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William Murray

William Murray

Expected Value of a Function of Random Variables

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Table of Contents

I. 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
II. 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
III. 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
IV. 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
V. 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
VI. 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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Expected Value of a Function of Random Variables

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Expected Value of a Function of Random Variables

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
  • Review of Single Variable Case 0:29
    • Expected Value of a Single Variable
    • Expected Value of a Function g(Y)
  • Bivariate Case 2:11
    • Expected Value of a Function g(Y₁, Y₂)
  • Linearity of Expectation 3:24
    • Linearity of Expectation 1
    • Linearity of Expectation 2
    • Linearity of Expectation 3: Additivity
  • 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

Transcription: Expected Value of a Function of Random Variables

Hello, welcome again to the probability lectures here on www.educator.com, my name is Will Murray.0000

We are working through a chapter on Bivariate density and distribution functions.0006

Everything in this chapter will have two variables, Y1 and Y2.0011

Today, we are going to talk about the expected value of the function of random variables.0015

That is something that we have seen in the single variable case.0021

We will see how the same ideas extend to the multivariable case.0025

I do want to start by reviewing the single variable case because0031

I think it will make the definitions make more sense for the multivariable case.0034

This kind of a review of stuff you seen before.0038

The expected value of a variable, if it is a discreet variable, what you do is you a sum up over all the possible values of the variable,0041

and then you just have Y by itself × the probability of that particular value of Y.0050

The analogous definition for the continuous case is that, instead of summing, you take an integral.0057

Instead of P of Y, you have F of Y which is the density function.0064

You still have this Y multiplied on it.0070

And then, we can also talk about the expected value.0073

Instead of Y itself, the expected value of G of Y which is some function.0076

For example, you might have G of Y to be Y².0081

The way you define that, if it is discreet is you sum up over all the values of Y,0086

you still have the probability of each value of Y there.0090

The difference is that, instead of the Y that we had before, we replace that with G of Y.0094

Then, the same kind of thing in a continuous case.0101

We take an integral, we still have the density function F of Y, and then0104

we replace the Y that we had before with the G of Y, and we have to solve that integral to find the expected value.0108

The multi variable cases, the new stuff that we are going to learn in this lecture.0116

The definitions are very similar, except that instead of single sums, we will have double sums.0120

Instead of single integrals, we will have double integrals.0126

Let us take a look at that.0131

In the Bivariate case, we will have a function of two variables.0133

G is now a function of Y1 and Y2.0136

The single sum, we have a double sum because there are two variables here.0140

There is a Y1 and there is a Y2.0144

We saw the probability function except it has two variables now.0147

P of Y1 Y2, and then the expected value of G of Y1 Y2, you will just multiply it on.0151

Instead of G of Y, we have G of Y1 Y2.0157

It is very similar to the single variable case, except that we have a Y1 and Y2.0161

In the Bivariate case, we still have that join density function F of Y1 Y2.0167

Again, we are integrating over both Y2 and Y1.0173

Instead of G of Y, we have G of Y1 Y2.0179

We will be doing a lot of double integrals in this lecture, I hope you are really fresh on your calculus 3,0185

your multivariable calculus because you will need to be able to do a lot of double integrals to solve the examples in this lecture.0191

Before we jump into the examples, there is a few more facts that I want to introduce you to.0200

In particular, linearity of expectation is a very useful principle to invoke, when you are calculating expected values.0204

The expected value of a constant is just a constant by itself.0216

Expectation is linear, in the sense that, if you have a constant multiplied by a function,0221

what you can do is just factor that constant on all of the expected value there.0228

Just factor out to the outside there.0233

That sometimes makes things a lot easier just because you can pull the constant outside,0236

not worry about them when you are doing your integral.0241

Even more useful is additivity.0244

If you have a sum of 2 functions and you want to find your expected value,0246

what you can do is calculate their expected value separately.0252

And then, just add them together.0255

That is extremely useful, if you try to find the expected value of something + something else,0257

just find your expected value separately and then, add them together.0262

We will see some examples, when we get to the problems of how that can be really useful.0265

It can save you doing a lot of very complicated integrals and sometimes reduce them to much simpler integrals.0270

