  William Murray

Beta 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 1 answer Last reply by: Dr. William MurrayMon Mar 9, 2015 9:32 PMPost by Nick Nick on March 6, 2015Why are we assuming theta2 = 1 and theta1 = 0 ?

### Beta 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
• Beta Function 0:29
• Fixed parameters
• Defining the Beta Function
• Relationship between the Gamma & Beta Functions
• Beta Distribution 3:31
• Density Function for the Beta Distribution
• Key Properties of the Beta Distribution 6:56
• Mean
• Variance
• Standard Deviation
• 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
• Example V: Morning Commute, Part A
• Example V: Morning Commute, Part B
• Example V: Summary

### Transcription: Beta Distribution

Hello, you are watching the probability lectures here on www.educator.com, my name is Will Murray.0000

Currently, we are working our way through the continuous distributions.0005

We have already had videos on the uniform distribution and the Gamma distribution0009

which included the exponential distribution, and the Chi square distribution.0014

Today, we are going to talk about the last of the continuous distributions which is the β distribution.0018

It is similar to the Gamma distribution, you will see some of the same elements but of course, it is also different.0024

Let us check it out.0030

Before we jump into the actual β distribution, we have to learn what the β function is.0032

The first thing that you have to keep straight here is that, there is a β function and then there is β distribution.0037

They are not exactly the same thing.0044

We are going to use the β function in the process of defining the β distribution.0046

In that sense, it is like the Gamma distribution where we had a Gamma function and then we also had a Gamma distribution.0051

The Gamma function was just one part of the Gamma distribution.0057

Let us see what the β function is.0061

We had two fixed parameters, there is always an Α and there is a β.0063

The β distribution and the β function, it is the whole family of things.0069

Because you could pick any different value for α you want and any different value for β you want.0075

Let us define the β function.0080

What we do is, you want to think of Α and β as being numbers.0083

You take these numbers α and β, and you plug them into this integral.0088

We have the integral of Y ⁺α – 1 × 1 – Y ⁺β – 1 DY, that is just some integral.0092

Remember, α and β are constants.0101

We plug in constant values and then, you have an integral that0105

you can solve using calculus 2 methods and you will get some constant number.0109

The idea is that you plug in Α and β, the β function will spit out a particular number, a constant value.0114

There is a relationship between the Γ and the β functions,0125

which often makes it quite easy to evaluate the β function, which is that B of Α × β is equal to Γ of Α × Γ β ÷ Γ of Α + β.0129

That is very convenient if you have whole numbers because this relationship between Γ of N and the factorial function.0145

Γ of a whole number is exactly equal to N -1!.0156

If you know the numbers for Α and β here, you can just drop them into these 3 Gamma functions,0162

and evaluate those using factorials, and it turns out to be something fairly easy to evaluate.0169

You can find B of α and β fairly quickly as a number, just by calculating out some factorials.0176

If they are not whole numbers then it is a much more difficult proposition.0183

We are not going to get into that.0186

Remember, this is just the β function, I have not talked yet about the β distribution.0188

β function is just something we plug in two numbers, Α and β, and it spits out a number as an answer.0193

Next to a step is to see how that is incorporated into the β density function,0201

which is the density function for the β distribution.0208

Now, we are going to talk about the β distribution.0213

Remember, we already talked about the β function, that is different from the β distribution but0215

it is part of the β distribution.0220

The idea here is that, we want to define the double polynomial distribution.0222

It is always on the interval from 0 to 1, we always have Y going from 0 to 1.0229

We want to define this double polynomial distribution using essentially the function Y ⁺α -1 and 1 - Y ⁺β -1.0232

The point there is that, it is symmetric between 0 and 1.0247

Whatever the function does at 0, the 1 - Y behaves similarly at 1, depending on what the values of α and β are.0253

That is the basic function that we want to look at.0263

The problem is that, if we integrate that, we might not necessarily get it exactly equal to 1.0265

The probability density function must always satisfy the property that, when you integrate it,0272

we take the integral over the whole domain with the answer has to come out to be 1.0279

In order to fix that, what we do is we take this Y ⁺α -1 and 1 – Y ⁺β-1.0285

We just divide by the value of the integral, in order to make the integral come out to be 1.0292

