence 
50:48  
 
 Step 2: Find the Distribution Function 
51:47  
 
 Step 3: Find the Density Function 
60:17  
 
 Summary 
62:05  

Transformations 
1:00:16 
 
Intro 
0:00  
 
Premise 
0:32  
 
 Premise 
0:33  
 
Goal 
1:37  
 
 Goal Number 1: Find the Full Distribution Function 
1:38  
 
 Goal Number 2: Find the Density Function 
1:49  
 
 Goal Number 3: Calculate Probabilities 
2:04  
 
Three Methods 
2:34  
 
 Method 1: Distribution Functions 
2:35  
 
 Method 2: Transformations 
2:57  
 
 Method 3: Momentgenerating Functions 
3:05  
 
Requirements for Transformation Method 
3:22  
 
 The Transformation Method Only Works for Singlevariable Situations 
3:23  
 
 Must be a Strictly Monotonic Function 
3:50  
 
 Example: Strictly Monotonic Function 
4:50  
 
 If the Function is Monotonic, Then It is Invertible 
5:30  
 
Formula for Transformations 
7:09  
 
 Formula for Transformations 
7:11  
 
Example I: Determine whether the Function is Monotonic, and if so, Find Its Inverse 
8:26  
 
Example II: Find the Density Function 
12:07  
 
Example III: Determine whether the Function is Monotonic, and if so, Find Its Inverse 
17:12  
 
Example IV: Find the Density Function for the Magnitude of the Next Earthquake 
21:30  
 
Example V: Find the Expected Magnitude of the Next Earthquake 
33:20  
 
Example VI: Find the Density Function, Including the Range of Possible Values for u 
47:42  

MomentGenerating Functions 
1:18:52 
 
Intro 
0:00  
 
Premise 
0:30  
 
 Premise 
0:31  
 
Goal 
1:40  
 
 Goal Number 1: Find the Full Distribution Function 
1:41  
 
 Goal Number 2: Find the Density Function 
1:51  
 
 Goal Number 3: Calculate Probabilities 
2:01  
 
Three Methods 
2:39  
 
 Method 1: Distribution Functions 
2:40  
 
 Method 2: Transformations 
2:50  
 
 Method 3: MomentGenerating Functions 
2:55  
 
Review of MomentGenerating Functions 
3:04  
 
 Recall: The MomentGenerating Function for a Random Variable Y 
3:05  
 
 The MomentGenerating Function is a Function of t (Not y) 
3:45  
 
MomentGenerating Functions for the Discrete Distributions 
4:31  
 
 Binomial 
4:50  
 
 Geometric 
5:12  
 
 Negative Binomial 
5:24  
 
 Hypergeometric 
5:33  
 
 Poisson 
5:42  
 
MomentGenerating Functions for the Continuous Distributions 
6:08  
 
 Uniform 
6:09  
 
 Normal 
6:17  
 
 Gamma 
6:29  
 
 Exponential 
6:34  
 
 Chisquare 
7:05  
 
 Beta 
7:48  
 
Useful Formulas with the MomentGenerating Functions 
8:48  
 
 Useful Formula 1 
8:49  
 
 Useful Formula 2 
9:51  
 
How to Use MomentGenerating Functions 
10:41  
 
 How to Use MomentGenerating Functions 
10:42  
 
Example I: Find the Density Function 
12:22  
 
Example II: Find the Density Function 
30:58  
 
Example III: Find the Probability Function 
43:29  
 
Example IV: Find the Probability Function 
51:43  
 
Example V: Find the Distribution 
60:14  
 
Example VI: Find the Density Function 
72:10  

Order Statistics 
1:04:56 
 
Intro 
0:00  
 
Premise 
0:11  
 
 Example Question: How Tall Will the Tallest Student in My Next Semester's Probability Class Be? 
0:12  
 