Let us go ahead and start in on the examples.0276

First example here, we have a joint density function of E ^- Y2.0280

Remember that, := just means it is defined to be.0285

Our joint density function is defined to be E ⁻Y2.0289

Y1 and Y2 are both between 0 and infinity, but Y2 is always bigger than Y1.0294

We want to calculate the expected value of Y1 + Y2.0300

Right away, what I'm going to do is draw a graph of this region because it is not a perfectly square region,0304

it is not totally obvious what the region is.0311

I think the easiest thing is to start out with a graph.0313

There is Y1 on my horizontal axis, there is Y2 on my vertical axis.0317

I'm going to graph the line Y1 = Y2.0321

If your X and Y will be the line Y = X.0326

That diagonal line right there, that is the line Y1 = Y2.0329

What we want to look at, is the region where Y2 is bigger than Y1.0338

That is the region above that line, let me go ahead and color that in.0343

There is that region right there.0348

I want to describe that region because I’m going to be setting up a double integral to calculate this expected value.0352

It looks to me like, let me draw this in red.0360

It is easiest to describe it listing my Y2 first and then my Y1, in terms of Y2.0367

The point of that is that, I will have one more 0 to deal with, when I set up my limits.0374

That will make things a little bit easier.0378

If I list Y2 first, my Y2 will go from 0 to infinity.0380

My Y1 goes from Y1 = 0, that is the vertical axis, up to that diagonal line, that is the line Y1 is equal to Y2.0388

That is a good way to describe the region, we will use that to set up a double integral to find this expected value.0400

This expected value of Y1 + Y2 is equal to the double integral,0409

I will fill in the limits in a second, of Y1 + Y2 × the joint density function, the F of Y1 Y2 which is E ⁻Y2.0414

Let me fill in the limits on the integral, Y2 goes from 0 to infinity, Y2 = 0 to Y2, I will take the limit as it goes to infinity.0427

Y1 goes from 0 to Y2.0444

That means, I’m going to integrate Y1 first because it is on the inside.0453

I will put DY1 on the inside and DY2 on the outside.0457

Let me go ahead and do that integral.0463

Of course, if you are fortunate enough that you can use software for your integrals, at this point,0465

you just drop the whole thing right into a computer algebra system,0472

something like mathlab or mathematica, or something like that.0477

It will just spit out an answer for you.0481

If you are fortunate to have access to that, then go ahead and do that.0483

I want to go ahead work out the integral, just to show you that it is not that bad.0487

It can be done by hand.0490

When we do that first integral, notice that the inside integral, the variable is Y1 so that means Y2 is a constant.0493

In particular, the E ⁻Y2 was a constant.0501

I’m going to separate out that E ⁻Y2.0504

Then, the integral of Y1 + Y2 with respect to Y1 is Y1²/2 + Y2 Y1.0508

I just did that inside integral with respect to Y1.0520

I have to evaluate that from Y1 = 0 to Y1 = Y2.0523

I get E ⁻Y2 and if I plug in Y1 = Y2, I will get Y2²/2 + Y2².0531

I see I can combine those into 3/2 E ⁻Y2 and Y2².0545

3/2 Y2² E - Y2.0554

And that was just doing the inside integrals, I still need to evaluate the outside integral.0559

Y2 = 0 to Y2 goes to infinity, of this expression here, this is DY2.0565

Basically, I’m integrating X² –X, that is a classic case for integration by parts.0573

Let me go ahead and set up a little table to do my integration by parts.0580

I will pull the 3/2 out because that is really not relevant, it is not going to change anything in the integration by parts.0588

E ⁻Y2, this is my little shorthand trick for doing integration by parts.0598

If you do not know this trick or if you are rusty on integration by parts, we got some lectures,0604

it is in the level 2 college calculus section here on www.educator.com.0610

There is a whole lecture on integration by parts, you can just check it out and get all caught up to speed.0617

For the time being, I’m just going to use my short hand trick which is to take derivatives on the left,0622

that is 2Y2, and then 2, and then 0.0627

Integrals on the right, -E ⁻Y2 and then +E ⁻Y2, and then –E ⁻Y2.0631

You make these little diagonal hashes with a + - +.0642

You multiply down the diagonals and read off the answer.0647

This is 3/2 ×, it is negative because there is a negative there.0651

- Y2² E ^- Y2 - 2Y2 E ⁻Y2 - 2 E ⁻Y2.0660

All of these need to be evaluated from Y2 starting at 0 and then, we will take the limit as it goes to infinity.0673