That is how we create the density function for the β distribution.0299

We start out with Y ⁺α -1 × 1 – Y ⁺β -1.0303

And then, we divide it by the integral of that function, in order to not make the total integral come out to be 1.0309

You really want to think of this B of Α β in the denominator here,0316

it is just a correction term that we put in there to make the total integral come out to be 1.0321

It is just a constant and it is just sort of a fudge factor really,0326

that is probably the best way to think of it as a fudge factor to make the integral come out to be 1,0336

the integral of F of Y DY equal to 1.0344

It is not really the most important part of the function here.0349

The most important part are these two polynomial terms, the Y ⁺α -1 and 1 – Y ⁺β -1.0354

You want to think of those as the really important part of the function.0362

Those are the ones that give its shape and we will explore some of the graphs later.0365

This denominator is just a constant, it is a fudge factor the gets thrown in there,0370

in order to make the total integral come out to be 1.0374

The denominator is exactly the β function that we learn on the previous slide.0379

It is read up exactly to be the integral of Y ⁺α -1, 1 –Y ⁺β -1.0385

We divide by that constant, the whole integral now comes out to be 1.0393

That is the density function for the β distribution.0397

Remember, it is a kind of a polynomial thing and also remember that,0401

the density function and the β function are two different things, let us try to keep those straight.0405

We have the β distribution, we should figure out the key properties, I have listed them here.0414

Remember, the mean is always the same as expected value.0419

Expected value and mean are synonymous, they mean the same thing.0423

The mean for the β distribution is Α/Α + β.0427

That is a fairly easy formula to keep track of.0433

The variance is much more complicated and much harder to remember.0437

The variance turns out to be Α × β ÷ α + β² × Α + β + 1.0440

Much messier formula for the variance, we will see an example of that in the problems later on.0450

You will see how that gets used.0456

The standard deviation is usually not worth remembering, because you can always figure it out,0458

if you remember the variance.0463

The standard deviation is always the square root of the variance.0465

That is true for any distribution not just the β distribution.0469

The standard deviation, what we do is we just take that complicated formula for the variance0472

and we slap a square root around everything.0477

It is really not that enlightening by itself, it is more useful to remember the mean and variance of the β distribution.0481

In example 1, we are going to calculate B of 3, 4.0492

Just little practice with the β function first, before we jump into actually solving any probability problems.0498

The solution here is to, there are two ways you can do it.0505

One way would be to solve this as an integral.0510

Here is α, and here is β.0516

You could solve this out as an integral, the integral from 0 to 1 of Y ⁺α – 1 1 – Y ⁺β-1 DY.0518

Because that is the definition of B of Α β, that is the definition there.0533

You can solve out this integral, it would not be too bad because you plug in Α = 3 and β = 4.0538

You just do a little calculus and it would come out to be some number.0547

I'm not going to do it that way because I do not want to do that much calculus.0552

Let me show you another method to do it, instead.0555

The other method is, to remember this relationship between the β function and the Gamma function.0559

We learn that several slides ago, you can go back and check that if you did not pick up on that and register it the first time.0567

But, the relationship between the β function and the Gamma function is that, B of α and β is Γ of Α × Γ of β ÷ Γ of Α + β.0574

That is where it translates everything here into Gamma function.0591

I will go ahead and fill in α = 3 and β = 4.0595

Our Gamma of α + β will be Γ of 3 + 4 is 7.0602

I have to solve some Gamma functions but remember that Gamma function is,0609

it can be thought of as a generalization of the factorial function.0615

Γ of N is just N -1!.0619

This is, Γ of 3 would be 2!, Γ of 4 would be 3!, and Γ of 7 would be 6!.0624

Now, it is very easy just to calculate those factorials.0635

2! Is just 2, 3! Is 6, 6! is 9 × 2 × 3 × 4 × 5 × 6.0638

I'm writing it like that because now, I can cancel some things, I will not have to multiply by big numbers.0654

I will cancel the 6 there and I will cancel that 2 with that 2.0659

And now, I have got 1/3 × 4 × 5 which is 3 × 4 is 12 × 5 is 60, 1/60.0664

To remind you how that worked.0681

There are two ways I could have solve this.0683

I could use the original definition of the β function which is an integral formula.0685

And then, I would just plug in the values of the Α and β and done some calculus to work out the integral formula.0691

I see that with the benefit of having gotten the other way, that my integral better has worked out to be 1/60.0698

