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Table of Contents
I. Probability by Counting
Experiments, Outcomes, Samples, Spaces, Events
59m 30s
 Intro0:00
 Terminology0:19
 Experiment0:26
 Outcome0:56
 Sample Space1:16
 Event1:55
 Key Formula2:47
 Formula for Finding the Probability of an Event2:48
 Example: Drawing a Card3:36
 Example I5:01
 Experiment5:38
 Outcomes5:54
 Probability of the Event8:11
 Example II12:00
 Experiment12:17
 Outcomes12:34
 Probability of the Event13:49
 Example III16:33
 Experiment17:09
 Outcomes17:33
 Probability of the Event18:25
 Example IV21:20
 Experiment21:21
 Outcomes22:00
 Probability of the Event23:22
 Example V31:41
 Experiment32:14
 Outcomes32:35
 Probability of the Event33:27
 Alternate Solution40:16
 Example VI43:33
 Experiment44:08
 Outcomes44:24
 Probability of the Event53:35
Combining Events: Multiplication & Addition
1h 2m 47s
 Intro0:00
 Unions of Events0:40
 Unions of Events0:41
 Disjoint Events3:42
 Intersections of Events4:18
 Intersections of Events4:19
 Conditional Probability5:47
 Conditional Probability5:48
 Independence8:20
 Independence8:21
 Warning: Independent Does Not Mean Disjoint9:53
 If A and B are Independent11:20
 Example I: Choosing a Number at Random12:41
 Solving by Counting12:52
 Solving by Probability17:26
 Example II: Combination22:07
 Combination Deal at a Restaurant22:08
 Example III: Rolling Two Dice24:18
 Define the Events24:20
 Solving by Counting27:35
 Solving by Probability29:32
 Example IV: Flipping a Coin35:07
 Flipping a Coin Four Times35:08
 Example V: Conditional Probabilities41:22
 Define the Events42:23
 Calculate the Conditional Probabilities46:21
 Example VI: Independent Events53:42
 Define the Events53:43
 Are Events Independent?55:21
Choices: Combinations & Permutations
56m 3s
 Intro0:00
 Choices: With or Without Replacement?0:12
 Choices: With or Without Replacement?0:13
 Example: With Replacement2:17
 Example: Without Replacement2:55
 Choices: Ordered or Unordered?4:10
 Choices: Ordered or Unordered?4:11
 Example: Unordered4:52
 Example: Ordered6:08
 Combinations9:23
 Definition & Equation: Combinations9:24
 Example: Combinations12:12
 Permutations13:56
 Definition & Equation: Permutations13:57
 Example: Permutations15:00
 Key Formulas17:19
 Number of Ways to Pick r Things from n Possibilities17:20
 Example I: Five Different Candy Bars18:31
 Example II: Five Identical Candy Bars24:53
 Example III: Five Identical Candy Bars31:56
 Example IV: Five Different Candy Bars39:21
 Example V: Pizza & Toppings45:03
Inclusion & Exclusion
43m 40s
 Intro0:00
 Inclusion/Exclusion: Two Events0:09
 Inclusion/Exclusion: Two Events0:10
 Inclusion/Exclusion: Three Events2:30
 Inclusion/Exclusion: Three Events2:31
 Example I: Inclusion & Exclusion6:24
 Example II: Inclusion & Exclusion11:01
 Example III: Inclusion & Exclusion18:41
 Example IV: Inclusion & Exclusion28:24
 Example V: Inclusion & Exclusion39:33
Independence
46m 9s
 Intro0:00
 Formula and Intuition0:12
 Definition of Independence0:19
 Intuition0:49
 Common Misinterpretations1:37
 Myth & Truth 11:38
 Myth & Truth 22:23
 Combining Independent Events3:56
 Recall: Formula for Conditional Probability3:58
 Combining Independent Events4:10
 Example I: Independence5:36
 Example II: Independence14:14
 Example III: Independence21:10
 Example IV: Independence32:45
 Example V: Independence41:13
Bayes' Rule
1h 2m 10s
 Intro0:00
 When to Use Bayes' Rule0:08
 When to Use Bayes' Rule: Disjoint Union of Events0:09
 Bayes' Rule for Two Choices2:50
 Bayes' Rule for Two Choices2:51
 Bayes' Rule for Multiple Choices5:03
 Bayes' Rule for Multiple Choices5:04
 Example I: What is the Chance that She is Diabetic?6:55
 Example I: Setting up the Events6:56
 Example I: Solution11:33
 Example II: What is the chance that It Belongs to a Woman?19:28
 Example II: Setting up the Events19:29