Let us sort that out, we still have this 3/2.0684

When we plug in Y2 going to infinity, all of these terms are going to disappear.0689

There is as little bit of Patel’s rule in there but I'm not really showing you the details.0695

But, basically, the E⁻Y2 term dominates and it takes everything to 0.0700

Even though, it is being multiplied by infinity.0708

If you want to check the details of that, just go through Patel’s rule and check that out.0710

All the terms at infinity are going to 0, so 0 -0 -0.0716

I’m going to plug in Y2 = 0 and I will get, - and - so +, but it is 0 anyway, - and – so it is +2 E⁰ which is just 1, +2.0721

All the 0 disappear, I got 3/2 × 2.0740

My answer, my expected value of Y1 + Y2 is 3, that finishes that problem.0744

Let me go back over the steps and make sure that everybody is clear on everything.0753

The first thing to do here is to graph the region.0757

I looked at those limits there, Y1 and Y2 both go to infinity, but Y1 is always less than Y2.0762

That is how I got this region right here, this triangular region that goes on infinitely far.0770

And then, I want to describe that region in terms of Y1 and Y2.0776

I thought it would be easier to list Y2 first, because then I can get a 0 for the lower bound on Y1.0781

That makes my life slightly easier to have that 0 there.0789

That is why I picked Y2 to list first.0792

You can also done it with Y1 first, but it will made it a little more messy.0795

I’m going to use those limits to set up this integral, right here.0800

Because it is Y1 + Y2, that is why I multiplied everything by Y1 + Y2.0805

And then, I use the density function to get this E ⁻Y2.0811

Now, it is just a calculus 3 problem.0816

The only thing you have to be careful about is which variable you are integrating with respect to.0819

At first, I’m integrating with respect to Y1 which means Y2 is a constant.0825

That is why I got these different answers for Y1 and Y2 because the variable of integration was Y1.0829

Plugged in my values for Y1 and now it simplifies down to an integral in terms of Y2,0835

which is something that I needed parts for.0845

That is where I kind of outsourced to this tabular integration technique to do integration by parts.0847

That came back and gave me the answer, this long answer.0857

But, I plugged in my bounds, 0 and infinity.0861

The infinity terms all dropped out, that is really showing some Patel’s rule there.0864

I did not show the Patel’s rule but that is kind of what was going on in the background there.0870

The infinity terms all dropped out, and most of the 0 term dropout but this 0 term gave me a value of 2.0874

When I multiply that by 3/2, that is how I got my expected value of 3.0881

In example 2 here, we have got a joint density function of 2 × 1 - Y2 and Y1 and Y2 are both bigger than 0 and less than 1.0889

Let me graph that, before we go any farther, that is a simpler region than we had in example 1,0901

because that is a square region.0907

Here is Y1 and here is Y2, and there is 0 and there is 1, and there is 1.0910

We are just looking at a square region here.0918

Let me color than in and we will need to integrate that.0922

We will need to describe that region.0926

It is very easy to describe, as Y1 goes from 0 to 1 and Y2 goes from 0 to 1.0932

Let us go ahead and figure out what we are calculating.0941

E of Y1 Y2, the expected value of Y1 × Y2.0944

Remember, the way you calculate that is you set up a double integral on your region, Y1 = 0 to 1.0951

Maybe, I will list the Y2 on the outside, they are both constant so it will work the same way, either way.0964

Y2 = 0 to Y2 = 1, Y1 = 0 to Y1 = 1.0972

The function that I'm trying to find the expected value of, is Y1 Y2.0982

I will multiply Y1 Y2 in there and then I will put in the density function that we are given, which is 2 × 1 - Y2.0987

While, I got to setup my first integral as DY1 and my outside integral is DY2.0996

I’m going to go ahead and solve that integral.1004

The first one on the inside, I’m integrating with respect to Y1.1007

I see I have a 2Y1, I’m going to use those together and just get Y1².1013

And then, everything else is a constant because I’m integrating with respect to Y1.1019

