Join Dr. William Murray in his Introduction to Probability online course where every lesson begins with a clear overview of formulas followed by step-by-step examples. Dr. Murray focuses on the understanding and application of formulas rather than derivations to help you save time and ace your class.

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## I. Probability by Counting

Experiments, Outcomes, Samples, Spaces, Events 59:30
Intro 0:00
Terminology 0:19
Experiment 0:26
Outcome 0:56
Sample Space 1:16
Event 1:55
Key Formula 2:47
Formula for Finding the Probability of an Event 2:48
Example: Drawing a Card 3:36
Example I 5:01
Experiment 5:38
Outcomes 5:54
Probability of the Event 8:11
Example II 12:00
Experiment 12:17
Outcomes 12:34
Probability of the Event 13:49
Example III 16:33
Experiment 17:09
Outcomes 17:33
Probability of the Event 18:25
Example IV 21:20
Experiment 21:21
Outcomes 22:00
Probability of the Event 23:22
Example V 31:41
Experiment 32:14
Outcomes 32:35
Probability of the Event 33:27
Alternate Solution 40:16
Example VI 43:33
Experiment 44:08
Outcomes 44:24
Probability of the Event 53:35
Combining Events: Multiplication & Addition 1:02:47
Intro 0:00
Unions of Events 0:40
Unions of Events 0:41
Disjoint Events 3:42
Intersections of Events 4:18
Intersections of Events 4:19
Conditional Probability 5:47
Conditional Probability 5:48
Independence 8:20
Independence 8:21
Warning: Independent Does Not Mean Disjoint 9:53
If A and B are Independent 11:20
Example I: Choosing a Number at Random 12:41
Solving by Counting 12:52
Solving by Probability 17:26
Example II: Combination 22:07
Combination Deal at a Restaurant 22:08
Example III: Rolling Two Dice 24:18
Define the Events 24:20
Solving by Counting 27:35
Solving by Probability 29:32
Example IV: Flipping a Coin 35:07
Flipping a Coin Four Times 35:08
Example V: Conditional Probabilities 41:22
Define the Events 42:23
Calculate the Conditional Probabilities 46:21
Example VI: Independent Events 53:42
Define the Events 53:43
Are Events Independent? 55:21
Choices: Combinations & Permutations 56:03
Intro 0:00
Choices: With or Without Replacement? 0:12
Choices: With or Without Replacement? 0:13
Example: With Replacement 2:17
Example: Without Replacement 2:55
Choices: Ordered or Unordered? 4:10
Choices: Ordered or Unordered? 4:11
Example: Unordered 4:52
Example: Ordered 6:08
Combinations 9:23
Definition & Equation: Combinations 9:24
Example: Combinations 12:12
Permutations 13:56
Definition & Equation: Permutations 13:57
Example: Permutations 15:00
Key Formulas 17:19
Number of Ways to Pick r Things from n Possibilities 17:20
Example I: Five Different Candy Bars 18:31
Example II: Five Identical Candy Bars 24:53
Example III: Five Identical Candy Bars 31:56
Example IV: Five Different Candy Bars 39:21
Example V: Pizza & Toppings 45:03
Inclusion & Exclusion 43:40
Intro 0:00
Inclusion/Exclusion: Two Events 0:09
Inclusion/Exclusion: Two Events 0:10
Inclusion/Exclusion: Three Events 2:30
Inclusion/Exclusion: Three Events 2:31
Example I: Inclusion & Exclusion 6:24
Example II: Inclusion & Exclusion 11:01
Example III: Inclusion & Exclusion 18:41
Example IV: Inclusion & Exclusion 28:24
Example V: Inclusion & Exclusion 39:33
Independence 46:09
Intro 0:00
Formula and Intuition 0:12
Definition of Independence 0:19
Intuition 0:49
Common Misinterpretations 1:37
Myth & Truth 1 1:38
Myth & Truth 2 2:23
Combining Independent Events 3:56
Recall: Formula for Conditional Probability 3:58
Combining Independent Events 4:10
Example I: Independence 5:36
Example II: Independence 14:14
Example III: Independence 21:10
Example IV: Independence 32:45
Example V: Independence 41:13
Bayes' Rule 1:02:10
Intro 0:00
When to Use Bayes' Rule 0:08
When to Use Bayes' Rule: Disjoint Union of Events 0:09
Bayes' Rule for Two Choices 2:50
Bayes' Rule for Two Choices 2:51
Bayes' Rule for Multiple Choices 5:03
Bayes' Rule for Multiple Choices 5:04
Example I: What is the Chance that She is Diabetic? 6:55
Example I: Setting up the Events 6:56
Example I: Solution 11:33
Example II: What is the chance that It Belongs to a Woman? 19:28
Example II: Setting up the Events 19:29
Example II: Solution 21:45
Example III: What is the Probability that She is a Democrat? 27:31
Example III: Setting up the Events 27:32
Example III: Solution 32:08
Example IV: What is the chance that the Fruit is an Apple? 39:11
Example IV: Setting up the Events 39:12
Example IV: Solution 43:50
Example V: What is the Probability that the Oldest Child is a Girl? 51:16
Example V: Setting up the Events 51:17
Example V: Solution 53:07