 Setting 
0:56  
 
 Definition 1 
1:49  
 
 Definition 2 
2:01  
 
 Question: What are the Distributions & Densities? 
4:08  
 
Formulas 
4:47  
 
 Distribution of Max 
5:11  
 
 Density of Max 
6:00  
 
 Distribution of Min 
7:08  
 
 Density of Min 
7:18  
 
Example I: Distribution & Density Functions 
8:29  
 
 Example I: Distribution 
8:30  
 
 Example I: Density 
11:07  
 
 Example I: Summary 
12:33  
 
Example II: Distribution & Density Functions 
14:25  
 
 Example II: Distribution 
14:26  
 
 Example II: Density 
17:21  
 
 Example II: Summary 
19:00  
 
Example III: Mean & Variance 
20:32  
 
 Example III: Mean 
20:33  
 
 Example III: Variance 
25:48  
 
 Example III: Summary 
30:57  
 
Example IV: Distribution & Density Functions 
35:43  
 
 Example IV: Distribution 
35:44  
 
 Example IV: Density 
43:03  
 
 Example IV: Summary 
46:11  
 
Example V: Find the Expected Time Until the Team's First Injury 
51:14  
 
 Example V: Solution 
51:15  
 
 Example V: Summary 
61:11  

Sampling from a Normal Distribution 
1:00:07 
 
Intro 
0:00  
 
Setting 
0:36  
 
 Setting 
0:37  
 
Assumptions and Notation 
2:18  
 
 Assumption Forever 
2:19  
 
 Assumption for this Lecture Only 
3:21  
 
 Notation 
3:49  
 
The Sample Mean 
4:15  
 
 Statistic We'll Study the Sample Mean 
4:16  
 
 Theorem 
5:40  
 
Standard Normal Distribution 
7:03  
 
 Standard Normal Distribution 
7:04  
 
Converting to Standard Normal 
10:11  
 
 Recall 
10:12  
 
 Corollary to Theorem 
10:41  
 
Example I: Heights of Students 
13:18  
 
Example II: What Happens to This Probability as n → ∞ 
22:36  
 
Example III: Units at a University 
32:24  
 
Example IV: Probability of Sample Mean 
40:53  
 
Example V: How Many Samples Should We Take? 
48:34  

The Central Limit Theorem 
1:09:55 
 
Intro 
0:00  
 
Setting 
0:52  
 
 Setting 
0:53  
 
Assumptions and Notation 
2:53  
 
 Our Samples are Independent (Independent Identically Distributed) 
2:54  
 
 No Longer Assume that the Population is Normally Distributed 
3:30  
 
The Central Limit Theorem 
4:36  
 
 The Central Limit Theorem Overview 
4:38  
 
 The Central Limit Theorem in Practice 
6:24  
 
Standard Normal Distribution 
8:09  
 
 Standard Normal Distribution 
8:13  
 
Converting to Standard Normal 
10:13  
 
 Recall: If Y is Normal, Then … 
10:14  
 
 Corollary to Theorem 
11:09  
 
Example I: Probability of Finishing Your Homework 
12:56  
 
 Example I: Solution 
12:57  
 
 Example I: Summary 
18:20  
 
 Example I: Confirming with the Standard Normal Distribution Chart 
20:18  
 
Example II: Probability of Selling Muffins 
21:26  
 
 Example II: Solution 
21:27  
 
 Example II: Summary 
29:09  
 
 Example II: Confirming with the Standard Normal Distribution Chart 
31:09  
 
Example III: Probability that a Soda Dispenser Gives the Correct Amount of Soda 
32:41  
 
 Example III: Solution 
32:42  
 
 Example III: Summary 
38:03  
 
 Example III: Confirming with the Standard Normal Distribution Chart 
40:58  
 
Example IV: How Many Samples Should She Take? 
42:06  
 
 Example IV: Solution 
42:07  
 
 Example IV: Summary 
49:18  
 
 Example IV: Confirming with the Standard Normal Distribution Chart 
51:57  
 
Example V: Restaurant Revenue 
54:41  
 
 Example V: Solution 
54:42  
 
 Example V: Summary 
64:21  
 
 Example V: Confirming with the Standard Normal Distribution Chart 
66:48  