Y2 × 1 - Y2, and I evaluate that from Y1 = 0 to Y1 = 1.1023

If I plug in Y1 = 1, I just get Y2 × 1 - Y2 and Y1 = 0 drops out.1033

This is what I’m integrating, I’m integrating this DY2.1041

That is Y2 - Y2², the integral of Y2 is Y2²/2.1048

The integral of Y2² is Y2³/3.1055

I need to evaluate that from Y2 = 0 to Y2 = 1.1062

If I plug in Y2 = 1, I get ½ -1/3.1071

Plug in Y2 = 0, they are both 0.1075

½ - 1/3, if you put that over a common denominator is 3/6 -2/6.1080

My expected value of Y1 × Y2 is exactly 1/6.1087

That finishes that problem, let me recap the steps.1093

First thing I did was, I looked at the region there and I drew a graph there Y1 and Y21097

are both between 0 and 1 so I got a square.1104

And then, I describe that region in terms of Y1 and Y2, they are both constants because it is a square region.1107

And then, I use that to set up a double integral.1114

The Y1 Y2 here, that came from the fact that we are trying to find the expected value of Y1 Y2.1121

The 2 × 1 - Y2 came from the density function that we are given.1128

At that point, it is just a calculus 3 problem, multivariable calculus, solving a double integral.1134

The important thing to notice is the first variable of integration, the inside one is DY1.1140

That is why all the Y2 were just constants.1146

I integrated 2Y1 to get Y1², dropped in my values for Y1 and it simplified down to something, in terms of Y2.1149

I integrated that with respect to Y2, dropped in my values for Y2,1161

and got a number that represents the expected value of Y1 × Y2.1167

In example 3, we have two variables Y1 and Y2, we have not been told the joint density function,1175

but what we have been told is that the mean of Y1 is 7 and the mean of Y2 is 5.1183

We have a couple of functions defined here, U1 is Y1 + 2Y2 and U2 is Y1 - Y2.1192

We have to calculate the expected values of the U.1201

The point of this problem is to link linearity of expectation.1206

If you do not remember that, go back to the introductory slides for this lecture and1211

look for the one called linearity of expectation.1215

That is what we are going to use very heavily to solve this problem.1219

Linearity of expectation is the key to solving this problem.1223

The expected value of U1 is equal to the expected value, just by definition of U1 of Y1 + 2Y2.1234

But then, expectation is linear, we can separate this out into the expected value of Y1 + 2 × the expected value of Y2.1244

We can just plug in the means that we have been given.1256

The mean of Y1 was 7 and the mean of Y2 is 5/7 +, 7 + 2 × 5 is 7 + 10 that is 17.1260

I almost got tripped up on my arithmetic, at the end there.1276

U2 behaves exactly the same way.1279

The expected value of U2 is the expected value of Y1 and - Y2.1282

Again, it splits up using linearity there.1293

That is the expected value of Y1 - the expected value of Y2 which is 7 -5.1296

Our expected value of Y1 - Y2 is 2.1305

We have an answer for both of those.1312

The key to finding those answers was really the fact about linearity of expectation.1314

But, if you have the expected value, you can split it up and take expected value separately.1320

You can also pull out constants.1325

That is true for expectation, by the way, that is not true for variance.1328

You do not want to be monkeying around the linearity of variance because there is no such thing.1331

But for expectation, that is true.1336

What we did here was, I just plug in what U1 was, Y1 + 2Y2.1340

Use linearity to split up into expected values of Y1 and Y2, then I just dropped in those expected values.1345

Remember, expected value and mean are the same thing.1352

I just dropped in the values of 7 and 5, and I got 17 there.1355

And then, the same thing for U2, I plug in Y1 - Y2, split up, and dropped in my expected values of 7 and 5,1359

and that is where that 2 came from.1368

In example 4, we have got joint density function F of Y1 Y2 defined to be 6 × 1 - Y2,1372

where Y1 and Y2 are both between 0 and 1, but Y2 is bigger than Y1.1383

Let me graph out this region, before we go any farther with that.1389

There is Y2, here is Y1, and we are looking between 0 and 1 on both axis.1394

But, we only want to take the region where Y1 is bigger than Y2.1405

That is the area above the line Y = X.1410

Let me draw that region, color that a bit.1414

That blue region is what we are going to be looking at.1420

We are going to end up doing a double integral for this one, actually, two double integrals.1423

I think it is going to be useful to describe the limits of this region.1428

I switched my Y1 and Y2 on the axis, not sure why I seemed to like to do that by looking1436

but I know I have done that several times.1442

Y1 is always the horizontal axis and Y2 is always the vertical axis.1444

This should be if I done it correctly.1449

I think the best way to describe this is in terms of Y2 first.1452

Y2 has to be constant, goes from 0 to 1.1456

And then Y1, Y1 we cannot use constants because we will get the whole square.1460

