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Introduction to Probability Online Course Dr. William Murray
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35 Lessons (30hr : 47min)
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4.8 372 Ratings & 13 Reviews
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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.

Table of Contents
Course Details
About Dr. Murray

Section 1: Probability by Counting
	Experiments, Outcomes, Samples, Spaces, Events			59:30
		Intro		0:00
		Terminology		0:19
			Experiment	0:26
			Outcome	0:56
			Sample Space	1:16
			Event	1:55
		Key Formula		2:47
			Formula for Finding the Probability of an Event	2:48
			Example: Drawing a Card	3:36
		Example I		5:01
			Experiment	5:38
			Outcomes	5:54
			Probability of the Event	8:11
		Example II		12:00
			Experiment	12:17
			Outcomes	12:34
			Probability of the Event	13:49
		Example III		16:33
			Experiment	17:09
			Outcomes	17:33
			Probability of the Event	18:25
		Example IV		21:20
			Experiment	21:21
			Outcomes	22:00
			Probability of the Event	23:22
		Example V		31:41
			Experiment	32:14
			Outcomes	32:35
			Probability of the Event	33:27
			Alternate Solution	40:16
		Example VI		43:33
			Experiment	44:08
			Outcomes	44:24
			Probability of the Event	53:35
	Combining Events: Multiplication & Addition			1:02:47
		Intro		0:00
		Unions of Events		0:40
			Unions of Events	0:41
			Disjoint Events	3:42
		Intersections of Events		4:18
			Intersections of Events	4:19
		Conditional Probability		5:47
			Conditional Probability	5:48
		Independence		8:20
			Independence	8:21
			Warning: Independent Does Not Mean Disjoint	9:53
			If A and B are Independent	11:20
		Example I: Choosing a Number at Random		12:41
			Solving by Counting	12:52
			Solving by Probability	17:26
		Example II: Combination		22:07
			Combination Deal at a Restaurant	22:08
		Example III: Rolling Two Dice		24:18
			Define the Events	24:20
			Solving by Counting	27:35
			Solving by Probability	29:32
		Example IV: Flipping a Coin		35:07
			Flipping a Coin Four Times	35:08
		Example V: Conditional Probabilities		41:22
			Define the Events	42:23
			Calculate the Conditional Probabilities	46:21
		Example VI: Independent Events		53:42
			Define the Events	53:43
			Are Events Independent?	55:21
	Choices: Combinations & Permutations			56:03
		Intro		0:00
		Choices: With or Without Replacement?		0:12
			Choices: With or Without Replacement?	0:13
			Example: With Replacement	2:17
			Example: Without Replacement	2:55
		Choices: Ordered or Unordered?		4:10
			Choices: Ordered or Unordered?	4:11
			Example: Unordered	4:52
			Example: Ordered	6:08
		Combinations		9:23
			Definition & Equation: Combinations	9:24
			Example: Combinations	12:12
		Permutations		13:56
			Definition & Equation: Permutations	13:57
			Example: Permutations	15:00
		Key Formulas		17:19
			Number of Ways to Pick r Things from n Possibilities	17:20
		Example I: Five Different Candy Bars		18:31
		Example II: Five Identical Candy Bars		24:53
		Example III: Five Identical Candy Bars		31:56
		Example IV: Five Different Candy Bars		39:21
		Example V: Pizza & Toppings		45:03
	Inclusion & Exclusion			43:40
		Intro		0:00
		Inclusion/Exclusion: Two Events		0:09
			Inclusion/Exclusion: Two Events	0:10
		Inclusion/Exclusion: Three Events		2:30
			Inclusion/Exclusion: Three Events	2:31
		Example I: Inclusion & Exclusion		6:24
		Example II: Inclusion & Exclusion		11:01
		Example III: Inclusion & Exclusion		18:41
		Example IV: Inclusion & Exclusion		28:24
		Example V: Inclusion & Exclusion		39:33
	Independence			46:09
		Intro		0:00
		Formula and Intuition		0:12
			Definition of Independence	0:19
			Intuition	0:49
		Common Misinterpretations		1:37
			Myth & Truth 1	1:38
			Myth & Truth 2	2:23
		Combining Independent Events		3:56
			Recall: Formula for Conditional Probability	3:58
			Combining Independent Events	4:10
		Example I: Independence		5:36
		Example II: Independence		14:14
		Example III: Independence		21:10
		Example IV: Independence		32:45