 Example II: Solution21:45
 Example III: What is the Probability that She is a Democrat?27:31
 Example III: Setting up the Events27:32
 Example III: Solution32:08
 Example IV: What is the chance that the Fruit is an Apple?39:11
 Example IV: Setting up the Events39:12
 Example IV: Solution43:50
 Example V: What is the Probability that the Oldest Child is a Girl?51:16
 Example V: Setting up the Events51:17
 Example V: Solution53:07
II. Random Variables
Random Variables & Probability Distribution
38m 21s
 Intro0:00
 Intuition0:15
 Intuition for Random Variable0:16
 Example: Random Variable0:44
 Intuition, Cont.2:52
 Example: Random Variable as Payoff2:57
 Definition5:11
 Definition of a Random Variable5:13
 Example: Random Variable in Baseball6:02
 Probability Distributions7:18
 Probability Distributions7:19
 Example I: Probability Distribution for the Random Variable9:29
 Example II: Probability Distribution for the Random Variable14:52
 Example III: Probability Distribution for the Random Variable21:52
 Example IV: Probability Distribution for the Random Variable27:25
 Example V: Probability Distribution for the Random Variable34:12
Expected Value (Mean)
46m 14s
 Intro0:00
 Definition of Expected Value0:20
 Expected Value of a (Discrete) Random Variable or Mean0:21
 Indicator Variables3:03
 Indicator Variable3:04
 Linearity of Expectation4:36
 Linearity of Expectation for Random Variables4:37
 Expected Value of a Function6:03
 Expected Value of a Function6:04
 Example I: Expected Value7:30
 Example II: Expected Value14:14
 Example III: Expected Value of Flipping a Coin21:42
 Example III: Part A21:43
 Example III: Part B30:43
 Example IV: Semester Average36:39
 Example V: Expected Value of a Function of a Random Variable41:28
Variance & Standard Deviation
47m 23s
 Intro0:00
 Definition of Variance0:11
 Variance of a Random Variable0:12
 Variance is a Measure of the Variability, or Volatility1:06
 Most Useful Way to Calculate Variance2:46
 Definition of Standard Deviation3:44
 Standard Deviation of a Random Variable3:45
 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 Deviation11:48
 Example III: Mean12:56
 Example III: Variance14:06
 Example III: Standard Deviation15:42
 Example IV: Calculate the Mean, Variance, & Standard Deviation17:54
 Example IV: Mean18:47
 Example IV: Variance20:36
 Example IV: Standard Deviation25:34
 Example V: Calculate the Mean, Variance, & Standard Deviation29:56
 Example V: Mean30:13
 Example V: Variance33:28
 Example V: Standard Deviation34:48
 Example VI: Calculate the Mean, Variance, & Standard Deviation37:29
 Example VI: Possible Outcomes38:09
 Example VI: Mean39:29
 Example VI: Variance41:22
 Example VI: Standard Deviation43:28
Markov's Inequality
26m 45s
 Intro0:00
 Markov's Inequality0:25
 Markov's Inequality: Definition & Condition0:26
 Markov's Inequality: Equation1:15
 Markov's Inequality: Reverse Equation2:48
 Example I: Money4:11
 Example II: Rental Car9:23
 Example III: Probability of an Earthquake12:22
 Example IV: Defective Laptops16:52
 Example V: Cans of Tuna21:06
Tchebysheff's Inequality
42m 11s
 Intro0:00
 Tchebysheff's Inequality (Also Known as Chebyshev's Inequality)0:52
 Tchebysheff's Inequality: Definition0:53
 Tchebysheff's Inequality: Equation1:19
 Tchebysheff's Inequality: Intuition3:21
 Tchebysheff's Inequality in Reverse4:09
 Tchebysheff's Inequality in Reverse4:10
 Intuition5:13
 Example I: Money5:55
 Example II: College Units13:20
 Example III: Using Tchebysheff's Inequality to Estimate Proportion16:40
 Example IV: Probability of an Earthquake25:21
 Example V: Using Tchebysheff's Inequality to Estimate Proportion32:57
III. Discrete Distributions
Binomial Distribution (Bernoulli Trials)
52m 36s
 Intro0:00
 Binomial Distribution0:29
 Binomial Distribution (Bernoulli Trials) Overview0:30
 Prototypical Examples: Flipping a Coin n Times1:36