We just want the region from Y1 = 0 up to Y1, that line there is the line Y1 = Y2.1466

Y1 stays less than or equal to Y2, that describes that triangular region.1480

We use that to set up a double integral or a couple of double integral, since we have two expected values to solve here.1486

The expected value of Y1 is the double integral and I will follow those limits there.1494

Y2 goes from 0 to 1 and Y1 goes from 0 up to Y2.1500

Since, we are finding the expected value of Y1, I'm going to multiply in a Y1 × the density function, 6 × 1 - Y2.1511

And then, the inside integral is DY1, the outside integral is DY2.1522

Now, it is just a double integral problem, we can solve it using what we learned in calculus 3.1526

The first variable, the inside one is DY1 which means that 1 - Y2 is just a big old constant.1534

The integral of 6 × Y1 is 3Y1².1544

We want to integrate that or evaluate that from Y1 = 0 to Y1 = Y2.1554

If I plug in my limits for Y1, 3Y1² will give me 3Y2².1563

3Y2² × 1 - Y2, the Y1 = 0 just drops out.1572

That one does not play a role.1578

That is finishing the first integral, I still need to integrate that with respect to Y2.1581

I changed a Y1 to a Y2, that is very important.1590

3Y2² × 1 is just 3Y2², that integrates to Y2³ -3Y2³.1594

I have to integrate that, that would integrate to 3 × Y2⁴/4.1607

¾ Y2⁴, I have to evaluate that from Y2 = 0 to Y2 = 1.1617

If I plug in 1 to both of those, I get 1 - ¾.1628

If I plug in Y2 = 0 to both of those, they both drop out, -0.1634

That simplifies down to 1 - ¾ is ¼, that is my expected value for Y1.1640

I still have to calculate the expected value of Y2, because the problem asks for both of those.1647

I have already done some of the work, I already described the region.1652

It will be the same integral, Y2 = 0 to Y2 = 1.1655

Y1 = 0 to Y1 = Y2, the same integral except that, instead of multiplying by Y1,1662

I’m going to multiply by Y2, because that is what I'm finding the expected value of.1671

Y2 × 6 × 1 - Y2, DY1 and DY2.1676

Still doing the inside integral, since I’m integrating with respect to Y1, all the variables here are in terms of Y2.1686

I'm integrating just a huge constant.1697

It is 6Y2 × 1 - Y2, that is all a constant just × Y1.1700

I need to evaluate that from Y1 = 0 to Y1 = Y2.1708

Let us see, if I plug in Y1 = Y2, I will get 6Y2² × 1 - Y2.1716

There is a Y2 there and that Y1 becomes a Y2, when I evaluate it.1728

If I plug in Y1 = 0, it just drops out.1734

I have the integral from Y2 = 0 to 1 of 6Y2² DY2.1738

What I notice here is that, that is the same as the integral I had above, 3Y2² 1 – Y2 except that,1748

I have a 6 instead of 3, that is 2 × the same integral as above.1757

Maybe, I can plug that here with a star.1763

This is 2 × integral *.1767

If I evaluate that integral, what I will just get is 2 × the answer that I got above.1769

2 × ¼ and 2 × ¼ is just ½.1774

That gives me an answer for the expected value of Y2 of being ½.1780

If you do not like the way I did that, by sort of citing the integral above, just go ahead and work out this integral.1787

It is just a calculus 1 problem, it should not take you very long.1792

You should end up getting ½, it should work out.1795

That side gives us both the answers that we are looking for.1801

Let me remind you how we set that up.1804

The first thing I did, like in all of these problems is, I looked at the region that was given to me and1807

I graphed the regions.1814

In this case, Y1 and Y2 are both between 0 and 1, but Y2 is bigger than Y1.1815

I just looked at the region above that diagonal line, the Y = X line.1821

I decided that it was easier to describe that region, if we kind of march horizontally.1827

That means describing the Y2 first, in terms of constants.1833

And then, describing Y1 going from 0 to Y2.1838

If you just said Y1 goes from 0 to 1, all of the sudden you described a square, and that is not the region that we are looking at.1843

