text:
 <NAME>

Random Variables & Probability Distribution

Slide Duration:

Section 1: Probability by Counting
Experiments, Outcomes, Samples, Spaces, Events

59m 30s

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

1h 2m 47s

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

56m 3s

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

43m 40s

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

46m 9s

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

1h 2m 10s

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

38m 21s

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)

46m 14s

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

47m 23s

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

26m 45s

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

42m 11s

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)

52m 36s

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

52m 50s

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

51m 39s

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

36m 27s

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

52m 19s

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

57m 17s

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

36m 18s

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

32m 49s

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

1h 3m 54s

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)

1h 8m 27s

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
1:03:54
Beta Distribution

52m 45s

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

51m 58s

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

50m 52s

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

42m 38s

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

1h 2m 24s

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

51m 39s

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

37m 7s

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

59m 50s

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

1h 7m 35s

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
1:00:17
Summary
1:02:05
Transformations

1h 16s

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

1h 18m 52s

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
1:00:14
Example VI: Find the Density Function
1:12:10
Order Statistics

1h 4m 56s

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
1:01:11
Sampling from a Normal Distribution

1h 7s

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

1h 9m 55s

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
1:04:21
Example V: Confirming with the Standard Normal Distribution Chart
1:06:48
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• ## Transcription

### Random Variables & Probability Distribution

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Intuition 0:15
• Intuition for Random Variable
• Example: Random Variable
• Intuition, Cont. 2:52
• Example: Random Variable as Payoff
• Definition 5:11
• Definition of a Random Variable
• Example: Random Variable in Baseball
• Probability Distributions 7:18
• Probability Distributions
• 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

### Transcription: Random Variables & Probability Distribution

Hi and welcome back to the probability lectures here on www.educator.com.0000

My name is <NAME>, and today, we are going to talk about random variables.0004

The idea that goes along with that is the probability distribution.0008

We are going to learn what those terms mean, let us jump right into it.0012

I'm going to start with the intuition for random variable0017

because a formal definition of random variable is not really that illuminating.0021

We will start out with the intuition, try to give you an idea of roughly what they mean,0026

then I will give you the formal definition.0030

The intuition for a random variable is it is a quantity you keep track of during an experiment.0032

We are going to use Y for random variable.0040

I want to start out with an example right away.0045

Let us say we are going to play the World Series and the Yankees are going to play the Giants for 7 games.0048

That is not quite how the actual world series run.0054

Before I get a bunch of angry comments from a bunch of sports fans,0056

I know that some× in the World Series, one team wins all four games right away.0060

It does not actually run for 7 games.0068

We are not playing under those rules.0070

We are going to play 7 games, all 7 games no matter what, no matter who wins the first 2 games.0072

What can happen there is, there are7 different games,0078

each one can have 2 possible outcomes because the Yankees can win and the Giants can win.0082

There are really 2⁷ possible outcomes.0086

There are 128 possible sequences of events in the World Series.0089

Honestly, what we have been really only care about is the number of games that one team wins vs.0095

the number of games that the other team wins.0101

We could say, for example, that Y is going to be the number of games that the Yankees win.0104

What we are really care about is whether Y is more or less than 40111

because whichever team wins 4 more games wins the world series.0114

We certainly do not really care who wins the first game vs. the second game,0118

what we care about is the total number of games that each team wins.0123

For example, one possible way the World Series can go would be,0128

if you are keeping track of the Yankees, you can have win-win, lose-lose, win-lose-win.0133

That is the possible outcome of the World Series.0143

What we would keep track of their though is that there are 4 wins.0146

We would say Y of that outcome would be 4.0152

That is really all that matters, it does not matter the order in which the wins occur.0155

That is the intuition for random variable is you are keeping track of a certain number during an experiment.0160

At the end, your outcome gives you a certain value.0167

Let us do another example to try to understand that intuition a little more before I give you the formal definition.0171

Another way to think about random variables is to think about a payoff on an experiment.0179

Meaning depending on what happens, you get paid a certain amount of money.0185

This is often useful if you are thinking about gambling games in a casino.0190

Depending on how the cards come out, the dice come out, whatever, you get paid a certain amount.0194

For example, here is a very simple gambling game.0199

You are going to draw a card from a 52 card deck, if you drawn an ace then I will pay you $1.00.0203 If you draw a 2 then I will pay you$2.00, all the way up to I you draw a 9 then I will pay you $9.00.0209 If it is a 10, you have to pay me$10.00.0217

