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William Murray

Expected Value of a Function of Random Variables

Slide Duration: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: Pogo-sticking Championship, Part A33:25
- Example IV: Pogo-sticking 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 & Chi-square)

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
- Chi-square Distribution14:34
- Chi-square Distribution: Overview14:35
- Chi-square Distribution: Mean16:27
- Chi-square Distribution: Variance16:37
- Chi-square 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: Summary1:03:54

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

Moment-Generating Functions

51m 58s

- Intro0:00
- Moments0:30
- Definition of Moments0:31
- Moment-Generating Functions (MGFs)3:53
- Moment-Generating Functions3:54
- Using the MGF to Calculate the Moments5:21
- Moment-Generating Functions for the Discrete Distributions8:22
- Moment-Generating Functions for Binomial Distribution8:36
- Moment-Generating Functions for Geometric Distribution9:06
- Moment-Generating Functions for Negative Binomial Distribution9:28
- Moment-Generating Functions for Hypergeometric Distribution9:43
- Moment-Generating Functions for Poisson Distribution9:57
- Moment-Generating Functions for the Continuous Distributions11:34
- Moment-Generating Functions for the Uniform Distributions11:43
- Moment-Generating Functions for the Normal Distributions12:24
- Moment-Generating Functions for the Gamma Distributions12:36
- Moment-Generating Functions for the Exponential Distributions12:44
- Moment-Generating Functions for the Chi-square Distributions13:11
- Moment-Generating Functions for the Beta Distributions13:48
- Useful Formulas with Moment-Generating Functions15:02
- Useful Formulas with Moment-Generating Functions 115:03
- Useful Formulas with Moment-Generating Functions 216:21
- Example I: Moment-Generating 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 Moment-generating 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: Moment-generating 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 Function1:00:17
- Summary1:02:05

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: Moment-generating Functions3:05
- Requirements for Transformation Method3:22
- The Transformation Method Only Works for Single-variable 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

Moment-Generating 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: Moment-Generating Functions2:55
- Review of Moment-Generating Functions3:04
- Recall: The Moment-Generating Function for a Random Variable Y3:05
- The Moment-Generating Function is a Function of t (Not y)3:45
- Moment-Generating Functions for the Discrete Distributions4:31
- Binomial4:50
- Geometric5:12
- Negative Binomial5:24
- Hypergeometric5:33
- Poisson5:42
- Moment-Generating Functions for the Continuous Distributions6:08
- Uniform6:09
- Normal6:17
- Gamma6:29
- Exponential6:34
- Chi-square7:05
- Beta7:48
- Useful Formulas with the Moment-Generating Functions8:48
- Useful Formula 18:49
- Useful Formula 29:51
- How to Use Moment-Generating Functions10:41
- How to Use Moment-Generating 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 Distribution1:00:14
- Example VI: Find the Density Function1:12:10

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: Summary1:01:11

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: Summary1:04:21
- Example V: Confirming with the Standard Normal Distribution Chart1:06:48

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