 Process with Two Outcomes: Games Between Teams2:12
 Process with Two Outcomes: Rolling a Die to Get a 62:42
 Formula for the Binomial Distribution3:45
 Fixed Parameters3:46
 Formula for the Binomial Distribution6:27
 Key Properties of the Binomial Distribution9:54
 Mean9:55
 Variance10:56
 Standard Deviation11:13
 Example I: Games Between Teams11:36
 Example II: Exam Score17:01
 Example III: Expected Grade & Standard Deviation25:59
 Example IV: Pogosticking Championship, Part A33:25
 Example IV: Pogosticking Championship, Part B38:24
 Example V: Expected Championships Winning & Standard Deviation45:22
Geometric Distribution
52m 50s
 Intro0:00
 Geometric Distribution0:22
 Geometric Distribution: Definition0:23
 Prototypical Example: Flipping a Coin Until We Get a Head1:08
 Geometric Distribution vs. Binomial Distribution.1:31
 Formula for the Geometric Distribution2:13
 Fixed Parameters2:14
 Random Variable2:49
 Formula for the Geometric Distribution3:16
 Key Properties of the Geometric Distribution6:47
 Mean6:48
 Variance7:10
 Standard Deviation7:25
 Geometric Series7:46
 Recall from Calculus II: Sum of Infinite Series7:47
 Application to Geometric Distribution10:10
 Example I: Drawing Cards from a Deck (With Replacement) Until You Get an Ace13:02
 Example I: Question & Solution13:03
 Example II: Mean & Standard Deviation of Winning Pin the Tail on the Donkey16:32
 Example II: Mean16:33
 Example II: Standard Deviation18:37
 Example III: Rolling a Die22:09
 Example III: Setting Up22:10
 Example III: Part A24:18
 Example III: Part B26:01
 Example III: Part C27:38
 Example III: Summary32:02
 Example IV: Job Interview35:16
 Example IV: Setting Up35:15
 Example IV: Part A37:26
 Example IV: Part B38:33
 Example IV: Summary39:37
 Example V: Mean & Standard Deviation of Time to Conduct All the Interviews41:13
 Example V: Setting Up42:50
 Example V: Mean46:05
 Example V: Variance47:37
 Example V: Standard Deviation48:22
 Example V: Summary49:36
Negative Binomial Distribution
51m 39s
 Intro0:00
 Negative Binomial Distribution0:11
 Negative Binomial Distribution: Definition0:12
 Prototypical Example: Flipping a Coin Until We Get r Successes0:46
 Negative Binomial Distribution vs. Binomial Distribution1:04
 Negative Binomial Distribution vs. Geometric Distribution1:33
 Formula for Negative Binomial Distribution3:39
 Fixed Parameters3:40
 Random Variable4:57
 Formula for Negative Binomial Distribution5:18
 Key Properties of Negative Binomial7:44
 Mean7:47
 Variance8:03
 Standard Deviation8:09
 Example I: Drawing Cards from a Deck (With Replacement) Until You Get Four Aces8:32
 Example I: Question & Solution8:33
 Example II: Chinchilla Grooming12:37
 Example II: Mean12:38
 Example II: Variance15:09
 Example II: Standard Deviation15:51
 Example II: Summary17:10
 Example III: Rolling a Die Until You Get Four Sixes18:27
 Example III: Setting Up19:38
 Example III: Mean19:38
 Example III: Variance20:31
 Example III: Standard Deviation21:21
 Example IV: Job Applicants24:00
 Example IV: Setting Up24:01
 Example IV: Part A26:16
 Example IV: Part B29:53
 Example V: Mean & Standard Deviation of Time to Conduct All the Interviews40:10
 Example V: Setting Up40:11
 Example V: Mean45:24
 Example V: Variance46:22
 Example V: Standard Deviation47:01
 Example V: Summary48:16
Hypergeometric Distribution
36m 27s
 Intro0:00
 Hypergeometric Distribution0:11
 Hypergeometric Distribution: Definition0:12
 Random Variable1:38
 Formula for the Hypergeometric Distribution1:50
 Fixed Parameters1:51
 Formula for the Hypergeometric Distribution2:53
 Key Properties of Hypergeometric6:14
 Mean6:15
 Variance6:42
 Standard Deviation7:16
 Example I: Students Committee7:30
 Example II: Expected Number of Women on the Committee in Example I11:08
 Example III: Pairs of Shoes13:49
 Example IV: What is the Expected Number of Left Shoes in Example III?20:46