If it is a jack, queen, king, those are called face cards then you have to pay me $10.00 for any of those cards as well.0221 The quantity we can keep track of here is the amount of money that you stand to make on this experiment.0232 Notice here that this Y could be positive or negative.0238 For example, Y if you draw an ace then you stand to make a$1.00 off this experiment, that is a positive outcome for you.0242

Y if you draw a 2, would be $2.00, all the way up to Y of 9, if you draw a 9, you get$9.00.0252

But if you draw 10 then you have to pay me $10.00.0264 From your perspective, that is -$10.00 for you.0269

Y of 10 would be -10 and Y of a jack would be -10, and so on.0273

Any face card, you have to pay me $10.00.0282 We think of the payoff and from your perspective that is a negative payoff for you.0285 The way you want to think about this random variable is it is the amount of money that you make from this experiment0291 which could be positive or negative, depending on whether you go away richer or whether you have to pay me and you walk away poor.0299 Those are some rough, intuitive ideas to keep in mind as we get to the formal definition of random variable,0306 which is that it is a function from the sample space to R.0314 R is the set of real numbers here.0319 Remember, the sample space is the set of outcomes for an experiment.0322 The random variable number, if you want to think about it as a payoff then for each outcome there is some kind of payoff.0340 There is a real numbers worth a payoff.0346 It is a function that takes in an outcome and it gives you back a number which you can think of is the amount you get paid.0349 Or it is the quantity that you are keeping track of during the experiment.0359 To return to our first example there which was the World Series, how many games do the Yankees win?0364 I have listed some different possible outcomes of the World Series.0370 We are assuming that we are going to play all 7 games in the World Series,0374 even if somebody has already wrapped up the best of 7 early.0378 We are going to play all 7 games, no matter what.0383 If we are kind of looking at it from the perspective of the Yankees, if it is win-win-win, lose-lose-win-lose, there are 4 wins there.0385 The Y of that outcome is 4.0396 If we lose the first 6 games and we win the last game, the Y of that outcome is 1.0399 If we alternate winning and losing, win-lose, win-lose, win-lose-win, that is also 4.0406 In the sense of comparing that to the first outcome there, they are the same as far as Y is concerned0412 because the Y is going to be the same number either way.0419 Remember, there is 2⁷ possible ways that the World Series could go.0423 I’m not going to write them all down.0427 Each one of them has a number associated to it, from 0 to 7 depending on how many games the Yankees win there.0429 I want to move on to my next definition which is the probability distribution.0439 The notation gets a little confusing here because there is a lowercase letters and capital letters.0443 Here you want to think of y here as a number, that is a possible value of the random variable.0449 The Y here is the actual random variable.0458 This is what gets a little confusing because we will say Y=y.0468 What we are really asking there is when is the random variable going to take on that particular number or that particular value?0472 You think of the y as being the value and Y is the actual variable.0482 What is the probability that the random variable has a particular value?0488 This P is the probability that the random variable takes on a particular number or takes on a particular value.0494 We try to find that probability and we add up all those probabilities over all the outcomes that lead to that value.0505 We say that that is the probability of that value.0513 This problem will make more sense after we do some examples.0516 Stick around for examples, if it is a little confusing to you right now.0519 What we are doing here, what this notation means is now the sample space is the set of all the outcomes.0523 We look at all the outcomes in the sample space for which the random variable has that particular value0531 and then we add up the probabilities of all those outcomes.0538 That is what this formula means.0542 It will make more sense after we do some examples.0543 This function, we think of this as being a function P of y.0548 It is the probability distribution of the random variable Y.0552 We will talk about calculating P of different numbers.0556 I think that will make some sense after you see some of these examples.0560 Let us jump in and do some examples in all of these notations to make a little more sense.0566 In our first example, we are going back to the one that I mentioned earlier in the lecture0571 where you draw a card from a standard 52 card deck.0576 If it is an ace -9, I will pay you that amount.0580 If you draw an ace, I’m going to pay you$1.00.0583

If you draw a 2, I will pay $2.00.0585 All the way up to, if you draw a 9 I will pay$9.00.0587

If it is a 10 then all of a sudden the tables are turned and you have to pay me $10.00.0591 From your perspective, that is -$10.00.0597

If it is a jack, queen, king, those are called face cards, then you still have to pay me $10.00.0599 The question here is what is the probability distribution for this random variable?0605 What I'm really asking here, what this kind of question is asking is0612 what is the probability of getting different possible values for the random variable?0616 What is the probability that you will collect exactly$0.00 from this experiment?0624