## II. Random Variables

Random Variables & Probability Distribution 38:21
Intro 0:00
Intuition 0:15
Intuition for Random Variable 0:16
Example: Random Variable 0:44
Intuition, Cont. 2:52
Example: Random Variable as Payoff 2:57
Definition 5:11
Definition of a Random Variable 5:13
Example: Random Variable in Baseball 6:02
Probability Distributions 7:18
Probability Distributions 7:19
Example I: Probability Distribution for the Random Variable 9:29
Example II: Probability Distribution for the Random Variable 14:52
Example III: Probability Distribution for the Random Variable 21:52
Example IV: Probability Distribution for the Random Variable 27:25
Example V: Probability Distribution for the Random Variable 34:12
Expected Value (Mean) 46:14
Intro 0:00
Definition of Expected Value 0:20
Expected Value of a (Discrete) Random Variable or Mean 0:21
Indicator Variables 3:03
Indicator Variable 3:04
Linearity of Expectation 4:36
Linearity of Expectation for Random Variables 4:37
Expected Value of a Function 6:03
Expected Value of a Function 6:04
Example I: Expected Value 7:30
Example II: Expected Value 14:14
Example III: Expected Value of Flipping a Coin 21:42
Example III: Part A 21:43
Example III: Part B 30:43
Example IV: Semester Average 36:39
Example V: Expected Value of a Function of a Random Variable 41:28
Variance & Standard Deviation 47:23
Intro 0:00
Definition of Variance 0:11
Variance of a Random Variable 0:12
Variance is a Measure of the Variability, or Volatility 1:06
Most Useful Way to Calculate Variance 2:46
Definition of Standard Deviation 3:44
Standard Deviation of a Random Variable 3:45
Example I: Which of the Following Sets of Data Has the Largest Variance? 5:34
Example II: Which of the Following Would be the Least Useful in Understanding a Set of Data? 9:02
Example III: Calculate the Mean, Variance, & Standard Deviation 11:48
Example III: Mean 12:56
Example III: Variance 14:06
Example III: Standard Deviation 15:42
Example IV: Calculate the Mean, Variance, & Standard Deviation 17:54
Example IV: Mean 18:47
Example IV: Variance 20:36
Example IV: Standard Deviation 25:34
Example V: Calculate the Mean, Variance, & Standard Deviation 29:56
Example V: Mean 30:13
Example V: Variance 33:28
Example V: Standard Deviation 34:48
Example VI: Calculate the Mean, Variance, & Standard Deviation 37:29
Example VI: Possible Outcomes 38:09
Example VI: Mean 39:29
Example VI: Variance 41:22
Example VI: Standard Deviation 43:28
Markov's Inequality 26:45
Intro 0:00
Markov's Inequality 0:25
Markov's Inequality: Definition & Condition 0:26
Markov's Inequality: Equation 1:15
Markov's Inequality: Reverse Equation 2:48
Example I: Money 4:11
Example II: Rental Car 9:23
Example III: Probability of an Earthquake 12:22
Example IV: Defective Laptops 16:52
Example V: Cans of Tuna 21:06
Tchebysheff's Inequality 42:11
Intro 0:00
Tchebysheff's Inequality (Also Known as Chebyshev's Inequality) 0:52
Tchebysheff's Inequality: Definition 0:53
Tchebysheff's Inequality: Equation 1:19
Tchebysheff's Inequality: Intuition 3:21
Tchebysheff's Inequality in Reverse 4:09
Tchebysheff's Inequality in Reverse 4:10
Intuition 5:13
Example I: Money 5:55
Example II: College Units 13:20
Example III: Using Tchebysheff's Inequality to Estimate Proportion 16:40
Example IV: Probability of an Earthquake 25:21
Example V: Using Tchebysheff's Inequality to Estimate Proportion 32:57