 Example V: Using Indicator Variables & Linearity of Expectation25:40
Poisson Distribution
52m 19s
 Intro0:00
 Poisson Distribution0:18
 Poisson Distribution: Definition0:19
 Formula for the Poisson Distribution2:16
 Fixed Parameter2:17
 Formula for the Poisson Distribution2:59
 Key Properties of the Poisson Distribution5:30
 Mean5:34
 Variance6:07
 Standard Deviation6:27
 Example I: Forest Fires6:41
 Example II: Call Center, Part A15:56
 Example II: Call Center, Part B20:50
 Example III: Confirming that the Mean of the Poisson Distribution is λ26:53
 Example IV: Find E (Y²) for the Poisson Distribution35:24
 Example V: Earthquakes, Part A37:57
 Example V: Earthquakes, Part B44:02
IV. Continuous Distributions
Density & Cumulative Distribution Functions
57m 17s
 Intro0:00
 Density Functions0:43
 Density Functions0:44
 Density Function to Calculate Probabilities2:41
 Cumulative Distribution Functions4:28
 Cumulative Distribution Functions4:29
 Using F to Calculate Probabilities5:58
 Properties of the CDF (Density & Cumulative Distribution Functions)7:27
 F(∞) = 07:34
 F(∞) = 18:30
 F is Increasing9:14
 F'(y) = f(y)9:21
 Example I: Density & Cumulative Distribution Functions, Part A9:43
 Example I: Density & Cumulative Distribution Functions, Part B14:16
 Example II: Density & Cumulative Distribution Functions, Part A21:41
 Example II: Density & Cumulative Distribution Functions, Part B26:16
 Example III: Density & Cumulative Distribution Functions, Part A32:17
 Example III: Density & Cumulative Distribution Functions, Part B37:08
 Example IV: Density & Cumulative Distribution Functions43:34
 Example V: Density & Cumulative Distribution Functions, Part A51:53
 Example V: Density & Cumulative Distribution Functions, Part B54:19
Mean & Variance for Continuous Distributions
36m 18s
 Intro0:00
 Mean0:32
 Mean for a Continuous Random Variable0:33
 Expectation is Linear2:07
 Variance2:55
 Variance for Continuous random Variable2:56
 Easier to Calculate Via the Mean3:26
 Standard Deviation5:03
 Standard Deviation5:04
 Example I: Mean & Variance for Continuous Distributions5:43
 Example II: Mean & Variance for Continuous Distributions10:09
 Example III: Mean & Variance for Continuous Distributions16:05
 Example IV: Mean & Variance for Continuous Distributions26:40
 Example V: Mean & Variance for Continuous Distributions30:12
Uniform Distribution
32m 49s
 Intro0:00
 Uniform Distribution0:15
 Uniform Distribution0:16
 Each Part of the Region is Equally Probable1:39
 Key Properties of the Uniform Distribution2:45
 Mean2:46
 Variance3:27
 Standard Deviation3:48
 Example I: Newspaper Delivery5:25
 Example II: Picking a Real Number from a Uniform Distribution8:21
 Example III: Dinner Date11:02
 Example IV: Proving that a Variable is Uniformly Distributed18:50
 Example V: Ice Cream Serving27:22
Normal (Gaussian) Distribution
1h 3m 54s
 Intro0:00
 Normal (Gaussian) Distribution0:35
 Normal (Gaussian) Distribution & The Bell Curve0:36
 Fixed Parameters0:55
 Formula for the Normal Distribution1:32
 Formula for the Normal Distribution1:33
 Calculating on the Normal Distribution can be Tricky3:32
 Standard Normal Distribution5:12
 Standard Normal Distribution5:13
 Graphing the Standard Normal Distribution6:13
 Standard Normal Distribution, Cont.8:30
 Standard Normal Distribution Chart8:31
 Nonstandard Normal Distribution14:44
 Nonstandard Normal Variable & Associated Standard Normal14:45
 Finding Probabilities for Z15:39
 Example I: Chance that Standard Normal Variable Will Land Between 1 and 2?16:46
 Example I: Setting Up the Equation & Graph16:47
 Example I: Solving for z Using the Standard Normal Chart19:05
 Example II: What Proportion of the Data Lies within Two Standard Deviations of the Mean?20:41
 Example II: Setting Up the Equation & Graph20:42
 Example II: Solving for z Using the Standard Normal Chart24:38