The probability that this random variable will take the value 0.0631

In this case, there is no way that it is going to be exactly 0 because if you get a certain set of cards,0634

you are going to have to pay me.0641

If we get a different set of cards, I'm going to have to pay you.0643

There is no way that this experiment can wind up being worth $0.00 from you.0647 You are going to win something, you are going to lose something, you are not going to break even on this experiment.0652 Let us look at the probability of 1.0657 P of 1 is the probability that our random variable takes the value 1.0661 Let us think about all the ways that you can make exactly$1.00 on this, that means you have to draw an ace.0669

There are 4 aces in a 52 card deck and the probability that you are going to make exactly $1.00 on this is exactly 1/13.0676 The probability that you make$2.00, that is the probability that the random variable takes the value 2.0689

There are four 2s in the deck, you have a 4/52 chance which simplifies again down to 1/13.0698

It is similar to that all the way up through 3, 4, 5, 6, up to 9, the probability of 9, again it is the probability that Y is equal to 9.0708

There are four 9 in a 52 cards and that is 1/13.0720

It suddenly changes, there is no way you can get 10 because if you draw a 10 from the deck then you have to pay me.0730

That does not count as getting $10.00 for you.0739 That would be the probability that Y is equal to 10 and there is no way that you can get$10.00 out of this game, that is 0.0743

The probability, there is one more possible value that this game can take for you know which is -10.0752

That happens when you draw a 10 or you draw a face card, any of those, the result in you having to pay me $10.00.0759 How many different cards are there?0773 There are four 10, jacks, queens, kings, there are 16 cards here that will give you a net loss of$10.000775

That simplifies down to 4/13.0788

That is your probability of getting -$10.00 out of this game, if you are losing$10.00 as you play this game.0793

That is our full probability distribution, I will put a box around the whole thing here because really that whole thing is our answer.0801

The probability distribution means you are thinking of all the different possible values of y and0811

you are calculating the probability of Y, of each one of those values.0818

And that is what we did here.0822

To recap that, the possible values of y, I threw in 0.0824

It turned out that there is no way that you can make exactly $0.00 out of this game.0828 The probability of getting 0 is 0.0833 Then, I calculated the probability of 1 through 9.0837 For each one of those, there were 4 cards that can give you that particular payoff and0840 that is 4/52 giving us 1/13 for each one of those.0847 There are really 9 different possibilities there.0855 They all have probability 1/13.0858 And then, I went ahead and included the probability of 10 although there is no way that you can get$10.00 out of this game.0860

That is why we got 0 there.0868

I also had to include -10 because there is a possibility that you might lose $10.00 off this game.0870 There are 16 cards in the deck that will give you a loss of$10.00.0877

When I simplify 16/52, I get 4 out of 13.0881

That is the probability distribution for that random variable on that experiment.0886

In our second example here, we are going to flip a fair coin 10 ×.0894

Y is going to be the number of heads and I want to find the probability distribution for this random variable.0897

We are going to calculate P of y, where y is all the possible values that Y could be.0906

It is the probability that Y could be equal to that particular value y.0915

Let me note before we start, the possible values that y could be, that is the number of heads that we can see here.0923

And the fewest heads, we could possibly get be 0.0931

The most heads we can possibly get would be 10, if all 10 flips come out to be heads.0934

Let us think of on the probability of any particular value.0939

Let me think about that, to get exactly y heads we must,0948

Here is how you can think about that, we must fill in 10 blanks 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 with tails and heads.0959

We know how many of the each because we would have to fill in y heads, yh.0984

That would mean that the remaining 10 - y blanks, 10 – yt would all have to be tails.0993

Let us think about that, we are choosing y of those 10 blanks to be heads.1002

We must choose y blanks to be heads.1012

Once we made that choice, there is nothing more to decide because automatically then, all the remaining blanks would have to be tails.1022

Let us think about how many ways there are to do that.1033

There are 10 choose y ways to do that.1039

This is something we learn earlier on in the probability lecture series here on www.educator.com.1050

We had a chapter on making choices.1056

Probably, that chapter was on ordered vs. unordered, with replacement vs. without replacement.1059

This is really an unordered, without replacement selection.1066

The formula that you use for that kind of choice is 10 choose y.1075

You can look that up on the earlier lectures, if you do not remember how to do that.1081

But there is 10 choose y ways to do that.1086

Suppose you have a particular way of doing that, each particular way of filling in the blanks,1089

let us say for example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.1102

Let us say, for example y =3.1108

You choose 3 places that mean all the others have to be tails.1115

What is the probability of that exact arrangement of heads and tails coming up in 10 flips?1119

It means you got to get a tail the first flip then a tail, then a head, then tail-tail, head-head, tail-tail-tail.1127

Each one of those has ½ chance of occurring.1133

To get that entire sequence, that exact sequence, each one has a ½ ⁺10 or 1/2 ⁺10 chance.1136