## III. Discrete Distributions

Binomial Distribution (Bernoulli Trials) 52:36
Intro 0:00
Binomial Distribution 0:29
Binomial Distribution (Bernoulli Trials) Overview 0:30
Prototypical Examples: Flipping a Coin n Times 1:36
Process with Two Outcomes: Games Between Teams 2:12
Process with Two Outcomes: Rolling a Die to Get a 6 2:42
Formula for the Binomial Distribution 3:45
Fixed Parameters 3:46
Formula for the Binomial Distribution 6:27
Key Properties of the Binomial Distribution 9:54
Mean 9:55
Variance 10:56
Standard Deviation 11:13
Example I: Games Between Teams 11:36
Example II: Exam Score 17:01
Example III: Expected Grade & Standard Deviation 25:59
Example IV: Pogo-sticking Championship, Part A 33:25
Example IV: Pogo-sticking Championship, Part B 38:24
Example V: Expected Championships Winning & Standard Deviation 45:22
Geometric Distribution 52:50
Intro 0:00
Geometric Distribution 0:22
Geometric Distribution: Definition 0:23
Prototypical Example: Flipping a Coin Until We Get a Head 1:08
Geometric Distribution vs. Binomial Distribution. 1:31
Formula for the Geometric Distribution 2:13
Fixed Parameters 2:14
Random Variable 2:49
Formula for the Geometric Distribution 3:16
Key Properties of the Geometric Distribution 6:47
Mean 6:48
Variance 7:10
Standard Deviation 7:25
Geometric Series 7:46
Recall from Calculus II: Sum of Infinite Series 7:47
Application to Geometric Distribution 10:10
Example I: Drawing Cards from a Deck (With Replacement) Until You Get an Ace 13:02
Example I: Question & Solution 13:03
Example II: Mean & Standard Deviation of Winning Pin the Tail on the Donkey 16:32
Example II: Mean 16:33
Example II: Standard Deviation 18:37
Example III: Rolling a Die 22:09
Example III: Setting Up 22:10
Example III: Part A 24:18
Example III: Part B 26:01
Example III: Part C 27:38
Example III: Summary 32:02
Example IV: Job Interview 35:16
Example IV: Setting Up 35:15
Example IV: Part A 37:26
Example IV: Part B 38:33
Example IV: Summary 39:37
Example V: Mean & Standard Deviation of Time to Conduct All the Interviews 41:13
Example V: Setting Up 42:50
Example V: Mean 46:05
Example V: Variance 47:37
Example V: Standard Deviation 48:22
Example V: Summary 49:36
Negative Binomial Distribution 51:39
Intro 0:00
Negative Binomial Distribution 0:11
Negative Binomial Distribution: Definition 0:12
Prototypical Example: Flipping a Coin Until We Get r Successes 0:46
Negative Binomial Distribution vs. Binomial Distribution 1:04
Negative Binomial Distribution vs. Geometric Distribution 1:33
Formula for Negative Binomial Distribution 3:39
Fixed Parameters 3:40
Random Variable 4:57
Formula for Negative Binomial Distribution 5:18
Key Properties of Negative Binomial 7:44
Mean 7:47
Variance 8:03
Standard Deviation 8:09
Example I: Drawing Cards from a Deck (With Replacement) Until You Get Four Aces 8:32
Example I: Question & Solution 8:33
Example II: Chinchilla Grooming 12:37
Example II: Mean 12:38
Example II: Variance 15:09
Example II: Standard Deviation 15:51
Example II: Summary 17:10
Example III: Rolling a Die Until You Get Four Sixes 18:27
Example III: Setting Up 19:38
Example III: Mean 19:38
Example III: Variance 20:31
Example III: Standard Deviation 21:21
Example IV: Job Applicants 24:00
Example IV: Setting Up 24:01
Example IV: Part A 26:16
Example IV: Part B 29:53
Example V: Mean & Standard Deviation of Time to Conduct All the Interviews 40:10
Example V: Setting Up 40:11
Example V: Mean 45:24
Example V: Variance 46:22
Example V: Standard Deviation 47:01
Example V: Summary 48:16
Hypergeometric Distribution 36:27
Intro 0:00
Hypergeometric Distribution 0:11
Hypergeometric Distribution: Definition 0:12
Random Variable 1:38
Formula for the Hypergeometric Distribution 1:50
Fixed Parameters 1:51
Formula for the Hypergeometric Distribution 2:53
Key Properties of Hypergeometric 6:14
Mean 6:15
Variance 6:42
Standard Deviation 7:16
Example I: Students Committee 7:30
Example II: Expected Number of Women on the Committee in Example I 11:08
Example III: Pairs of Shoes 13:49
Example IV: What is the Expected Number of Left Shoes in Example III? 20:46
Example V: Using Indicator Variables & Linearity of Expectation 25:40
Poisson Distribution 52:19
Intro 0:00
Poisson Distribution 0:18
Poisson Distribution: Definition 0:19
Formula for the Poisson Distribution 2:16
Fixed Parameter 2:17
Formula for the Poisson Distribution 2:59
Key Properties of the Poisson Distribution 5:30
Mean 5:34
Variance 6:07
Standard Deviation 6:27
Example I: Forest Fires 6:41
Example II: Call Center, Part A 15:56
Example II: Call Center, Part B 20:50
Example III: Confirming that the Mean of the Poisson Distribution is λ 26:53
Example IV: Find E (Y²) for the Poisson Distribution 35:24
Example V: Earthquakes, Part A 37:57
Example V: Earthquakes, Part B 44:02