The way I actually calculated it, was to use this relationship between0708

the β function and Gamma function which I gave you a couple slides ago.0713

Translates a β function into three computations using Gamma functions.0719

I dropped in the values for Α and β, and converted those into factorials.0725

Remember, the Gamma function is just like the factorial function except offset by 1,0730

when you have a whole number in there.0735

I plugged in those factorials, calculated it out, and simplified it down to a relatively nice fraction there.0738

In example 2, we are asked to graph several density functions for the β distribution,0747

using several combinations of Α and β.0753

Let me first remind you what the density function is for the β distribution.0756

The density function for the β distribution is F of Y is equal to Y ⁺α – 1 1- Y ⁺β -1.0763

Those are really the important factors in the density function for the β distribution.0777

There is one other factor but it is just a constant, and it is the sort of correction term.0782

This fudge factor that gets introduced, in order to make the total integral be 1.0787

I'm just going to write that as a constant here.0793

I’m not going to even bother to work that out for the different combinations here because0795

that is not so important to determining the shape.0800

What is really important to determining the shape is the values of Α and β,0803

and those factors of Y ⁺α -1 and 1 - Y ⁺β -1 in the numerator.0808

Let me set up some axis here and we will take a look at these different combinations of Α and β.0817

Here are my axes.0826

Remember, the β distribution is always defined from Y equal 0 to Y = 1.0828

Up at Y here, it is a little confusing on the X axis.0835

But here is Y = 0 and here is Y = 1.0839

The important thing to look at here, are the different values of Α and β.0845

In particular, whether they are greater than 1 or less than 1.0850

Here is why, let us look at the key part of the density function here is Y ⁺α – 1.0856

Let us think about what that does for different values of Α.0868

At Y = 0, let us think about what that does.0876

If Α is bigger than 1, that means our exponent Y ⁺α -1 is positive.0882

Y ⁺α-1 will go to 0.0890

If Α is equal to 1 then Y ⁺Α -1 would just be Y at 0 go to 1.0900

If Α is less than 1 then Y ⁺α -1 will be Y to a negative number.0910

0 to negative power, it is like trying to divide by 0, that will go to positive infinity.0917

That helps me characterize the behavior at Y = 0 depending on the different values of Α.0927

The same kind of thing happens, if we look at 1 - Y ⁺β -1.0934

This similar kinds of phenomenon will occur at Y = 1, because if Y goes to 1 then 1 - Y goes to 0.0944

If β is bigger than 1 then 1 - Y ⁺β -1 goes to 0.0956

If β is equal to 1 then 1 - Y ⁺β -1 goes to 1.0964

If β is less than 1 then 1 - Y ⁺β -1 goes to infinity, because we are trying to take 0 to a negative exponent there.0972

Now, we will look at the different combinations of α and β.0985

In the first combination here, I see I got an Α of ½ and a β of 2.0988

Α of ½ is less than 1 that means it is going to go to infinity at Y = 0.0994

A β of 2 is bigger than 1, that means that Y = 1, it is going to go to 0.1002

I got something that goes to infinity as Y goes to 0.1012

And then, it comes down, let me make it a little steeper and in it is declined1021

because we are only allowed at the total area 1 here.1027

We are allowed to have total area 1.1033

It has to come down and hit 0 at Y = 1.1037

What we got right there is the graph of Α = ½ and β is equal to 2.1043

Let me do the next one in a different color.1052

I will do 1 and 4 in green, that Α is 1 and β is 4.1055

I see when α is equal to 1 and Y is equal to 0, it goes to 1.1064

Now, that is a little bit misleading.1070

I cannot say it is exactly equal to 1 because I'm kind of ignoring the effect of this constant here.1072

It is going to be 1 ÷ some constant.1079

Let me just show it at some finite value.1084

The important thing there is, it is not going 0 and it is not going to infinity.1088

I see that, as it goes to Y = 1, the β value is still bigger than 1.1093

It still going to go down to 0 there.1100

Let me make that a little more rounded, more curved, it is not a completely straight line.1116

That is my combination for Α is equal to 1 and β is equal to 4.1128

And that means that, at 0 it is going to go to a finite limit and at 1 it is going to go down to 0.1140

I will do the next one in red, α is 10 and β is 2.1151

What is that mean, Α is 10, that is way bigger than 1 which means at Y = 0, it is definitely going to go to 0.1159