Any particular exact sequence has a 1/2 ⁺10 chance of occurring.1151

The probability of getting all those sequences is 10 choose y × 1/2 ⁺10.1161

10 choose y, these are combinations, that is the notation for combinations that I'm using there.1173

You could also use the notation C of 10 choose y, if you prefer that notation.1180

We use that also in the earlier lecture here on www.educator.com.1188

10 choose y divided by 2 ⁺10, that is the total probability that we are going to get exactly y heads.1193

Let me remind you the range here.1203

This is for all possible values of y from 0 up to 10.1205

That is our probability distribution, that is our probability of getting any particular value of y, as y ranges from 0 to 10.1211

Let me go back over that and make sure that everything is still clear.1224

We are trying to find the probability of any particular value of y, which means1228

the probability that our random variable will take the value exactly y,1232

which means it will get exactly y heads when we flip the coin 10 ×.1238

When we think about that, it means that you are filling in 10 blanks with y heads and 10 - y tails.1244

If you think about the number of ways to do that, there are 10 choose y ways to fill in exactly y heads.1253

Once you fill in the heads then you have to fill in all the remaining places with tails.1263

That was an unordered choice, that was without replacement.1270

If you do not remember what those words mean, there is an earlier video here on the educator series on probability.1273

You can just go back and look those up and you will see that this is what we are talking about.1280

Each one of those ways, each one of those exact sequences has a 1/ 2 ⁺101283

probability of coming up when you flip a coin 10 ×.1290

The total probability is 10 choose y divided by 2 ⁺10 or 10 choose y × 1/ 2 ⁺10.1294

That is our probability distribution, that is our probability of getting exactly y heads for any number y between 0 and 10.1302

In our third example here, we are going to roll a dice repeatedly until we get a 6 showing.1314

We want to let y be the number of rolls that it takes for us to see our first 6.1320

We want to find the probability distribution for this random variable.1326

Let us go ahead and calculate.1330

First of all, the range that we can get then different values.1332

It could be that we get very lucky that we get a 6 right away.1337

Let me write this in that form because our y could be 1, if we get 6 right away.1342

If we miss the 6 in the first roll, we get it on the second roll, then it could be 2, it could be 3.1350

Actually, this could go on indefinitely because we really do not know how many rolls it might take.1357

If we are very unlucky, it could take as 150 rolls before we see the first 6.1363

Another way to say this is, 1 is less than or equal to y less than infinity.1369

Y could be potentially any positive integer here.1375

We are going to have to investigate the probability of each one of those values.1379

Let us think about what those probabilities are.1384

The probability of getting 1, let me write that in P notation.1387

The probability that the random variable is equal to 1.1392

That means you get a 6 on your very first roll.1396

There is a 1/6 chance that you get a 6 on your very first roll.1400

What about the probability that you get 2?1405

That is the probability that Y is equal to 2.1409

That means you get a 6 on the second roll.1413

Think about that, that means you must not have gotten the 6 on the first roll1416

because if you got a 6 on the first roll, you would have stop.1421

In order to get a 6 exactly on the second roll then you must get a 6 on the first roll,1424

there is a 5/6 chance that you are going to fail on that first roll.1432

And then, you must get a 6 on the second roll which means there is a 1/6 chance of getting that.1436

The probability of taking exactly 2 rolls here is 5/6 × 1/6.1444

Or about the probability of getting exactly 3 rolls.1452

The probability that our random variable takes the value 3.1456

To get 3 rolls that mean you missed getting a 6 on the first 2 rolls.1462

It is 5/6 to miss it on the first roll.1465

It is 5/6 again, to miss it on the second roll because if we get it on the first and second roll,1469

you are not going to get on to 3 rolls, × you have to get it in the 3rd roll so there is 1/6 there.1475

This continues here, the probability of taking exactly y rolls which is the probability that Y is equal to the number of y.1484

You are going to multiply together a bunch of 5/6 and how many you are going to multiply that, represents losing not getting a 6 on the first y -1 rolls1497

because you want to get a 6 exactly on the last roll, on the yth roll.1509

It is 5/6 ⁺y -1 × 1/6.1515

This represents failing on the first y -1 rolls and this represents succeeding on the yth roll.1523

That is what it takes, in order to get exactly y rolls.1545

That is our generic formula and let me go ahead and remind you that1551

the range for y was any number bigger than or equal to 1 and less than infinity.1556

It could be arbitrarily big, it could take thousands and thousands of rolls, if you are very unlucky to get our first 6 here.1565

Let me remind you of the steps there.1576

We want the probability that is going to take exactly y rolls to get a 6.1578

For example, to get exactly one roll, to get a 6, there is a 1/6 chance that we are going to get a 6 on the very first roll.1585