## IV. Continuous Distributions

Density & Cumulative Distribution Functions 57:17
Intro 0:00
Density Functions 0:43
Density Functions 0:44
Density Function to Calculate Probabilities 2:41
Cumulative Distribution Functions 4:28
Cumulative Distribution Functions 4:29
Using F to Calculate Probabilities 5:58
Properties of the CDF (Density & Cumulative Distribution Functions) 7:27
F(-∞) = 0 7:34
F(∞) = 1 8:30
F is Increasing 9:14
F'(y) = f(y) 9:21
Example I: Density & Cumulative Distribution Functions, Part A 9:43
Example I: Density & Cumulative Distribution Functions, Part B 14:16
Example II: Density & Cumulative Distribution Functions, Part A 21:41
Example II: Density & Cumulative Distribution Functions, Part B 26:16
Example III: Density & Cumulative Distribution Functions, Part A 32:17
Example III: Density & Cumulative Distribution Functions, Part B 37:08
Example IV: Density & Cumulative Distribution Functions 43:34
Example V: Density & Cumulative Distribution Functions, Part A 51:53
Example V: Density & Cumulative Distribution Functions, Part B 54:19
Mean & Variance for Continuous Distributions 36:18
Intro 0:00
Mean 0:32
Mean for a Continuous Random Variable 0:33
Expectation is Linear 2:07
Variance 2:55
Variance for Continuous random Variable 2:56
Easier to Calculate Via the Mean 3:26
Standard Deviation 5:03
Standard Deviation 5:04
Example I: Mean & Variance for Continuous Distributions 5:43
Example II: Mean & Variance for Continuous Distributions 10:09
Example III: Mean & Variance for Continuous Distributions 16:05
Example IV: Mean & Variance for Continuous Distributions 26:40
Example V: Mean & Variance for Continuous Distributions 30:12
Uniform Distribution 32:49
Intro 0:00
Uniform Distribution 0:15
Uniform Distribution 0:16
Each Part of the Region is Equally Probable 1:39
Key Properties of the Uniform Distribution 2:45
Mean 2:46
Variance 3:27
Standard Deviation 3:48
Example I: Newspaper Delivery 5:25
Example II: Picking a Real Number from a Uniform Distribution 8:21
Example III: Dinner Date 11:02
Example IV: Proving that a Variable is Uniformly Distributed 18:50
Example V: Ice Cream Serving 27:22
Normal (Gaussian) Distribution 1:03:54
Intro 0:00
Normal (Gaussian) Distribution 0:35
Normal (Gaussian) Distribution & The Bell Curve 0:36
Fixed Parameters 0:55
Formula for the Normal Distribution 1:32
Formula for the Normal Distribution 1:33
Calculating on the Normal Distribution can be Tricky 3:32
Standard Normal Distribution 5:12
Standard Normal Distribution 5:13
Graphing the Standard Normal Distribution 6:13
Standard Normal Distribution, Cont. 8:30
Standard Normal Distribution Chart 8:31
Nonstandard Normal Distribution 14:44
Nonstandard Normal Variable & Associated Standard Normal 14:45
Finding Probabilities for Z 15:39
Example I: Chance that Standard Normal Variable Will Land Between 1 and 2? 16:46
Example I: Setting Up the Equation & Graph 16:47
Example I: Solving for z Using the Standard Normal Chart 19:05
Example II: What Proportion of the Data Lies within Two Standard Deviations of the Mean? 20:41
Example II: Setting Up the Equation & Graph 20:42
Example II: Solving for z Using the Standard Normal Chart 24:38
Example III: Scores on an Exam 27:34
Example III: Setting Up the Equation & Graph, Part A 27:35
Example III: Setting Up the Equation & Graph, Part B 33:48
Example III: Solving for z Using the Standard Normal Chart, Part A 38:23
Example III: Solving for z Using the Standard Normal Chart, Part B 40:49
Example IV: Temperatures 42:54
Example IV: Setting Up the Equation & Graph 42:55
Example IV: Solving for z Using the Standard Normal Chart 47:03
Example V: Scores on an Exam 48:41
Example V: Setting Up the Equation & Graph, Part A 48:42
Example V: Setting Up the Equation & Graph, Part B 53:20
Example V: Solving for z Using the Standard Normal Chart, Part A 57:45
Example V: Solving for z Using the Standard Normal Chart, Part B 59:17