 Example III: Scores on an Exam27:34
 Example III: Setting Up the Equation & Graph, Part A27:35
 Example III: Setting Up the Equation & Graph, Part B33:48
 Example III: Solving for z Using the Standard Normal Chart, Part A38:23
 Example III: Solving for z Using the Standard Normal Chart, Part B40:49
 Example IV: Temperatures42:54
 Example IV: Setting Up the Equation & Graph42:55
 Example IV: Solving for z Using the Standard Normal Chart47:03
 Example V: Scores on an Exam48:41
 Example V: Setting Up the Equation & Graph, Part A48:42
 Example V: Setting Up the Equation & Graph, Part B53:20
 Example V: Solving for z Using the Standard Normal Chart, Part A57:45
 Example V: Solving for z Using the Standard Normal Chart, Part B59:17
Gamma Distribution (with Exponential & Chisquare)
1h 8m 27s
 Intro0:00
 Gamma Function0:49
 The Gamma Function0:50
 Properties of the Gamma Function2:07
 Formula for the Gamma Distribution3:50
 Fixed Parameters3:51
 Density Function for Gamma Distribution4:07
 Key Properties of the Gamma Distribution7:13
 Mean7:14
 Variance7:25
 Standard Deviation7:30
 Exponential Distribution8:03
 Definition of Exponential Distribution8:04
 Density11:23
 Mean13:26
 Variance13:48
 Standard Deviation13:55
 Chisquare Distribution14:34
 Chisquare Distribution: Overview14:35
 Chisquare Distribution: Mean16:27
 Chisquare Distribution: Variance16:37
 Chisquare Distribution: Standard Deviation16:55
 Example I: Graphing Gamma Distribution17:30
 Example I: Graphing Gamma Distribution17:31
 Example I: Describe the Effects of Changing α and β on the Shape of the Graph23:33
 Example II: Exponential Distribution27:11
 Example II: Using the Exponential Distribution27:12
 Example II: Summary35:34
 Example III: Earthquake37:05
 Example III: Estimate Using Markov's Inequality37:06
 Example III: Estimate Using Tchebysheff's Inequality40:13
 Example III: Summary44:13
 Example IV: Finding Exact Probability of Earthquakes46:45
 Example IV: Finding Exact Probability of Earthquakes46:46
 Example IV: Summary51:44
 Example V: Prove and Interpret Why the Exponential Distribution is Called 'Memoryless'52:51
 Example V: Prove52:52
 Example V: Interpretation57:44
 Example V: Summary03:54
Beta Distribution
52m 45s
 Intro0:00
 Beta Function0:29
 Fixed parameters0:30
 Defining the Beta Function1:19
 Relationship between the Gamma & Beta Functions2:02
 Beta Distribution3:31
 Density Function for the Beta Distribution3:32
 Key Properties of the Beta Distribution6:56
 Mean6:57
 Variance7:16
 Standard Deviation7:37
 Example I: Calculate B(3,4)8:10
 Example II: Graphing the Density Functions for the Beta Distribution12:25
 Example III: Show that the Uniform Distribution is a Special Case of the Beta Distribution24:57
 Example IV: Show that this Triangular Distribution is a Special Case of the Beta Distribution31:20
 Example V: Morning Commute37:39
 Example V: Identify the Density Function38:45
 Example V: Morning Commute, Part A42:22
 Example V: Morning Commute, Part B44:19
 Example V: Summary49:13
MomentGenerating Functions
51m 58s
 Intro0:00
 Moments0:30
 Definition of Moments0:31
 MomentGenerating Functions (MGFs)3:53
 MomentGenerating Functions3:54
 Using the MGF to Calculate the Moments5:21
 MomentGenerating Functions for the Discrete Distributions8:22
 MomentGenerating Functions for Binomial Distribution8:36
 MomentGenerating Functions for Geometric Distribution9:06
 MomentGenerating Functions for Negative Binomial Distribution9:28
 MomentGenerating Functions for Hypergeometric Distribution9:43
 MomentGenerating Functions for Poisson Distribution9:57
 MomentGenerating Functions for the Continuous Distributions11:34
 MomentGenerating Functions for the Uniform Distributions11:43
 MomentGenerating Functions for the Normal Distributions12:24
 MomentGenerating Functions for the Gamma Distributions12:36
 MomentGenerating Functions for the Exponential Distributions12:44