It is so much bigger than 1 that, that term is going to sort of drag it down for a long time.1168

Here it is at 0, at Y = 0 and it is going to drag along near 0 for quite a long time.1174

Since β is 2, that means when it gets to the right hand and point here, it is still going down to 0.1184

It does have to have area equal to 1, the area underneath the curve has to be equal 1 because all of these functions do.1193

At some point, it is going to get some area.1200

Let me show it getting some area right at, as it approaches 1 there.1203

It is skewed to the right hand side, that Α = 10 drags it down on the left.1213

The β = 2 does drag it down at Y = 1 but we have to have an area of 1 in there, somewhere.1220

That is my 10, 2, α = 10 and β = 2.1228

I’m going to try out my new purple marker for that.1237

Here is α and that looks a lot like a blue to me.1243

Α is 1.1 and β is 2.1247

What that means, α is 1.1 and it is going to behave very similarly to the Α = 1,1251

which means it is going to trying to go into a finite limit.1262

Β = 2 means it is also going to be tied down at 0.1267

Let me show this graph that sort of trying to go a finite limit, as it approaches Y is equal to 0.1272

There it is, as it approaches Y = 0, it is trying to go to a positive limit.1281

What happens is, when it gets right down to Y = 0, that 1.1 is still bigger than 1 which means it is forced to go to 0.1287

Here is this graph that sort of trying to go to a finite limit and then at the last moment,1298

it has to turn sharply downwards and go to 0.1303

This purple graph, if you can tell the difference between the purple and the blue,1310

this purple graph is Α = 1.1 and β is equal to 2.1313

Let me recap all the different of things we are exploring here.1323

I have to highlight them in yellow, as I go along.1327

The important thing is to remember the general form of the β density function.1329

Here it is right here, Y ⁺α-1 1 - Y ⁺β -1.1335

And then, that constant in the denominator is really not important.1341

I’m not going to worry about it, it does not affect the shape of the graph.1345

There are two sub components there, the Y ⁺Α -1 determines what happens at Y = 0.1351

You want to look at the value of Α to determine what happens at Y= 0.1358

If Α is bigger than 1, then you got a positive exponent on Y, it is going to go to 0.1363

If α is equal to 1, that terms drops out and it goes to a finite limit.1369

If α is less than 1, then you got 0 to a negative number which means you are trying to divide by 0,1374

which means you are going to go to infinity.1381

If you look at the first graph here, there is an Α less than 1 and that is why it is going to infinity.1383

The second graph, α is equal to 1, goes to a finite limit.1391

The third graph, Α is much bigger than 1 which is why it starts off at 0 and stays at 0 for a long time.1396

Finally, the 4th graph, Α is bigger than 1 but very close to 1, that is why it is sort of1408

trying to go to a finite limit and then at the last minute, it has to drop down to 0.1414

The second component of this function is the 1 - Y ⁺β -1.1420

That determines what the graph does at Y = 1, in very much asymmetric fashion to the Y = 0,1426

except that it is looking at the value of β, instead.1434

If β is bigger than 1, you got something going to 0.1437

If β is equal to 1, you got something going to 1, and or at least a constant.1441

If β is less than 1, then you are trying to divide by 0, you are going to go to infinity again.1447

In this case all, of the β that we looked at were all bigger than 1, which means all of these graphs sort of tied down to 0 at Y = 1.1455

That is why all these graphs tie down here, but they all exhibit different behavior on their way to getting to that point,1466

which is why we get this sort of interesting of differences in all these graphs up to that point.1475

It is worth trying some of these out in your calculator, if you want to try graphing some of these in your calculator,1481

just throw these values of α and β into the density function, and put the graphs on your calculator,1486

and see what kinds of shapes you get.1492

You get quite a lot of riots, it is kind of fun.1494

In example 3 here, we are going to make a connection to a previous distribution that1498

we learn which was the uniform distribution.1504

It turns out that it is a special case of the β distribution.1507

I did a whole video on the uniform distribution.1511

If you do not remember the definition of the uniform distribution, or if you do not know what is it all,1515

you can go back and look at the previous video covering the uniform distribution.1521