Gamma Distribution (with Exponential & Chi-square) 1:08:27
Intro 0:00
Gamma Function 0:49
The Gamma Function 0:50
Properties of the Gamma Function 2:07
Formula for the Gamma Distribution 3:50
Fixed Parameters 3:51
Density Function for Gamma Distribution 4:07
Key Properties of the Gamma Distribution 7:13
Mean 7:14
Variance 7:25
Standard Deviation 7:30
Exponential Distribution 8:03
Definition of Exponential Distribution 8:04
Density 11:23
Mean 13:26
Variance 13:48
Standard Deviation 13:55
Chi-square Distribution 14:34
Chi-square Distribution: Overview 14:35
Chi-square Distribution: Mean 16:27
Chi-square Distribution: Variance 16:37
Chi-square Distribution: Standard Deviation 16:55
Example I: Graphing Gamma Distribution 17:30
Example I: Graphing Gamma Distribution 17:31
Example I: Describe the Effects of Changing α and β on the Shape of the Graph 23:33
Example II: Exponential Distribution 27:11
Example II: Using the Exponential Distribution 27:12
Example II: Summary 35:34
Example III: Earthquake 37:05
Example III: Estimate Using Markov's Inequality 37:06
Example III: Estimate Using Tchebysheff's Inequality 40:13
Example III: Summary 44:13
Example IV: Finding Exact Probability of Earthquakes 46:45
Example IV: Finding Exact Probability of Earthquakes 46:46
Example IV: Summary 51:44
Example V: Prove and Interpret Why the Exponential Distribution is Called 'Memoryless' 52:51
Example V: Prove 52:52
Example V: Interpretation 57:44
Example V: Summary 63:54
Beta Distribution 52:45
Intro 0:00
Beta Function 0:29
Fixed parameters 0:30
Defining the Beta Function 1:19
Relationship between the Gamma & Beta Functions 2:02
Beta Distribution 3:31
Density Function for the Beta Distribution 3:32
Key Properties of the Beta Distribution 6:56
Mean 6:57
Variance 7:16
Standard Deviation 7:37
Example I: Calculate B(3,4) 8:10
Example II: Graphing the Density Functions for the Beta Distribution 12:25
Example III: Show that the Uniform Distribution is a Special Case of the Beta Distribution 24:57
Example IV: Show that this Triangular Distribution is a Special Case of the Beta Distribution 31:20
Example V: Morning Commute 37:39
Example V: Identify the Density Function 38:45
Example V: Morning Commute, Part A 42:22
Example V: Morning Commute, Part B 44:19
Example V: Summary 49:13
Moment-Generating Functions 51:58
Intro 0:00
Moments 0:30
Definition of Moments 0:31
Moment-Generating Functions (MGFs) 3:53
Moment-Generating Functions 3:54
Using the MGF to Calculate the Moments 5:21
Moment-Generating Functions for the Discrete Distributions 8:22
Moment-Generating Functions for Binomial Distribution 8:36
Moment-Generating Functions for Geometric Distribution 9:06
Moment-Generating Functions for Negative Binomial Distribution 9:28
Moment-Generating Functions for Hypergeometric Distribution 9:43
Moment-Generating Functions for Poisson Distribution 9:57
Moment-Generating Functions for the Continuous Distributions 11:34
Moment-Generating Functions for the Uniform Distributions 11:43
Moment-Generating Functions for the Normal Distributions 12:24
Moment-Generating Functions for the Gamma Distributions 12:36
Moment-Generating Functions for the Exponential Distributions 12:44
Moment-Generating Functions for the Chi-square Distributions 13:11
Moment-Generating Functions for the Beta Distributions 13:48
Useful Formulas with Moment-Generating Functions 15:02
Useful Formulas with Moment-Generating Functions 1 15:03
Useful Formulas with Moment-Generating Functions 2 16:21
Example I: Moment-Generating Function for the Binomial Distribution 17:33
Example II: Use the MGF for the Binomial Distribution to Find the Mean of the Distribution 24:40
Example III: Find the Moment Generating Function for the Poisson Distribution 29:28
Example IV: Use the MGF for Poisson Distribution to Find the Mean and Variance of the Distribution 36:27
Example V: Find the Moment-generating Function for the Uniform Distribution 44:47