 MomentGenerating Functions for the Chisquare Distributions13:11
 MomentGenerating Functions for the Beta Distributions13:48
 Useful Formulas with MomentGenerating Functions15:02
 Useful Formulas with MomentGenerating Functions 115:03
 Useful Formulas with MomentGenerating Functions 216:21
 Example I: MomentGenerating Function for the Binomial Distribution17:33
 Example II: Use the MGF for the Binomial Distribution to Find the Mean of the Distribution24:40
 Example III: Find the Moment Generating Function for the Poisson Distribution29:28
 Example IV: Use the MGF for Poisson Distribution to Find the Mean and Variance of the Distribution36:27
 Example V: Find the Momentgenerating Function for the Uniform Distribution44:47
V. Multivariate Distributions
Bivariate Density & Distribution Functions
50m 52s
 Intro0:00
 Bivariate Density Functions0:21
 Two Variables0:23
 Bivariate Density Function0:52
 Properties of the Density Function1:57
 Properties of the Density Function 11:59
 Properties of the Density Function 22:20
 We Can Calculate Probabilities2:53
 If You Have a Discrete Distribution4:36
 Bivariate Distribution Functions5:25
 Bivariate Distribution Functions5:26
 Properties of the Bivariate Distribution Functions 17:19
 Properties of the Bivariate Distribution Functions 27:36
 Example I: Bivariate Density & Distribution Functions8:08
 Example II: Bivariate Density & Distribution Functions14:40
 Example III: Bivariate Density & Distribution Functions24:33
 Example IV: Bivariate Density & Distribution Functions32:04
 Example V: Bivariate Density & Distribution Functions40:26
Marginal Probability
42m 38s
 Intro0:00
 Discrete Case0:48
 Marginal Probability Functions0:49
 Continuous Case3:07
 Marginal Density Functions3:08
 Example I: Compute the Marginal Probability Function5:58
 Example II: Compute the Marginal Probability Function14:07
 Example III: Marginal Density Function24:01
 Example IV: Marginal Density Function30:47
 Example V: Marginal Density Function36:05
Conditional Probability & Conditional Expectation
1h 2m 24s
 Intro0:00
 Review of Marginal Probability0:46
 Recall the Marginal Probability Functions & Marginal Density Functions0:47
 Conditional Probability, Discrete Case3:14
 Conditional Probability, Discrete Case3:15
 Conditional Probability, Continuous Case4:15
 Conditional Density of Y₁ given that Y₂ = y₂4:16
 Interpret This as a Density on Y₁ & Calculate Conditional Probability5:03
 Conditional Expectation6:44
 Conditional Expectation: Continuous6:45
 Conditional Expectation: Discrete8:03
 Example I: Conditional Probability8:29
 Example II: Conditional Probability23:59
 Example III: Conditional Probability34:28
 Example IV: Conditional Expectation43:16
 Example V: Conditional Expectation48:28
Independent Random Variables
51m 39s
 Intro0:00
 Intuition0:55
 Experiment with Two Random Variables0:56
 Intuition Formula2:17
 Definition and Formulas4:43
 Definition4:44
 Short Version: Discrete5:10
 Short Version: Continuous5:48
 Theorem9:33
 For Continuous Random Variables, Y₁ & Y₂ are Independent If & Only If: Condition 19:34
 For Continuous Random Variables, Y₁ & Y₂ are Independent If & Only If: Condition 211:22
 Example I: Use the Definition to Determine if Y₁ and Y₂ are Independent12:49
 Example II: Use the Definition to Determine if Y₁ and Y₂ are Independent21:33
 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
37m 7s
 Intro0:00
 Review of Single Variable Case0:29
 Expected Value of a Single Variable0:30
 Expected Value of a Function g(Y)1:12
 Bivariate Case2:11
 Expected Value of a Function g(Y₁, Y₂)2:12
 Linearity of Expectation3:24
 Linearity of Expectation 13:25
 Linearity of Expectation 23:38
 Linearity of Expectation 3: Additivity4:03
 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
59m 50s
 Intro0:00
 Definition and Formulas for Covariance0:38