You will see lots of information about the uniform distribution.1525

You just need a quick refresher on the uniform distribution.1528

I will remind you that the uniform distribution, the density function is just F of Y1531

is always equal to 1/θ2 - θ1, where θ1 and θ2 are constants here.1539

Those tripe lines there, the triple = means it is always equal to something.1548

It is not varying at all, it is equal to this constant the whole way.1554

That is where Y varies between θ1 and θ2.1560

What we are going to do now is show how the β distribution, if you choose the right parameters1569

turns into the uniform distribution.1575

That was the uniform distribution that I showed you up above.1578

Now, let me show you the formula for the β distribution which is much more complicated.1581

F of Y is equal to Y ⁺Α -1 × 1 - Y ⁺β -1 ÷ B of Α β and that goes from Y between 0 and 1.1589

I want to show you how you can choose the right parameters and turn the β distribution into the uniform distribution.1615

I'm going to choose my Α is equal to 1 and my β is also equal to 1.1621

Let us see how that worked out.1630

First of all, let us find the constant value B of Α β.1631

I'm going to use the relationship with the Gamma function, in order to figure that out.1637

That is Γ of 1 × Γ of 1 ÷ Γ of 1 + 1.1643

This is the relationship between the β function and the Gamma function,1653

that I mentioned back in one of the earlier slides in the lecture.1656

If you do not remember this, just go back and check your earlier slide and you will see the β and Γ relationship.1659

Also, remember here that Γ of N is N -1!, when N is a whole number.1667

This is 0! × 0! ÷ Γ of 2 would be 1!.1676

Of course, all those factorials, 0 and 1 are just 1, this is all just 1.1684

The β and density function is F of Y, that denominator is now going to be 1, we just work that out.1690

Y ⁺α -1, α is 1 so α-1 is 0 × 1 - Y ⁺β -1 is also 0.1698

And, that just comes out to be 1.1710

It looks like it is going to be constant.1714

Notice, by the way that, if it is constant then I can put three lines there,1717

that is equal to 1/1 -0 that is 1/θ2 – θ1, where θ1 is 0 and θ2 is 1.1723

What I found is that, I got the same density function for the β distribution as I would have gotten for the uniform distribution.1738

This is the same as the uniform distribution.1757

I discovered the uniform distribution as a special distinguished member of the β family.1769

If you choose your α and β right, the β distribution just turns into the uniform distribution.1775

Let me recap that, first of all, I recalled the uniform distribution and we did have a whole lecture on the uniform distribution.1783

You can check the video on that, if you do not really remember how that worked out.1791

The idea is you take Y between two values, θ1 and θ2.1794

F of Y is just 1 ÷ θ2 – θ1.1800

It is a constant distribution, there is no Y term appearing in there.1804

And then, I reminded myself of the density function for the β distribution.1809

That is our density function for the β distribution.1814

I picked good values in the parameters, I pick α to be 1 and β to be 1.1817

Then, I just plug those in.1823

First, I had to calculate B of Α β and I turn that into an expression using Gamma functions.1825

In turn, the Gamma functions turn into factorial.1832

It just reduced down to 1 which why my denominator here was 1, that is where that one came from.1835

And then, I plugged in my values of Α and β into the exponent, I got 0 is for the exponents,1843

which why everything just disintegrated into big old 1 here.1848

We just got that constant distribution F of Y is equal to 1, all the way across,1854

which is the same as the uniform distribution on the interval 0-1.1859

If you take θ1 = 0 and θ2 = 1, then we get the uniform distribution of 1 on that interval.1866

The uniform distribution, it turns out, is a special case of the β distribution.1875

In example 4, we are going to look at the triangular distribution F of Y = 2Y from 0 to 1,1882

and show that that is also a special case of the β distribution.1889

Let me just draw a quick graph of that triangular distribution.1893

It is obvious why we are calling it the triangle distribution.1896

Let me make my axis in black, I think that will show things a little better.1901

We are going from Y = 0 to Y = 1 here.1909

You always do that with the β distribution, it always goes from 0 to 1.1912

F of Y = 2Y, that is just a straight line is not it.1917

Probably, a little bit steeper than that, let me make that a little bit steeper.1923

F of Y = 2Y, and then, the challenge here is to recognize that as a special case of the β distribution.1932

Let me remind you of the density function for the β distribution.1944

And then, we will take a look at it and see if we can make it match what we have here.1948