## V. Multivariate Distributions

Bivariate Density & Distribution Functions 50:52
Intro 0:00
Bivariate Density Functions 0:21
Two Variables 0:23
Bivariate Density Function 0:52
Properties of the Density Function 1:57
Properties of the Density Function 1 1:59
Properties of the Density Function 2 2:20
We Can Calculate Probabilities 2:53
If You Have a Discrete Distribution 4:36
Bivariate Distribution Functions 5:25
Bivariate Distribution Functions 5:26
Properties of the Bivariate Distribution Functions 1 7:19
Properties of the Bivariate Distribution Functions 2 7:36
Example I: Bivariate Density & Distribution Functions 8:08
Example II: Bivariate Density & Distribution Functions 14:40
Example III: Bivariate Density & Distribution Functions 24:33
Example IV: Bivariate Density & Distribution Functions 32:04
Example V: Bivariate Density & Distribution Functions 40:26
Marginal Probability 42:38
Intro 0:00
Discrete Case 0:48
Marginal Probability Functions 0:49
Continuous Case 3:07
Marginal Density Functions 3:08
Example I: Compute the Marginal Probability Function 5:58
Example II: Compute the Marginal Probability Function 14:07
Example III: Marginal Density Function 24:01
Example IV: Marginal Density Function 30:47
Example V: Marginal Density Function 36:05
Conditional Probability & Conditional Expectation 1:02:24
Intro 0:00
Review of Marginal Probability 0:46
Recall the Marginal Probability Functions & Marginal Density Functions 0:47
Conditional Probability, Discrete Case 3:14
Conditional Probability, Discrete Case 3:15
Conditional Probability, Continuous Case 4:15
Conditional Density of Y₁ given that Y₂ = y₂ 4:16
Interpret This as a Density on Y₁ & Calculate Conditional Probability 5:03
Conditional Expectation 6:44
Conditional Expectation: Continuous 6:45
Conditional Expectation: Discrete 8:03
Example I: Conditional Probability 8:29
Example II: Conditional Probability 23:59
Example III: Conditional Probability 34:28
Example IV: Conditional Expectation 43:16
Example V: Conditional Expectation 48:28
Independent Random Variables 51:39
Intro 0:00
Intuition 0:55
Experiment with Two Random Variables 0:56
Intuition Formula 2:17
Definition and Formulas 4:43
Definition 4:44
Short Version: Discrete 5:10
Short Version: Continuous 5:48
Theorem 9:33
For Continuous Random Variables, Y₁ & Y₂ are Independent If & Only If: Condition 1 9:34
For Continuous Random Variables, Y₁ & Y₂ are Independent If & Only If: Condition 2 11:22
Example I: Use the Definition to Determine if Y₁ and Y₂ are Independent 12:49
Example II: Use the Definition to Determine if Y₁ and Y₂ are Independent 21:33
Example III: Are Y₁ and Y₂ Independent? 27:01
Example IV: Are Y₁ and Y₂ Independent? 34:51
Example V: Are Y₁ and Y₂ Independent? 43:44
Expected Value of a Function of Random Variables 37:07
Intro 0:00
Review of Single Variable Case 0:29
Expected Value of a Single Variable 0:30
Expected Value of a Function g(Y) 1:12
Bivariate Case 2:11
Expected Value of a Function g(Y₁, Y₂) 2:12
Linearity of Expectation 3:24
Linearity of Expectation 1 3:25
Linearity of Expectation 2 3:38
Linearity of Expectation 3: Additivity 4:03
Example I: Calculate E (Y₁ + Y₂) 4:39
Example II: Calculate E (Y₁Y₂) 14:47
Example III: Calculate E (U₁) and E(U₂) 19:33
Example IV: Calculate E (Y₁) and E(Y₂) 22:50
Example V: Calculate E (2Y₁ + 3Y₂) 33:05
Covariance, Correlation & Linear Functions 59:50
Intro 0:00
Definition and Formulas for Covariance 0:38
Definition of Covariance 0:39
Formulas to Calculate Covariance 1:36
Intuition for Covariance 3:54
Covariance is a Measure of Dependence 3:55
Dependence Doesn't Necessarily Mean that the Variables Do the Same Thing 4:12
If Variables Move Together 4:47
If Variables Move Against Each Other 5:04
Both Cases Show Dependence! 5:30
Independence Theorem 8:10
Independence Theorem 8:11
The Converse is Not True 8:32
Correlation Coefficient 9:33
Correlation Coefficient 9:34
Linear Functions of Random Variables 11:57
Linear Functions of Random Variables: Expected Value 11:58
Linear Functions of Random Variables: Variance 12:58
Linear Functions of Random Variables, Cont. 14:30
Linear Functions of Random Variables: Covariance 14:35
Example I: Calculate E (Y₁), E (Y₂), and E (Y₁Y₂) 15:31
Example II: Are Y₁ and Y₂ Independent? 29:16
Example III: Calculate V (U₁) and V (U₂) 36:14
Example IV: Calculate the Covariance Correlation Coefficient 42:12
Example V: Find the Mean and Variance of the Average 52:19