		Example V: Independence		41:13
	Bayes' Rule			1:02:10
		Intro		0:00
		When to Use Bayes' Rule		0:08
			When to Use Bayes' Rule: Disjoint Union of Events	0:09
		Bayes' Rule for Two Choices		2:50
			Bayes' Rule for Two Choices	2:51
		Bayes' Rule for Multiple Choices		5:03
			Bayes' Rule for Multiple Choices	5:04
		Example I: What is the Chance that She is Diabetic?		6:55
			Example I: Setting up the Events	6:56
			Example I: Solution	11:33
		Example II: What is the chance that It Belongs to a Woman?		19:28
			Example II: Setting up the Events	19:29
			Example II: Solution	21:45
		Example III: What is the Probability that She is a Democrat?		27:31
			Example III: Setting up the Events	27:32
			Example III: Solution	32:08
		Example IV: What is the chance that the Fruit is an Apple?		39:11
			Example IV: Setting up the Events	39:12
			Example IV: Solution	43:50
		Example V: What is the Probability that the Oldest Child is a Girl?		51:16
			Example V: Setting up the Events	51:17
			Example V: Solution	53:07
Section 2: Random Variables
	Random Variables & Probability Distribution			38:21
		Intro		0:00
		Intuition		0:15
			Intuition for Random Variable	0:16
			Example: Random Variable	0:44
		Intuition, Cont.		2:52
			Example: Random Variable as Payoff	2:57
		Definition		5:11
			Definition of a Random Variable	5:13
			Example: Random Variable in Baseball	6:02
		Probability Distributions		7:18
			Probability Distributions	7:19
		Example I: Probability Distribution for the Random Variable		9:29
		Example II: Probability Distribution for the Random Variable		14:52
		Example III: Probability Distribution for the Random Variable		21:52
		Example IV: Probability Distribution for the Random Variable		27:25
		Example V: Probability Distribution for the Random Variable		34:12
	Expected Value (Mean)			46:14
		Intro		0:00
		Definition of Expected Value		0:20
			Expected Value of a (Discrete) Random Variable or Mean	0:21
		Indicator Variables		3:03
			Indicator Variable	3:04
		Linearity of Expectation		4:36
			Linearity of Expectation for Random Variables	4:37
		Expected Value of a Function		6:03
			Expected Value of a Function	6:04
		Example I: Expected Value		7:30
		Example II: Expected Value		14:14
		Example III: Expected Value of Flipping a Coin		21:42
			Example III: Part A	21:43
			Example III: Part B	30:43
		Example IV: Semester Average		36:39
		Example V: Expected Value of a Function of a Random Variable		41:28
	Variance & Standard Deviation			47:23
		Intro		0:00
		Definition of Variance		0:11
			Variance of a Random Variable	0:12
			Variance is a Measure of the Variability, or Volatility	1:06
			Most Useful Way to Calculate Variance	2:46
		Definition of Standard Deviation		3:44
			Standard Deviation of a Random Variable	3:45
		Example I: Which of the Following Sets of Data Has the Largest Variance?		5:34
		Example II: Which of the Following Would be the Least Useful in Understanding a Set of Data?		9:02
		Example III: Calculate the Mean, Variance, & Standard Deviation		11:48
			Example III: Mean	12:56
			Example III: Variance	14:06
			Example III: Standard Deviation	15:42
		Example IV: Calculate the Mean, Variance, & Standard Deviation		17:54
			Example IV: Mean	18:47
			Example IV: Variance	20:36
			Example IV: Standard Deviation	25:34
		Example V: Calculate the Mean, Variance, & Standard Deviation		29:56
			Example V: Mean	30:13
			Example V: Variance	33:28
			Example V: Standard Deviation	34:48
		Example VI: Calculate the Mean, Variance, & Standard Deviation		37:29
			Example VI: Possible Outcomes	38:09
			Example VI: Mean	39:29
			Example VI: Variance	41:22
			Example VI: Standard Deviation	43:28
	Markov's Inequality			26:45
		Intro		0:00
		Markov's Inequality		0:25
			Markov's Inequality: Definition & Condition	0:26
			Markov's Inequality: Equation	1:15
			Markov's Inequality: Reverse Equation	2:48
		Example I: Money		4:11
		Example II: Rental Car		9:23
		Example III: Probability of an Earthquake		12:22
		Example IV: Defective Laptops		16:52
		Example V: Cans of Tuna		21:06
	Tchebysheff's Inequality			42:11
		Intro		0:00