I set up those limits, by looking at the region.1851

And then, I pull those limits over and used them to set up a double integral.1855

I took the density function 6 × 1 - Y2, and I multiplied it by the thing we are finding the expected value of,1862

which in the first case is Y1, but in the second case is Y2.1870

That is because of the different requirements, the first one is Y1 and the second one is Y2.1874

In each case, I got a double integral to solve.1881

And then, I just did a double integral, integrated DY1 first which is why 6Y1 turn into 3Y1².1884

1 - Y 2 was just a big old constant.1893

Plugged in my values for Y1, got an integral in terms of Y2, pretty easy integral in the sense of,1896

probably you can hand that off to first semester calculus student, and they can solve it for you.1904

The answer that they hopefully come up with would be ¼.1910

For Y2, same kind of thing happens except that, when you are doing that first integral, there are no Y1 at all,1914

which means you are integrating a constant.1921

You will just get that constant × Y1, plug in the value of Y1 = Y2 and we get 6Y2² 1 - Y2.1924

Because I'm lazy, I noticed that that was the same integral that we had back here,1933

except it is multiplied by 2 which means we can just take the old answer multiplied by 2.1937

2 × ¼ is ½.1944

If you had not noticed that, that integral was the same as the previous one, that is okay.1946

Just go ahead and work it out, and do one or more steps of calculus 1, and you will get an answer.1953

I want you to really understand these two answers.1960

I want you to remember them because the next example, example 5,1964

we are going to be using these answers from example 4.1968

It is the same setup as in example 4, I’m going to use these answers to take it, separate farther.1972

Make sure you understand these, before you go on to example 5.1980

In example 5, we have got the same setup that we had from example 4.1987

We are going to be using the answers that we derived in example 4, to solve example 5.1993

If you have not just watched example 4, go back and watch example 4, or are work it on your own.1999

Just make sure your answers agree with mine.2006

You want to have those answers fresh in your mind and ready to go for example 5.2009

Let me solve example 5, we are given the same setup F of Y1 Y2 is 6 × 1 - Y2 and then, they describe the region for us.2015

Let me just emphasize that, that is all the same as example 4.2026

Because of that, I’m not even going to try to graph it, like I did with all the other examples.2034

If you want to kind of work that out and you want my help, just go back and watch example 4,2038

you will see that that carefully worked out, I drew the graph and everything.2046

In this case, we do not have to do that again, because we are finding the expected value of 2Y1 + 3Y2.2050

The expected value of 2Y1 + 3Y2, the trick here is not to do an integral but to use linearity of expectation.2060

That was something that I introduced you to, on the third introductory slide to this lecture, linearity of expectation.2070

It is super useful for problems like this.2079

I should spell expectation right, since it is such a useful concept.2084

Linearity of expectation says, you can distribute and split up this expected value into 2 ×2088

the expected value of Y1 + 3 × the expected value of Y2.2096

Both of those were things that we calculated in example 4.2103

I'm not going to recalculate those now because we did no little job of work in calculating those.2108

The expected value for Y1, back when work it out in example 4 was ¼.2117

The expected value for Y2 was ½.2125

Go back and watch example 4, if you are unsure where those numbers are coming from.2130

This is 2 × ¼ is ½ + 3/2, that is just 4/2 or 2.2134

We got an answer right away, we did not have to do another double integral.2143

The reason we did not have to do another integral is because this is the same setup as example 4.2149

We already worked out the basic values in example 4.2155

We can just extend that using linearity of expectation.2159

If the phrase linearity of expectation does not quite trip off your tongue yet,2163

it is worth going back and watching that third slide again.2168

You will see how nicely it helps us in this problem because it allows us to take 2Y1 + 3Y2,2171

and split up and just find the expected values of Y1 and Y2, and then combine them back together.2178

Those expected values we calculated in example 4, to be ¼ and ½, just drop those numbers in and then simplify the fractions.2184

We get our expected value of 2 for 2Y1 + 3Y2.2194

That wraps up this lecture on expected value of a function of random variables.2202

Next up, I got a lecture on Covariance, it is a big topic, there is a lot of stuff in there.2208

I hope you will stick around for that.2212

In the meantime, this is the chapter on Bivariate density functions and distributions.2214

This is all part of the larger lecture series on probability, here on www.educator.com.2219

My name is Will Murray, thank you for joining me today, bye.2225

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