## VI. Distributions of Functions of Random Variables

Distribution Functions 1:07:35
Intro 0:00
Premise 0:44
Premise 0:45
Goal 1:38
Goal Number 1: Find the Full Distribution Function 1:39
Goal Number 2: Find the Density Function 1:55
Goal Number 3: Calculate Probabilities 2:17
Three Methods 3:05
Method 1: Distribution Functions 3:06
Method 2: Transformations 3:38
Method 3: Moment-generating Functions 3:47
Distribution Functions 4:03
Distribution Functions 4:04
Example I: Find the Density Function 6:41
Step 1: Find the Distribution Function 6:42
Step 2: Find the Density Function 10:20
Summary 11:51
Example II: Find the Density Function 14:36
Step 1: Find the Distribution Function 14:37
Step 2: Find the Density Function 18:19
Summary 19:22
Example III: Find the Cumulative Distribution & Density Functions 20:39
Step 1: Find the Cumulative Distribution 20:40
Step 2: Find the Density Function 28:58
Summary 30:20
Example IV: Find the Density Function 33:01
Step 1: Setting Up the Equation & Graph 33:02
Step 2: If u ≤ 1 38:32
Step 3: If u ≥ 1 41:02
Step 4: Find the Distribution Function 42:40
Step 5: Find the Density Function 43:11
Summary 45:03
Example V: Find the Density Function 48:32
Step 1: Exponential 48:33
Step 2: Independence 50:48
Step 2: Find the Distribution Function 51:47
Step 3: Find the Density Function 60:17
Summary 62:05
Transformations 1:00:16
Intro 0:00
Premise 0:32
Premise 0:33
Goal 1:37
Goal Number 1: Find the Full Distribution Function 1:38
Goal Number 2: Find the Density Function 1:49
Goal Number 3: Calculate Probabilities 2:04
Three Methods 2:34
Method 1: Distribution Functions 2:35
Method 2: Transformations 2:57
Method 3: Moment-generating Functions 3:05
Requirements for Transformation Method 3:22
The Transformation Method Only Works for Single-variable Situations 3:23
Must be a Strictly Monotonic Function 3:50
Example: Strictly Monotonic Function 4:50
If the Function is Monotonic, Then It is Invertible 5:30
Formula for Transformations 7:09
Formula for Transformations 7:11
Example I: Determine whether the Function is Monotonic, and if so, Find Its Inverse 8:26
Example II: Find the Density Function 12:07
Example III: Determine whether the Function is Monotonic, and if so, Find Its Inverse 17:12
Example IV: Find the Density Function for the Magnitude of the Next Earthquake 21:30
Example V: Find the Expected Magnitude of the Next Earthquake 33:20
Example VI: Find the Density Function, Including the Range of Possible Values for u 47:42
Moment-Generating Functions 1:18:52
Intro 0:00
Premise 0:30
Premise 0:31
Goal 1:40
Goal Number 1: Find the Full Distribution Function 1:41
Goal Number 2: Find the Density Function 1:51
Goal Number 3: Calculate Probabilities 2:01
Three Methods 2:39
Method 1: Distribution Functions 2:40
Method 2: Transformations 2:50
Method 3: Moment-Generating Functions 2:55
Review of Moment-Generating Functions 3:04
Recall: The Moment-Generating Function for a Random Variable Y 3:05
The Moment-Generating Function is a Function of t (Not y) 3:45
Moment-Generating Functions for the Discrete Distributions 4:31
Binomial 4:50
Geometric 5:12
Negative Binomial 5:24
Hypergeometric 5:33
Poisson 5:42
Moment-Generating Functions for the Continuous Distributions 6:08
Uniform 6:09
Normal 6:17
Gamma 6:29
Exponential 6:34
Chi-square 7:05
Beta 7:48
Useful Formulas with the Moment-Generating Functions 8:48
Useful Formula 1 8:49
Useful Formula 2 9:51
How to Use Moment-Generating Functions 10:41
How to Use Moment-Generating Functions 10:42
Example I: Find the Density Function 12:22
Example II: Find the Density Function 30:58
Example III: Find the Probability Function 43:29
Example IV: Find the Probability Function 51:43
Example V: Find the Distribution 60:14
Example VI: Find the Density Function 72:10
Order Statistics 1:04:56
Intro 0:00
Premise 0:11
Example Question: How Tall Will the Tallest Student in My Next Semester's Probability Class Be? 0:12