		Tchebysheff's Inequality (Also Known as Chebyshev's Inequality)		0:52
			Tchebysheff's Inequality: Definition	0:53
			Tchebysheff's Inequality: Equation	1:19
			Tchebysheff's Inequality: Intuition	3:21
		Tchebysheff's Inequality in Reverse		4:09
			Tchebysheff's Inequality in Reverse	4:10
			Intuition	5:13
		Example I: Money		5:55
		Example II: College Units		13:20
		Example III: Using Tchebysheff's Inequality to Estimate Proportion		16:40
		Example IV: Probability of an Earthquake		25:21
		Example V: Using Tchebysheff's Inequality to Estimate Proportion		32:57
Section 3: Discrete Distributions
	Binomial Distribution (Bernoulli Trials)			52:36
		Intro		0:00
		Binomial Distribution		0:29
			Binomial Distribution (Bernoulli Trials) Overview	0:30
			Prototypical Examples: Flipping a Coin n Times	1:36
			Process with Two Outcomes: Games Between Teams	2:12
			Process with Two Outcomes: Rolling a Die to Get a 6	2:42
		Formula for the Binomial Distribution		3:45
			Fixed Parameters	3:46
			Formula for the Binomial Distribution	6:27
		Key Properties of the Binomial Distribution		9:54
			Mean	9:55
			Variance	10:56
			Standard Deviation	11:13
		Example I: Games Between Teams		11:36
		Example II: Exam Score		17:01
		Example III: Expected Grade & Standard Deviation		25:59
		Example IV: 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
Section 4: Continuous Distributions
	Density & Cumulative Distribution Functions			57:17
		Intro		0:00
		Density Functions		0:43
			Density Functions	0:44
			Density Function to Calculate Probabilities	2:41
		Cumulative Distribution Functions		4:28
			Cumulative Distribution Functions	4:29
			Using F to Calculate Probabilities	5:58
		Properties of the CDF (Density & Cumulative Distribution Functions)		7:27
			F(-∞) = 0	7:34
			F(∞) = 1	8:30
			F is Increasing	9:14
			F'(y) = f(y)	9:21
		Example I: Density & Cumulative Distribution Functions, Part A		9:43
		Example I: Density & Cumulative Distribution Functions, Part B		14:16
		Example II: Density & Cumulative Distribution Functions, Part A		21:41
		Example II: Density & Cumulative Distribution Functions, Part B		26:16
		Example III: Density & Cumulative Distribution Functions, Part A		32:17
		Example III: Density & Cumulative Distribution Functions, Part B		37:08
		Example IV: Density & Cumulative Distribution Functions		43:34
		Example V: Density & Cumulative Distribution Functions, Part A		51:53
		Example V: Density & Cumulative Distribution Functions, Part B		54:19
	Mean & Variance for Continuous Distributions			36:18
		Intro		0:00
		Mean		0:32
			Mean for a Continuous Random Variable	0:33
			Expectation is Linear	2:07
		Variance		2:55
			Variance for Continuous random Variable	2:56
			Easier to Calculate Via the Mean	3:26
		Standard Deviation		5:03
			Standard Deviation	5:04
		Example I: Mean & Variance for Continuous Distributions		5:43
		Example II: Mean & Variance for Continuous Distributions		10:09
		Example III: Mean & Variance for Continuous Distributions		16:05
		Example IV: Mean & Variance for Continuous Distributions		26:40
		Example V: Mean & Variance for Continuous Distributions		30:12
	Uniform Distribution			32:49
		Intro		0:00
		Uniform Distribution		0:15
			Uniform Distribution	0:16
			Each Part of the Region is Equally Probable	1:39
		Key Properties of the Uniform Distribution		2:45
			Mean	2:46
			Variance	3:27
			Standard Deviation	3:48
		Example I: Newspaper Delivery		5:25
		Example II: Picking a Real Number from a Uniform Distribution		8:21
		Example III: Dinner Date		11:02
		Example IV: Proving that a Variable is Uniformly Distributed		18:50
		Example V: Ice Cream Serving		27:22
	Normal (Gaussian) Distribution			1:03:54
		Intro		0:00
		Normal (Gaussian) Distribution		0:35
			Normal (Gaussian) Distribution & The Bell Curve	0:36
			Fixed Parameters	0:55
		Formula for the Normal Distribution		1:32
			Formula for the Normal Distribution	1:33
			Calculating on the Normal Distribution can be Tricky	3:32
		Standard Normal Distribution		5:12
			Standard Normal Distribution	5:13
