# Probability Expected Value of a Function of Random Variables

Section 5: Multivariate Distributions: Lecture 5 | 37:07 min

In this lesson, we are going to talk about the expected value of the function of random variables. That is something that we have seen in the single variable case. We will see how the same ideas extend to the multivariable case. We will start by reviewing the single variable case because it will make the definitions make more sense for the multivariable case. The definitions are very similar, except that instead of single sums, we will have double sums and instead of single integrals, we will have double integrals. We will also cover some important facts about linearity of expectation.

William Murray

Expected Value of a Function of Random Variables

Slide Duration:Table of Contents

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

For more information, please see full course syllabus of Probability

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