 Definition of Covariance0:39
 Formulas to Calculate Covariance1:36
 Intuition for Covariance3:54
 Covariance is a Measure of Dependence3:55
 Dependence Doesn't Necessarily Mean that the Variables Do the Same Thing4:12
 If Variables Move Together4:47
 If Variables Move Against Each Other5:04
 Both Cases Show Dependence!5:30
 Independence Theorem8:10
 Independence Theorem8:11
 The Converse is Not True8:32
 Correlation Coefficient9:33
 Correlation Coefficient9:34
 Linear Functions of Random Variables11:57
 Linear Functions of Random Variables: Expected Value11:58
 Linear Functions of Random Variables: Variance12:58
 Linear Functions of Random Variables, Cont.14:30
 Linear Functions of Random Variables: Covariance14:35
 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 Coefficient42:12
 Example V: Find the Mean and Variance of the Average52:19
VI. Distributions of Functions of Random Variables
Distribution Functions
1h 7m 35s
 Intro0:00
 Premise0:44
 Premise0:45
 Goal1:38
 Goal Number 1: Find the Full Distribution Function1:39
 Goal Number 2: Find the Density Function1:55
 Goal Number 3: Calculate Probabilities2:17
 Three Methods3:05
 Method 1: Distribution Functions3:06
 Method 2: Transformations3:38
 Method 3: Momentgenerating Functions3:47
 Distribution Functions4:03
 Distribution Functions4:04
 Example I: Find the Density Function6:41
 Step 1: Find the Distribution Function6:42
 Step 2: Find the Density Function10:20
 Summary11:51
 Example II: Find the Density Function14:36
 Step 1: Find the Distribution Function14:37
 Step 2: Find the Density Function18:19
 Summary19:22
 Example III: Find the Cumulative Distribution & Density Functions20:39
 Step 1: Find the Cumulative Distribution20:40
 Step 2: Find the Density Function28:58
 Summary30:20
 Example IV: Find the Density Function33:01
 Step 1: Setting Up the Equation & Graph33:02
 Step 2: If u ≤ 138:32
 Step 3: If u ≥ 141:02
 Step 4: Find the Distribution Function42:40
 Step 5: Find the Density Function43:11
 Summary45:03
 Example V: Find the Density Function48:32
 Step 1: Exponential48:33
 Step 2: Independence50:48
 Step 2: Find the Distribution Function51:47
 Step 3: Find the Density Function00:17
 Summary02:05
Transformations
1h 16s
 Intro0:00
 Premise0:32
 Premise0:33
 Goal1:37
 Goal Number 1: Find the Full Distribution Function1:38
 Goal Number 2: Find the Density Function1:49
 Goal Number 3: Calculate Probabilities2:04
 Three Methods2:34
 Method 1: Distribution Functions2:35
 Method 2: Transformations2:57
 Method 3: Momentgenerating Functions3:05
 Requirements for Transformation Method3:22
 The Transformation Method Only Works for Singlevariable Situations3:23
 Must be a Strictly Monotonic Function3:50
 Example: Strictly Monotonic Function4:50
 If the Function is Monotonic, Then It is Invertible5:30
 Formula for Transformations7:09
 Formula for Transformations7:11
 Example I: Determine whether the Function is Monotonic, and if so, Find Its Inverse8:26
 Example II: Find the Density Function12:07
 Example III: Determine whether the Function is Monotonic, and if so, Find Its Inverse17:12
 Example IV: Find the Density Function for the Magnitude of the Next Earthquake21:30
 Example V: Find the Expected Magnitude of the Next Earthquake33:20
 Example VI: Find the Density Function, Including the Range of Possible Values for u47:42
MomentGenerating Functions
1h 18m 52s
 Intro0:00
 Premise0:30
 Premise0:31
 Goal1:40
 Goal Number 1: Find the Full Distribution Function1:41
 Goal Number 2: Find the Density Function1:51
 Goal Number 3: Calculate Probabilities2:01
 Three Methods2:39
 Method 1: Distribution Functions2:40
 Method 2: Transformations2:50
 Method 3: MomentGenerating Functions2:55
 Review of MomentGenerating Functions3:04
 Recall: The MomentGenerating Function for a Random Variable Y3:05
 The MomentGenerating Function is a Function of t (Not y)3:45