For β distribution, F of Y is equal to Y ⁺Α -1 × 1 – Y ⁺β -1 ÷ B of Α, β.1954

I want that to match 2Y, and I think what I want to do there is I want to pick Α equal to 2.1967

That will make the exponent on the Y match.1978

That 1 - Y does not really seem to match.1981

In order to make that dropout, I'm going to take my β equal to 1.1983

Let us plug those in and see how it works out.1989

B of Α β, I could use the original integral definition to calculate that but I'm fond of the relationship1992

between the β function and the Gamma function.2000

I’m going to use that.2003

It is Γ of Α × Γ of β ÷ Γ of α + β.2004

Those are the relationship that we had back on one of the earlier slides in this lecture.2013

You can look that up, if you do not remember it.2017

Γ of Α is Γ of 2 × Γ of 1, β is 1 here.2020

Γ of 2 + 1 is Γ of 3.2028

Now, let us remember that the Gamma function is just a sort of2031

a generalized version of the factorial function, except it shifted over by 1.2036

This is 1! × 0! ÷ 2!.2041

I’m shifting everything back by 1 that is because Γ of N is equal to N -1!, for whole numbers there.2047

This is easy to solve, 1! Is 1, 0! Is 1, and 2! Is 2, that is ½.2057

F of Y, now I know what my constant is, it is ½, is Y ⁺Α – 1.2066

That is Y¹, 1 - Y ⁺β – 1, that is 1 - Y = 0 ÷ ½.2074

That simplifies, the 1 - Y = 0 is just 1, it goes away.2083

What we get here is Y by itself, but then, ÷ ½ is the same as multiplying by 2.2089

It is 2Y, and of course Y is trapped between 0 and 1 here.2101

That is very encouraging because that is the distribution, that is the density function that we are looking at.2107

We started out with F of Y is 2Y, we found that to occur, if we pick the right parameters in the β distribution.2119

Let me recap here, we started out wanting F of Y is equal 2Y.2130

I wrote down the density function for the β distribution, Y ⁺α -1 1- Y ⁺β – 1/B of α, β.2136

I’m trying to make it match to Y, I kind of looked at this exponent α -1.2145

And I said, how can I make that be equal to 1, that will work if α is equal to 2.2150

That 1 - Y really is not represented over here.2156

The 1 - Y has a power of 0 here, I will take β equal 1 to make that work out.2160

And then, I had to calculate the constant B of Α, β.2167

I did that, by converting that into Gamma functions, according to the formula2171

that we had on the earlier slides for this lecture.2175

It is Γ of α × Γ of β ÷ Γ of α + β.2178

If you drop the values of α and β data in there, then you will get something that we can simplify into factorials.2183

Remember, there is a relationship between the Gamma function and the factorial function.2190

Once, you simplified it into factorials, it simplifies quite nicely down into a fraction.2197

The F of Y, if I fill in my α is 2 then I get Y¹.2204

Β is 1 so I get 1 – Y = 0.2211

That Y ÷ /2 which is 2Y.2217

2Y from Y going from 0 to 1 is exactly the density function that we started with.2221

It does match what we are given, we do achieve this triangular distribution as a special case of the β distribution.2229

Let me show you quickly why it is called the triangular distribution.2241

If you look at all the area there, the area is a triangle.2244

The density is sort of spread out over a triangle there, which is why we call it the triangular distribution.2251

In example 5, we got morning commute, maybe this is you driving to work in the morning or driving to school in the morning.2261

It is a random variable, apparently, that has a β distribution with α and β both being 2.2269

This is measuring your commune in hour.2277

Remember, Y is always between 0 and 1 in the β distribution.2280

I guess that means that your commute could be 0 or it could be up to an hour.2286

In part A, we are going to find the chance that it will take longer than 30 minutes.2293

What we will actually be looking for there is the probability that Y is bigger than or equal2298

to 30 minutes is ½ an hour, it is bigger than or equal to ½.2304

In part B, apparently, you have a rage level which is a function of how long your commute is.2310

You want to find the expected value of rage that you will arrive at work tomorrow.2316

Let us work this out.2324

The first thing I'm going to do is try to identify the density function for this distribution.2326

We got F of Y, the generic density function for the β distribution is Y ⁺α – 1 1- Y ⁺β – 1 ÷ B of Α, β.2333