			Graphing the Standard Normal Distribution	6:13
		Standard Normal Distribution, Cont.		8:30
			Standard Normal Distribution Chart	8:31
		Nonstandard Normal Distribution		14:44
			Nonstandard Normal Variable & Associated Standard Normal	14:45
			Finding Probabilities for Z	15:39
		Example I: Chance that Standard Normal Variable Will Land Between 1 and 2?		16:46
			Example I: Setting Up the Equation & Graph	16:47
			Example I: Solving for z Using the Standard Normal Chart	19:05
		Example II: What Proportion of the Data Lies within Two Standard Deviations of the Mean?		20:41
			Example II: Setting Up the Equation & Graph	20:42
			Example II: Solving for z Using the Standard Normal Chart	24:38
		Example III: Scores on an Exam		27:34
			Example III: Setting Up the Equation & Graph, Part A	27:35
			Example III: Setting Up the Equation & Graph, Part B	33:48
			Example III: Solving for z Using the Standard Normal Chart, Part A	38:23
			Example III: Solving for z Using the Standard Normal Chart, Part B	40:49
		Example IV: Temperatures		42:54
			Example IV: Setting Up the Equation & Graph	42:55
			Example IV: Solving for z Using the Standard Normal Chart	47:03
		Example V: Scores on an Exam		48:41
			Example V: Setting Up the Equation & Graph, Part A	48:42
			Example V: Setting Up the Equation & Graph, Part B	53:20
			Example V: Solving for z Using the Standard Normal Chart, Part A	57:45
			Example V: Solving for z Using the Standard Normal Chart, Part B	59:17
	Gamma Distribution (with Exponential & 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
Section 5: Multivariate Distributions
	Bivariate Density & Distribution Functions			50:52
		Intro		0:00
		Bivariate Density Functions		0:21
			Two Variables	0:23
			Bivariate Density Function	0:52
		Properties of the Density Function		1:57
			Properties of the Density Function 1	1:59
			Properties of the Density Function 2	2:20
			We Can Calculate Probabilities	2:53
			If You Have a Discrete Distribution	4:36
		Bivariate Distribution Functions		5:25
			Bivariate Distribution Functions	5:26
			Properties of the Bivariate Distribution Functions 1	7:19
			Properties of the Bivariate Distribution Functions 2	7:36
		Example I: Bivariate Density & Distribution Functions		8:08
		Example II: Bivariate Density & Distribution Functions		14:40
		Example III: Bivariate Density & Distribution Functions		24:33
		Example IV: Bivariate Density & Distribution Functions		32:04
		Example V: Bivariate Density & Distribution Functions		40:26
	Marginal Probability			42:38
		Intro		0:00
		Discrete Case		0:48
			Marginal Probability Functions	0:49
		Continuous Case		3:07
			Marginal Density Functions	3:08
		Example I: Compute the Marginal Probability Function		5:58
		Example II: Compute the Marginal Probability Function		14:07
		Example III: Marginal Density Function		24:01
		Example IV: Marginal Density Function		30:47
		Example V: Marginal Density Function		36:05
	Conditional Probability & Conditional Expectation			1:02:24
		Intro		0:00
		Review of Marginal Probability		0:46
			Recall the Marginal Probability Functions & Marginal Density Functions	0:47
		Conditional Probability, Discrete Case		3:14
			Conditional Probability, Discrete Case	3:15
		Conditional Probability, Continuous Case		4:15
			Conditional Density of Y₁ given that Y₂ = y₂	4:16
			Interpret This as a Density on Y₁ & Calculate Conditional Probability	5:03
		Conditional Expectation		6:44
			Conditional Expectation: Continuous	6:45
			Conditional Expectation: Discrete	8:03
		Example I: Conditional Probability		8:29
		Example II: Conditional Probability		23:59
		Example III: Conditional Probability		34:28
		Example IV: Conditional Expectation		43:16
		Example V: Conditional Expectation		48:28
	Independent Random Variables			51:39
		Intro		0:00
		Intuition		0:55
			Experiment with Two Random Variables	0:56
			Intuition Formula	2:17
		Definition and Formulas		4:43
			Definition	4:44
			Short Version: Discrete	5:10
			Short Version: Continuous	5:48
		Theorem		9:33
			For Continuous Random Variables, Y₁ & Y₂ are Independent If & Only If: Condition 1	9:34