 MomentGenerating Functions for the Discrete Distributions4:31
 Binomial4:50
 Geometric5:12
 Negative Binomial5:24
 Hypergeometric5:33
 Poisson5:42
 MomentGenerating Functions for the Continuous Distributions6:08
 Uniform6:09
 Normal6:17
 Gamma6:29
 Exponential6:34
 Chisquare7:05
 Beta7:48
 Useful Formulas with the MomentGenerating Functions8:48
 Useful Formula 18:49
 Useful Formula 29:51
 How to Use MomentGenerating Functions10:41
 How to Use MomentGenerating Functions10:42
 Example I: Find the Density Function12:22
 Example II: Find the Density Function30:58
 Example III: Find the Probability Function43:29
 Example IV: Find the Probability Function51:43
 Example V: Find the Distribution00:14
 Example VI: Find the Density Function12:10
Order Statistics
1h 4m 56s
 Intro0:00
 Premise0:11
 Example Question: How Tall Will the Tallest Student in My Next Semester's Probability Class Be?0:12
 Setting0:56
 Definition 11:49
 Definition 22:01
 Question: What are the Distributions & Densities?4:08
 Formulas4:47
 Distribution of Max5:11
 Density of Max6:00
 Distribution of Min7:08
 Density of Min7:18
 Example I: Distribution & Density Functions8:29
 Example I: Distribution8:30
 Example I: Density11:07
 Example I: Summary12:33
 Example II: Distribution & Density Functions14:25
 Example II: Distribution14:26
 Example II: Density17:21
 Example II: Summary19:00
 Example III: Mean & Variance20:32
 Example III: Mean20:33
 Example III: Variance25:48
 Example III: Summary30:57
 Example IV: Distribution & Density Functions35:43
 Example IV: Distribution35:44
 Example IV: Density43:03
 Example IV: Summary46:11
 Example V: Find the Expected Time Until the Team's First Injury51:14
 Example V: Solution51:15
 Example V: Summary01:11
Sampling from a Normal Distribution
1h 7s
 Intro0:00
 Setting0:36
 Setting0:37
 Assumptions and Notation2:18
 Assumption Forever2:19
 Assumption for this Lecture Only3:21
 Notation3:49
 The Sample Mean4:15
 Statistic We'll Study the Sample Mean4:16
 Theorem5:40
 Standard Normal Distribution7:03
 Standard Normal Distribution7:04
 Converting to Standard Normal10:11
 Recall10:12
 Corollary to Theorem10:41
 Example I: Heights of Students13:18
 Example II: What Happens to This Probability as n → ∞22:36
 Example III: Units at a University32:24
 Example IV: Probability of Sample Mean40:53
 Example V: How Many Samples Should We Take?48:34
The Central Limit Theorem
1h 9m 55s
 Intro0:00
 Setting0:52
 Setting0:53
 Assumptions and Notation2:53
 Our Samples are Independent (Independent Identically Distributed)2:54
 No Longer Assume that the Population is Normally Distributed3:30
 The Central Limit Theorem4:36
 The Central Limit Theorem Overview4:38
 The Central Limit Theorem in Practice6:24
 Standard Normal Distribution8:09
 Standard Normal Distribution8:13
 Converting to Standard Normal10:13
 Recall: If Y is Normal, Then …10:14
 Corollary to Theorem11:09
 Example I: Probability of Finishing Your Homework12:56
 Example I: Solution12:57
 Example I: Summary18:20
 Example I: Confirming with the Standard Normal Distribution Chart20:18
 Example II: Probability of Selling Muffins21:26
 Example II: Solution21:27
 Example II: Summary29:09
 Example II: Confirming with the Standard Normal Distribution Chart31:09
 Example III: Probability that a Soda Dispenser Gives the Correct Amount of Soda32:41
 Example III: Solution32:42
 Example III: Summary38:03
 Example III: Confirming with the Standard Normal Distribution Chart40:58
 Example IV: How Many Samples Should She Take?42:06
 Example IV: Solution42:07
 Example IV: Summary49:18
 Example IV: Confirming with the Standard Normal Distribution Chart51:57
 Example V: Restaurant Revenue54:41
 Example V: Solution54:42
 Example V: Summary04:21
 Example V: Confirming with the Standard Normal Distribution Chart06:48
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