Let me go ahead and find the value of that constant, B of Α, β.2348

I think the easiest way to calculate these, if you have whole numbers, is to convert it into Γ.2355

Let me convert that into Γ of Α × Γ of β ÷ Γ of α + β.2364

That is a formula that we learn back on one of the first slides in this lecture.2374

In this case, we got Γ of 2 × Γ of 2 ÷ Γ of 4.2378

Because α and β are both 2, that was given in the problem.2385

This is 1! × 1!.2389

Γ of 4 would be 3!.2395

Remember, a part of the Gamma function is, it sort of the generalization of the factorial function but it is offset by 1.2398

Γ of 4 is 3!, 1! Is 1, 3! Is 6, this is 1/6 here.2406

This F of Y is Y ⁺α – 1, that is Y¹, 1 - Y ⁺β -1 is also 1.2415

And now, we know that we are dividing it by 1/6.2425

This simplifies down into 6Y × 1 - Y is Y - Y².2429

I can go ahead and distribute that, 6 Y - 6 Y².2439

That is my density function for this distribution.2445

We are going to go ahead and use that to solve these two problems.2450

I'm going to jump over onto next slide and use that density function to solve these two problems.2453

But, let me recap where these came from.2461

First, I was writing out the generic density function for the β distribution.2463

There is the formula there.2468

I have to figure out what B of Α, β was, I did that down below.2470

I converted that into a bunch of Γ because I know easily how to solve the Gamma function,2473

when you have a whole number in there.2481

It is just the corresponding factorial, except you have to shift it down by 1 to plug the number into factorial.2483

That is why B of Α, β turns out to be 1/6.2494

I plugged that into my density function.2498

I also plugged in the values of Α and β.2510

2 -1 is 1, and I got 6 × Y - Y², that is the density function that I'm going to work with,2513

in order to solve these two problems.2521

On the next page, I’m not going to copy the problems because I need the space.2526

We are going to solve the probability that Y is greater than ½.2529

And then, we are going to find the expected value of R.2533

Those are the two things that we are going to solve on the next page there.2537

We are still working on example 5, we have the setup on the previous side.2545

What we figured out was the density function is equal to 6 × Y - Y².2549

For part A, we want to find the probability that your commute will be longer than 30 minutes.2556

The probability that Y will be bigger than or equal to, we are measuring things in terms of hours.2562

I converted 30 minutes into an hour, it is ½, that is the integral.2570

Our range is from 0 to 1, ½ to 1 because you want it bigger than ½ of 6Y -6Y² DY.2576

We have a little calculus problem to solve and it is not a hard one at all.2589

Let us see, integral of 6Y is 3Y².2594

The integral of 6Y² is 2Y³.2602

We want to integrate that, to evaluate that from Y = ½ to Y = 1.2607

That is 3 -2 -3 × ½² is ¾.2614

+ 2 × ½³, ½³ is 1/8.2626

2 × 1/8 is ¼.2631

We get ¼ there.2635

3 -2 is 1 - ¾ is ¼.2641

This is ¼ + ¼, and that is ½, that is the probability that you are going to spend more than 30 minutes in traffic tomorrow morning.2647

For part B, we had this rage level, your rage level was R is equal to Y² + 2Y + 1.2661

What we want to calculate there was your expected rage level.2672

How angry you expected to be, when you get to work?2675

What we will use heavily here is the linearity of expectation.2679

The expected value of R is the expected value Y² + 2Y + 1.2683

But that is equal to the expected value of Y², this is linearity now, + 2 × the expected value of Y +,2693

We can say that the expected value of 1 is just going to be 1.2704

We need to find expected value of Y and Y².2710

In order to figure out those, I'm going to remember what we learned on one of the very early slide.2714

You can flip back and I think the slide was called key properties of β distribution.2721

What we learn was that the expected value of Y was equal to Α ÷ Α + β.2726

In this case, α and β are both 2.2735

This is 2 ÷ 2 + 2, that is 2 ÷ 4 is ½.2738

That was the easy one, the expected value of Y² is a little trickier.2745

What we are going to do is use the variance.2749

Σ², and that was kind of complicated formula, let me remind you what it was.2752

This is coming from, I think it was the third slide in this lecture,.2758

It was called key properties of the β distribution.2761

(α + β)² × α + β + 1.2763

We are going to work that out, α × β.2771

Α and β are both 2, that was given to us on the previous slide.2773

2 × 2 is 4 ÷ 2 + 2² is 16, and then 2 + 2 + 1 is 5.2779

This 4 ÷ 16 is ¼ × 1/5 is 1/20.2789

That is not the expected value of Y², that is the variance.2797

Let me remind you how we use to calculate the variance.2802

That is the expected value of Y² - the expected value of (Y)².2806

What we can do is, we can use this to solve for the expected value of Y².2814

This is a little old trick in probability is, if you know the variance, you can sort of reverse engineer for E of Y².2821