2 rolls, we have to get one of the other five numbers for the first roll and then get a 6 for the last roll, for the second roll.1593

Three rolls, twice in a row we have to get one of the other five numbers.1602

On the third roll, we have to get our 6.1607

In general, to get exactly y rolls, we would have to get one of the other five numbers,1610

in other words fail to get a 6 on the first y -1 rolls.1617

On the very last roll, the yth roll, we have to get a 6, there is a 1/6 chance.1622

We put those together as 5/6 ⁺y-1 × 1/6.1628

Our range there is from 1 up to potentially as large positive integer as you can imagine there.1634

In example 4, we are going to keep track of a soccer match here,1647

actually 3 soccer matches between Manchester United and Liverpool football club.1650

In any given match, it tells us Liverpool is the stronger team this season.1657

They are twice as likely to win as Manchester, I hope I’m not upsetting a huge range of football fans here.1662

There are no ties, I’m eliminating ties here.1668

We are going to play penalty kicks or something, until we have a winner of every match.1672

Let y be the number of matches that Liverpool wins.1677

We are going to look at things from Liverpool’s perspective.1680

We want to know what is the probability distribution for this random variable?1683

The first thing to notice here is that, since Liverpool is twice as likely to win as Manchester,1689

in any given match, the probability in each particular match, Liverpool wins with probability,1695

If Liverpool is twice as likely to win, they must have a 2/3 chance of winning1716

because that would give Manchester 1/3 chance of winning.1722

The 2/3 is twice as likely as 1/3.1725

Manchester wins with probability is 1/3.1730

That is for any particular match, let us try to figure out the probabilities that Liverpool will win a certain number of matches.1735

If they are playing three matches then Liverpool might lose all 3, it might win 1, it might win 2, it might win all 3.1742

We are going to have to calculate the value 0, 1, 2, and 3.1751

The probability of 0, P of 0, that is the probability that Liverpool wins exactly 0 matches.1755

In order to win 0 matches, they are going to have to lose all three matches.1765

Any given match, they lose a probability 1/3.1771

The odds of them losing 3 in a row is 1/3³ which is 1/27.1773

That is the probability that Liverpool walks away from three matches without a single win.1781

The probability of 1, that is the probability that their total winnings are one match.1787

Let us think about that, how can Liverpool win one match?1796

One way to do it would be if they win the first one and then lose twice.1799

They could lose-win-lose, they could lose-lose and win the final match.1804

There are three different possibilities there but each one of those has Liverpool winning one match and losing 2.1813

In each one of those three possibilities, there is a 1/3 chance that they will win their relevant match.1821

There is a 1/3² chance that they will lose the 2 relevant matches.1832

A 2/3 chance that they will win the match that they are supposed to win.1837

If we multiply those all together, the 3 and 2/3 give us just 2 × 1/3 × 1/3 that is 2/9.1844

2/9 is the total probability that Liverpool will win exactly one match there.1858

How about the probability that they will win exactly two matches?1863

What is the probability that y is equal 2?1869

How can we win two matches?1872

They could win the first 2 and then lose, they could win-lose-win, they could lose and then come back and win 2 in a row.1875

There are 3 ways that can happen.1886

Any particular one of those ways, the likelihood that I have to win 2 particular matches,1889

there is a 2/3 chance for each one of those.1895

We have to lose a particular match, there is 1/3 chance that they will lose whichever match they are supposed to lose.1898

If we multiply those together, the 3 and 1/3 cancel each other out.1905

We get 4/9 there, that is the probability that Liverpool will win exactly 2 matches out of their 3 match series there.1910

Finally, what is the probability that Liverpool is going to win all three of them?1921

Probability that y=3?1927

To do that, Liverpool would just have to win-win-win.1932

The only way to do that is to win three matches in a row.1937

Each one, they have a 2/3 chance, 2/3³ is 8/27.1940

8/27, we have our whole probability distribution here.1952

We have got the 4 values, y goes from 0 to 3, the four numbers of matches that Liverpool might win.1961

We have our probabilities for each one.1968

Let me remind you how we figure that out.1972

First of all, we know that Liverpool is twice as likely to win as Manchester,1975

which means that Liverpool is going to win with probability 2/3, Manchester with probability 1/3.1980

We said there are no ties here, we are not allowing ties, that would just make it too complicated.1986

We figure out the probability that Liverpool is going to win 0 matches that means they have to lose 3 × in a row.1992

There is a 1/27 chance of that.1998

How can Liverpool win exactly one match?2001

There is three different ways they can do that.2003

For each one of those ways, the probability is 1/3 × 1/3 × 2/3, and some of ordering of that.2005