			For Continuous Random Variables, Y₁ & Y₂ are Independent If & Only If: Condition 2	11:22
		Example I: Use the Definition to Determine if Y₁ and Y₂ are Independent		12:49
		Example II: Use the Definition to Determine if Y₁ and Y₂ are Independent		21:33
		Example III: Are Y₁ and Y₂ Independent?		27:01
		Example IV: Are Y₁ and Y₂ Independent?		34:51
		Example V: Are Y₁ and Y₂ Independent?		43:44
	Expected Value of a Function of Random Variables			37:07
		Intro		0:00
		Review of Single Variable Case		0:29
			Expected Value of a Single Variable	0:30
			Expected Value of a Function g(Y)	1:12
		Bivariate Case		2:11
			Expected Value of a Function g(Y₁, Y₂)	2:12
		Linearity of Expectation		3:24
			Linearity of Expectation 1	3:25
			Linearity of Expectation 2	3:38
			Linearity of Expectation 3: Additivity	4:03
		Example I: Calculate E (Y₁ + Y₂)		4:39
		Example II: Calculate E (Y₁Y₂)		14:47
		Example III: Calculate E (U₁) and E(U₂)		19:33
		Example IV: Calculate E (Y₁) and E(Y₂)		22:50
		Example V: Calculate E (2Y₁ + 3Y₂)		33:05
	Covariance, Correlation & Linear Functions			59:50
		Intro		0:00
		Definition and Formulas for Covariance		0:38
			Definition of Covariance	0:39
			Formulas to Calculate Covariance	1:36
		Intuition for Covariance		3:54
			Covariance is a Measure of Dependence	3:55
			Dependence Doesn't Necessarily Mean that the Variables Do the Same Thing	4:12
			If Variables Move Together	4:47
			If Variables Move Against Each Other	5:04
			Both Cases Show Dependence!	5:30
		Independence Theorem		8:10
			Independence Theorem	8:11
			The Converse is Not True	8:32
		Correlation Coefficient		9:33
			Correlation Coefficient	9:34
		Linear Functions of Random Variables		11:57
			Linear Functions of Random Variables: Expected Value	11:58
			Linear Functions of Random Variables: Variance	12:58
		Linear Functions of Random Variables, Cont.		14:30
			Linear Functions of Random Variables: Covariance	14:35
		Example I: Calculate E (Y₁), E (Y₂), and E (Y₁Y₂)		15:31
		Example II: Are Y₁ and Y₂ Independent?		29:16
		Example III: Calculate V (U₁) and V (U₂)		36:14
		Example IV: Calculate the Covariance Correlation Coefficient		42:12
		Example V: Find the Mean and Variance of the Average		52:19
Section 6: Distributions of Functions of Random Variables
	Distribution Functions			1:07:35
		Intro		0:00
		Premise		0:44
			Premise	0:45
		Goal		1:38
			Goal Number 1: Find the Full Distribution Function	1:39
			Goal Number 2: Find the Density Function	1:55
			Goal Number 3: Calculate Probabilities	2:17
		Three Methods		3:05
			Method 1: Distribution Functions	3:06
			Method 2: Transformations	3:38
			Method 3: 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

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.

Additional Features:

Free Sample Lessons
Downloadable Lecture Slides
Study Guides
Instructor Comments

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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“I'm really excited about this Probability course, especially after the wonderful Calculus 2 videos you made.” — Rasheed A.

Visit Dr. Murray’s page

Student Feedback

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By Setare Metanat•May 16, 2018

sorry Dr Murry I know it is not your fault ,but I am not happy with educator tech team .though I asked for fixing this problem weeks ago, yet it has not fixed yet! this kind of delay since we should pay each month for these courses is not acceptable.

By Hector Flores•November 28, 2016

Thank you Dr. Murray. Perhaps it's an ipad issue or the network I was connected to at the time. I'll have to try it on a desktop.

By Thuy Nguyen•November 4, 2016

Hi Dr. Murray, thank you for your lecture, you explain so well. On the integral of finding E(Y1), your 1 looks like a 2, and I was confused until the end where you circled the number and said it was a, "one".

By Ali AlSaedi•February 23, 2016

Thank you very much, you made it so clear and easy to understand.

By bo zhang•August 25, 2015

No Problem, i passed the exam two weeks ago,thank you anyway.