E of Y² is equal to 1/20 + (E of Y)².2828

We can write that as 1/20, we figure out what E of Y is, that is ½.2840

We figure that out up above, ½², this is 1/20 + ¼ which is my common denominator, there is 20, 1/20 + ¼ is 5/20.2845

I get here 6/20 and that reduces down to 3/10.2864

My rage level, I decomposed it into E of Y² + 2E of Y + E of 1.2870

Now, I can solve that, E of R is E of Y² + 2 E of Y + 1.2878

I can solve that E of Y² is 3/10.2890

2 E of Y is 2 × ½, I should have written E of 1 up above, here is E of 1.2896

E of 1 is just 1 because that is the expected value of a constant.2907

A constant is always going to be a constant, it is 1.2912

This is 3/10 + 1 + 1, 3/10 + 2 is 23/10.2916

All the Y is known as 2.3.2927

I do not know what the units are there, I’m just going to leave them.2930

I guess 2.3 range units is what you are going to carry into work,2934

or at least the expected value of your rage as you could work tomorrow morning.2939

2.3 is the expected value there.2945

That answers both things that we were looking for here in this problem.2950

Let me remind you where everything came from.2953

This F of Y, this density function is something we figure out on the previous side.2957

You can check back on the previous idea and see all the steps calculating that.2961

In part A, we had to find the probability that it will take longer than 30 minutes to drive to work.2965

30 minutes is ½ an hour and we want it to be longer than 30 minutes.2971

We are going to integrate from ½ to 1, the density function, and that turns out to be a fairly straightforward integral.2977

I’m going to keep the fractions straight.2985

We get that the probability there is exactly ½.2989

Half the time your commute will be longer than 30 minutes, half the time it would be shorter than 30 minutes.2993

In part B, we had to find the expected value of your rage level, where it is defined like this.3001

The expected value decomposes into these three terms.3008

That is using linearity of expectation, very useful for this kind of problem.3013

The expected value of Y, I am looking at the formulas for mean and variance on earlier slide from this talk.3019

If you go back, just scroll up and you will see a slide called key properties of β distribution,3027

and that is where I got these formulas, these complicated formulas using α and β.3032

But then, I just plug in Α = 2 and β = 2, that was given to me in the problem.3036

And, simplify those down into fractions ½ and 1/20.3042

However, this was the variance, it was not the expected value of Y² directly.3048

Remember, the old way to calculate variance is to find the expected value of Y² - the expected value of (Y)².3053

What we can do is reverse engineer this, in order to solve for the term we want.3062

Here is it, the term we want is the expected value of Y².3069

I bring this E of Y² over to the other side and that is where I get 1/20 + E of Y².3073

And then, this half right here is that half right there.3081

That is where that half comes from.3086

That is some more fractions, doing little simplification of the fractions.3088

We get that E of Y² is exactly 3/10.3092

I drop that into our expected value for R, and then this ½ is also that ½ from the E of Y.3096

The expected value of 1 is always 1 because 1 is constant.3109

You expect it to have its value.3113

And then, simplifying down 3/10 + 1 + 1 is 23/10 or 2.3.3116

That wraps up this lecture on the β distribution.3124

This was the last of the continuous distributions.3128

We worked through the uniform distribution, and the normal distribution, and the Gamma distribution.3131

The Γ, of course, includes the exponential and the Chi square distribution.3137

We had one big lecture on all of those.3142

Finally, we got the β distribution.3144

You are supposed to be an expert now on the continuous distributions.3147

I got one more lecture in this chapter, it is going to cover moment generating functions.3151

That is what you will see, if you stick around for the next lecture.3155

In the meantime, you have been watching the probability lectures here on www.educator.com.3158

My name is Will Murray, thank you for joining me today, bye.3163

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