You multiply those together, that is where the 2/9 comes from.2015

How can we win exactly 2 matches?2019

There is three ways they can win exactly 2 matches.2021

For any one of those configurations, the probability is 2/3 × 2/3 × 1/3.2024

That is where we get the 4/9.2031

Finally, the probability that they will win all three matches is they would have to win all three in a row.2033

There is a 2/3 × 2/3 × 2/3 chance that they will win all 3.2041

That gives us 8/27 for the last value in our probability distribution their.2046

In the last example here, you are going to play a game with a friend.2054

You are going to do a little gambling with your friend.2057

You and your friend are each going to flip a coin, you flip your coin and your friend flips his coin.2059

If both the flips come out heads, then your friend has to pay your $10.00.2065 If they are both tails then he pays you$5.00.2069

If the coins match each other then your friend is going to pay you.2073

That is a good thing for you.2077

If the coins do not match each other, meaning you get a tail and he gets a head, or you get ahead and he is a tail,2079

then he wins and you have to pay him $5.00.2085 We are going to look at this from your perspective.2088 Let y be the amount you win, we want to find the probability distribution for this random variable.2091 I think of all the possible values that y can take and we want to find the probability of each one.2099 I’m going to start with just 0 because I like to include it.2103 In this case, there is no way that the random variable can come out to be exactly 0 here.2107 Depending on what the coins show up, either your friend is going to pay you or you are going to pay your friend.2118 There is no outcome that leads to a 0 exchange of money, the probability there is 0.2124 The probability that you are going to make$5.00.2128

To make $5.00, we have to get both coins being tails.2136 What is the probability of tail-tail and the chance of both coins showing tails is ½ × ½ is ¼.2142 That is your chance of making$5.00 out of this experiment.2152

The probability of you making $10.00 out of this experiment, for that to happen both flips have to be head.2156 You got to see 2 heads there.2162 The probability of 2 heads is ½ × ½ = ¼.2166 If the coins do not match then you have to pay your friend$5.00.2171

From your perspective, we got to win -$5.00 but you are really losing$5.00.2176

The way that happens is, if you show a tail and he flips a head, or if you flip a head and your friend flips a tail.2186

There are 2 possibilities there but each one of those has probability ¼.2195

¼ + ¼ is ½.2201

Those are all the different outcomes that can happen and those are all the different values of the random variable.2209

Those are all the different payoffs that can happen in this experiment.2216

I listed 0 even though it is not really a legitimate outcome here.2219

I just want to calculate it through, there is a 0 probability that no money is going to change hands here.2223

The probability of getting $5.00, we are looking at this from your perspective.2230$5.00 win for you would happen if there are 2 tails.2235

To get 2 tails in a row or to get both your coins to come up tails, there is a ¼ chance of that.2240

To get 2 heads in a row which is what you need to make $10.00, there is also ¼ chance.2246 The probability of getting -$5.00 that means that your friend takes $5.00 from you,2252 that has to happen if you flip a head and he flips a tail, or vice versa.2259 Each one of those combinations has a 1/4 chance meaning that there is a ½ chance that you are going to end up paying him$5.00.2265

That is how we calculated each one of those probabilities for each one of those payoffs.2275

That is considered to be the probability distribution for this random variable.2280

That is the last example and that wraps up this lecture on random variables and probability distributions.2287

This is part of the larger probability series here on www.educator.com.2295

My name is <NAME>, thank you for watching, bye.2299

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text:
Statistica Sinica 4(1994), 361-372

ALMOST SURE CONVERGENCE OF STOCHASTIC

APPROXIMATION PROCEDURES

<NAME>

Ohmeda

Abstract: In this paper we investigate the convergence of the Robbins-Monro procedure Xn+1=Xnan(Y nα) . The following along with some related results are obtained.

Let ξj=YjM(Xj) be the error in the jth observation. A necessary and sufficient condition for the almost sure convergence of {Xn} is

a.s.

If {ξj} is an i.i.d. sequence, p≥1, j =0, and aj=j-1/p for j≥1, then the above is true if and only if E1|p <∞ .

Key words and phrases: Robbins-Monro procedure, almost sure convergence, martingale differences.

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# Calculus of Variations Boundary Terms in Higher Dimensions

In 2D I apply the calculus of variations to get an equation of the following form on a domain $$\Omega\subset\mathbb{R}^2$$, and I want to extract the Euler-Lagrange equations:

$$\int_{\Omega} f(p)\delta(p)\,dA + \int_{\partial \Omega} g(p)\delta(p)\,ds + \int_{\partial \Omega} h(p)\nabla \delta(p)\cdot \hat{n}\,ds = 0,$$ for $$\delta(p)$$ an arbitrary variation. Oftentimes it is given that $$\delta$$ or its derivative vanishes on the domain boundary, so that one of the boundary terms vanishes, but here $$\delta$$ is completely free.