Setting 0:56
Definition 1 1:49
Definition 2 2:01
Question: What are the Distributions & Densities? 4:08
Formulas 4:47
Distribution of Max 5:11
Density of Max 6:00
Distribution of Min 7:08
Density of Min 7:18
Example I: Distribution & Density Functions 8:29
Example I: Distribution 8:30
Example I: Density 11:07
Example I: Summary 12:33
Example II: Distribution & Density Functions 14:25
Example II: Distribution 14:26
Example II: Density 17:21
Example II: Summary 19:00
Example III: Mean & Variance 20:32
Example III: Mean 20:33
Example III: Variance 25:48
Example III: Summary 30:57
Example IV: Distribution & Density Functions 35:43
Example IV: Distribution 35:44
Example IV: Density 43:03
Example IV: Summary 46:11
Example V: Find the Expected Time Until the Team's First Injury 51:14
Example V: Solution 51:15
Example V: Summary 61:11
Sampling from a Normal Distribution 1:00:07
Intro 0:00
Setting 0:36
Setting 0:37
Assumptions and Notation 2:18
Assumption Forever 2:19
Assumption for this Lecture Only 3:21
Notation 3:49
The Sample Mean 4:15
Statistic We'll Study the Sample Mean 4:16
Theorem 5:40
Standard Normal Distribution 7:03
Standard Normal Distribution 7:04
Converting to Standard Normal 10:11
Recall 10:12
Corollary to Theorem 10:41
Example I: Heights of Students 13:18
Example II: What Happens to This Probability as n → ∞ 22:36
Example III: Units at a University 32:24
Example IV: Probability of Sample Mean 40:53
Example V: How Many Samples Should We Take? 48:34
The Central Limit Theorem 1:09:55
Intro 0:00
Setting 0:52
Setting 0:53
Assumptions and Notation 2:53
Our Samples are Independent (Independent Identically Distributed) 2:54
No Longer Assume that the Population is Normally Distributed 3:30
The Central Limit Theorem 4:36
The Central Limit Theorem Overview 4:38
The Central Limit Theorem in Practice 6:24
Standard Normal Distribution 8:09
Standard Normal Distribution 8:13
Converting to Standard Normal 10:13
Recall: If Y is Normal, Then … 10:14
Corollary to Theorem 11:09
Example I: Probability of Finishing Your Homework 12:56
Example I: Solution 12:57
Example I: Summary 18:20
Example I: Confirming with the Standard Normal Distribution Chart 20:18
Example II: Probability of Selling Muffins 21:26
Example II: Solution 21:27
Example II: Summary 29:09
Example II: Confirming with the Standard Normal Distribution Chart 31:09
Example III: Probability that a Soda Dispenser Gives the Correct Amount of Soda 32:41
Example III: Solution 32:42
Example III: Summary 38:03
Example III: Confirming with the Standard Normal Distribution Chart 40:58
Example IV: How Many Samples Should She Take? 42:06
Example IV: Solution 42:07
Example IV: Summary 49:18
Example IV: Confirming with the Standard Normal Distribution Chart 51:57
Example V: Restaurant Revenue 54:41
Example V: Solution 54:42
Example V: Summary 64:21
Example V: Confirming with the Standard Normal Distribution Chart 66:48

## Course Details:

Duration: 30 hours, 47 minutes

Number of Lessons: 35

This class is geared towards students who are taking an Introduction to Probability course. Dr. Murray utilizes his teaching experience to share all his insights and strategies essential to doing well in your own class. Each lesson also comes with downloadable study guide PDFs.

• Free Sample Lessons
• Study Guides

Topics Include:

• Combinations & Permutations
• Bayes’ Rule
• Expected Value
• Tchebysheff’s Inequality
• Geometric Distribution
• Poisson Distribution
• Gaussian Distribution
• Marginal Probability
• Moment-Generating Functions
• Central Limit Theorem

Dr. Murray received his Ph.D from UC Berkeley, his BS from Georgetown University, and has been teaching in the university setting for 15+ years.

### Student Testimonials:

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"Professor Dr.William Murray, you are the best Professor I have ever seen!" — Muhammad A.