Can I conclude that $$f(p) = g(p) = h(p)=0$$?

Clearly I can choose a bump function $$\delta(p)$$ at any interior point whose value and derivative vanishes on the boundary, so that $$f(p)=0$$. I'm less clear on whether I can independently also force $$g(p)=0$$ and $$h(p)=0$$. Intuitively, I ought to be able to take $$\delta(p)$$ to be a test function whose normal derivative is $$1$$ at a boundary point $$p$$, but whose normal derivative and value vanishes outside an arbitrarily small neighborhood?

• I'm only familiar with the volume integrals in calculus of variations, but won't you get a boundary condition if you apply partial integration on the last integral? – Botond May 16 at 0:35
• @Botond if you move the derivative on the last term and apply Stokes's theorem, you get a volume integral involving $\nabla \delta$, which is counterproductive I think. – user7530 May 16 at 1:27
• What is the Lagrangian of the system? – Qmechanic May 17 at 18:40


text:
Componendo and Dividendo Rule

Formula

$\dfrac{a}{b} = \dfrac{c}{d} \,\,\,$ $\Leftrightarrow \,\,\,$ $\dfrac{a+b}{a-b} = \dfrac{c+d}{c-d}$

Proof

$a$, $b$, $c$ and $d$ are four literals but $b$ and $d$ are non-zero literal numbers. Assume, the ratio of $a$ to $b$ is equal to the ratio of $c$ to $d$.

$\dfrac{a}{b} = \dfrac{c}{d}$

Step: 1

Add one both sides of the left and right hand side fractions.

$\implies \dfrac{a}{b} + 1 = \dfrac{c}{d} + 1$

$\implies \dfrac{a+b}{b} = \dfrac{c+d}{d}$

Step: 2

Subtract one both sides of the left and right hand side fractions.

$\implies \dfrac{a}{b} -1 = \dfrac{c}{d} -1$

$\implies \dfrac{a-b}{b} = \dfrac{c-d}{d}$

Step: 3

Divide the equation obtained in step 1 by the equation obtained in step 2.

$\implies \dfrac{\Bigg[\dfrac{a+b}{b}\Bigg]}{\Bigg[\dfrac{a-b}{b}\Bigg]}$ $=$ $\dfrac{\Bigg[\dfrac{c+d}{d}\Bigg]}{\Bigg[\dfrac{c-d}{d}\Bigg]}$

$\implies \Bigg[\dfrac{a+b}{b}\Bigg] \times \Bigg[\dfrac{b}{a-b}\Bigg]$ $=$ $\Bigg[\dfrac{c+d}{d}\Bigg] \times \Bigg[\dfrac{d}{c-d}\Bigg]$

$\implies \Bigg[\dfrac{a+b}{a-b}\Bigg] \times \Bigg[\dfrac{b}{b}\Bigg]$ $=$ $\Bigg[\dfrac{c+d}{c-d}\Bigg] \times \Bigg[\dfrac{d}{d}\Bigg]$

$\implies \require{cancel} \Bigg[\dfrac{a+b}{a-b}\Bigg] \times \Bigg[\dfrac{\cancel{b}}{\cancel{b}}\Bigg]$ $=$ $\Bigg[\dfrac{c+d}{c-d}\Bigg] \times \Bigg[\dfrac{\cancel{d}}{\cancel{d}}\Bigg]$

$\implies \Bigg[\dfrac{a+b}{a-b}\Bigg] \times 1$ $=$ $\Bigg[\dfrac{c+d}{c-d}\Bigg] \times 1$

$\therefore \,\,\,\,\,\, \dfrac{a+b}{a-b} = \dfrac{c+d}{c-d}$

Therefore, it is proved that if ratio of $a$ to $b$ is equal to the ratio of $c$ to $d$, then the ratio of $a+b$ to $a-b$ is equal to the ratio of $c+d$ to $c-d$. This property is called the componendo and dividendo rule.

Verification

Let us test this theorem by verifying it in number system. Take two equal fractions to understand this property.

$\dfrac{1}{2} = 0.5$ and $\dfrac{2}{4} = 0.5$

$\therefore \,\,\,\,\,\, \dfrac{1}{2} = \dfrac{2}{4}$

Now, verify the Componendo and Dividendo law.

$\dfrac{1+2}{1-2} = \dfrac{3}{-1} = -3$

$\require{cancel} \dfrac{2+4}{2-4} = \dfrac{\cancel{6}}{\cancel{-2}} = \dfrac{3}{-1} = -3$

$\therefore \,\,\,\,\,\, \dfrac{1+2}{1-2} = \dfrac{2+4}{2-4}$

This example has verified the Componendo and Dividendo identity. Hence, if the ratio of any two numbers is equal to ratio of another two numbers, then the ratios of sum of numerator and denominator to difference of numerator and denominator of both rational numbers are equal.

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# American Institute of Mathematical Sciences

2014, 13(5): 1685-1718. doi: 10.3934/cpaa.2014.13.1685

## Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation

 1 Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875

Received  December 2013 Revised  January 2014 Published  June 2014

One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to nonlinear Klein-Gordon equation, we derived relativistic Newton's law and Einstein's formula for inertial mass. Here we apply the same approach to nonlinear Schrodinger equation and derive non-relativistic Newton's law for the trajectory of concentration.
Citation: <NAME>, <NAME>. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685
##### References:
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[26] <NAME>, <NAME> and <NAME>, Nonlinear eigenvalue problems of Schrnodinger type admitting eigenfunctions with given spectral characteristics,, \emph{Math. Nachr.}, 242 (2002), 91. doi: 10.1002/1522-2616(200207)242:1<91::AID-MANA91>3.0.CO;2-Z. [27] <NAME>, Classical Electrodynamics,, 3rd Edition, (1999). [28] <NAME>, Nonlinear Schrödinger equations,, in \emph{Schr\, (1989). doi: 10.1007/3-540-51783-9_22. [29] <NAME>, Electromagnetic Field Theory without Divergence Problems 1. The Born Legacy,, \emph{J. Stat. Physics}, 116 (2004), 1057. doi: 10.1023/B:JOSS.0000037250.72634.2a. [30] <NAME>, Quantum Mechanics: Genesis and Achievements,, Springer, (2013). doi: 10.1007/978-94-007-5542-0. [31] <NAME>, <NAME> and <NAME>, Effective Dynamics for a mechanical particle coupled to a wave field,, \emph{Comm. Math. Phys.}, 203 (1999), 1. doi: 10.1007/s002200050023. [32] <NAME>, The Variational Principles of Mechanics,, 4th ed., (1986). 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##### References:
 [1] <NAME> and <NAME>, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134 (2004), 893. doi: 10.1017/S030821050000353X. [2] <NAME> and <NAME>, Soliton dynamics in a potential,, \emph{Math. Res. Lett.}, 7 (2000), 329. doi: 10.4310/MRL.2000.v7.n3.a7. [3] <NAME> and <NAME>, Mass and spin renormalization in lorentz electrodynamics,, \emph{Ann. Phys.}, 289 (2001), 24. doi: 10.1006/aphy.2000.6119. [4] <NAME> and <NAME>, Wavepacket preservation under nonlinear evolution,, \emph{Commun. Math. Phys.}, 278 (2008), 329. doi: 10.1007/s00220-007-0406-0. [5] <NAME> and <NAME>, Nonlinear dynamics of a system of particle-like wavepackets,, in \emph{Instability in Models Connected with Fluid Flows (<NAME> and <NAME>}), (2008). doi: 10.1007/978-0-387-75217-4_3. [6] <NAME> and <NAME>, Wave-corpuscle mechanics for electric charges,, \emph{J. Stat. Phys.}, 138 (2010), 912. doi: 10.1007/s10955-009-9877-z. [7] <NAME> and <NAME>, Some mathematical problems in a neoclassical theory of electric charges,, \emph{Discrete and Continuous Dynamical Systems A}, 27 (2010), 1283. doi: 10.3934/dcds.2010.27.1283. [8] <NAME> and <NAME>, Electrodynamics of balanced charges,, \emph{Found. Phys.}, 41 (2011), 242. doi: 10.1007/s10701-010-9502-7. [9] <NAME> and <NAME>, Relativistic dynamics of accelerating particles derived from field equations,, \emph{Found. Phys.}, 42 (2012), 996. doi: 10.1007/s10701-012-9642-z. [10] <NAME> and <NAME>, Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation,, \emph{Comm. Math. Phys.}, 322 (2013), 453. doi: 10.1007/s00220-013-1732-z. [11] <NAME> and <NAME>, Some rigorous results on the Pauli-Fierz model of classical electrodynamics,, \emph{Ann. Inst. H. Poincar\'e, 58 (1993), 155. [12] <NAME>, Electrodynamics and Classical Theory of Fields and Particles,, Dover, (1980). 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