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William Murray

William Murray

Variance & Standard Deviation

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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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Lecture Comments (2)

1 answer

Last reply by: Dr. William Murray
Mon Sep 15, 2014 6:18 PM

Post by Christian Faya on September 13, 2014

In example V, when calculating the mean, I'm confused as to why we multiply the p(y) times the number showing on the die. To my understanding, each face on the die has the same probability, so why multiply the value of each facet of the die? Maybe I'm not understating this particular question. I had no trouble understanding the other problems, but this threw me off. By the way the lectures have been of great help!

Variance & Standard Deviation

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Variance & Standard Deviation

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  • Intro 0:00
  • Definition of Variance 0:11
    • Variance of a Random Variable
    • Variance is a Measure of the Variability, or Volatility
    • Most Useful Way to Calculate Variance
  • Definition of Standard Deviation 3:44
    • Standard Deviation of a Random Variable
  • 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
    • Example III: Variance
    • Example III: Standard Deviation
  • Example IV: Calculate the Mean, Variance, & Standard Deviation 17:54
    • Example IV: Mean
    • Example IV: Variance
    • Example IV: Standard Deviation
  • Example V: Calculate the Mean, Variance, & Standard Deviation 29:56
    • Example V: Mean
    • Example V: Variance
    • Example V: Standard Deviation
  • Example VI: Calculate the Mean, Variance, & Standard Deviation 37:29
    • Example VI: Possible Outcomes
    • Example VI: Mean
    • Example VI: Variance
    • Example VI: Standard Deviation

Transcription: Variance & Standard Deviation

Hi and welcome back to the probability lectures here on www.educator.com, my name is Will Murray.0000

Today, we are going to talk about variance and standard deviation, two very closely related concepts.0006

Let us check those out.0011

Let us jump right in with the definition of variance.0013

The variance, you want to start with the random variable Y.0016

The variance of that random variable is by definition, it is the expected value of the quantity Y - μ².0019

Where μ here is the mean of the random variable, also known as the expected value of the random variable.0028

That is the definition of variance.0036

We use 2 different notations for that.0038

Sometimes, we use V of Y for variance, and some×, we use σ² of Y.0041

Or let us just say σ², we are not even including the Y.0046

Those mean exactly the same thing.0050

If you see σ² and if you see V of Y, those are exactly the same.0052

It is just 2 different notations for the same concepts.0058

That is the first thing you want to get straight.0061

Σ² and V of Y are the same thing.0063

What that is really measuring is you are measuring how far the variable Y deviates from its own mean.0067

Remember, μ is the mean of the variable.0076

It is a measure of how much the variable wobbles around and how far it deviates from its own mean.0078

Another way to put that is to say it is a matter of the volatility of Y.0087

How volatile is the variable?0093

Let me give you some mini examples here.0095

Let us suppose that that is the mean of the variable.0099

A variable that does not deviate that much would have a small variance.0103

Another variable that could have the exact same mean, it could get to that mean by doing much more serious wobbles.0110

It can wobble much farther from the mean, that variable would have a much higher variance.0121

This would have the same mean on both of these variables but this would have a very small variance.0128

And then this variable would have a very high variance because it is wobbling around so much,0139

it is a very volatile random variable.0145

This is high variance here.0148

That is kind of the intuitive idea of what variance means.0152

It intuitively means how much is that variable wobbling around? How much is your dataset varying?0159

The most useful way to calculate the variance is using this formula.0167

The variance, you get it by calculating the expected value of the function Y²0173

and then it looks like I'm talking about the same thing but I'm not.0179

You subtract off the expected value of (Y²).0183

You might think that these 2 things are the same but they really are not.0188

Remember, the expected value of Y is also known as μ so this is μ².0193

The expected value of Y² here, in general, it is not equal to the expected value of (Y²).0197

That is not something that you can say that those 2 are the same.0207

You have to be very careful there.0211

The expected value of Y² is something you want to calculate separately.0213

We will see in the examples how you calculate that separately.0217

I got one more definition for you, before we jump into the example which is standard deviation.0223

The way you get to standard deviation of a random variable is you start with the variance and then you just take its square root.0229

That is the definition of standard deviation.0237

It is by definition, the square root of the variance.0239

That is what this := means.0245

This is the definition here, we are defining it.0247

Sometimes, we use σ to represent the standard deviation of a random variable.0251

The first thing you notice about standard deviation is that it is defined directly from variance.0256

If you tell me the variance, I know immediately the standard deviation.0264

If you tell me the standard deviation, I know immediately the variance.0268

You can just go back and forth from one to the other.0273

Essentially, they measure the same thing.0276

If one of them is big then the other one is going to be big and vice versa.0279

The standard deviation is also a measure of how much the variable wobbles around, how volatile the variable is.0284

You can use variance to keep track of that, or standard deviation to keep track of that.0292

Essentially, those are 2 different ways of measuring the same idea, standard deviation and variance.0297

Essentially, the reason we have 2 different notions for the same idea is that in computations,0305

It is sometimes more useful to use one of the other but they are not really measuring 2 distinct concepts there,0311

as mean and variance really are measuring 2 distinct concepts.0318

Let us see how we can calculate some variances and also I want to kind of test whether you have a good intuition for these.0324

We are going to start out with that in the examples.0332

First example, I'm giving you 3 datasets and I'm saying which one has the largest variance.0335

You can choose A which is a bunch of a 123, B is 11, 11, 13, 15, 15, C is 20, 21, 22, 23, and 24.0341

You might want to think about that for a little bit to see if you have a guess as to which one0353

has the largest variance, before I jump in and answer it for you.0357

Let me try to answer that.0361

I'm not going to calculate the variance on each one, I’m just going to graph them and I hope that when I graph them,0363

if you have a good intuition for what variance means then on you will understand quickly0369

which of these sets of data has the largest variance.0377

In this first one, we got 5 copies of 123, 5 copies of the same point there.0380

The mean of that set of data would just be a line going through all of them.0389

Let me make those dots a little bigger there so you can still see them even after I draw the mean.0396

The second one, the second set of data here, we got 11 those are much smaller numbers 11, 11, 13, and then 15 and 15,0402

let me draw the mean for that.0421

The mean runs right through the middle there.0423

There is the mean of that set of data.0426

I’m drawing my points a little bigger here.0428

For the third one, the 20, 21, 22, 23, and 24 here, I will graph that set of data.0431

We have slowly increasing set a data there.0439

I will draw the mean of that set of data, that goes right through the middle there, there is the mean right there.0447

The question is which of the sets of data has the highest variance?0455

If you look at these, I hope that it is clear from the graphs because if you look at the first set of data,0459

there is no variance at all because all those points are exactly on the mean line.0466

In the second set of data, there is a quite a high variance because most of the points are quite far from the mean lines.0474

In the third set of data, there is a lower variance because some of the points are still quite close to the mean line.0488

The highest variance that we see there is in the middle set of data because the points there are farther from the mean line.0500

I did this intuitively, I did not actually calculate the variance on any of these 3 sets of data.0508

But I can tell just by looking at them that the data points are farthest from the mean for set B there.0515

That is clear to me that that is going to have the highest variance,0527

just by knowing kind of intuitively what variance and tails, and I do not even have to calculate the variance on those.0531

Let us go ahead and look at the next example.0541

It is another multiple choice question.0543

Which of the following would be the least useful in understanding a set of data, knowing the mean and the standard deviation,0545

knowing the mean and variance, or knowing the standard deviation and the variance?0552

The key point here is to remember that the standard deviation is always equal to the square root of the variance which means.0557

If you tell me the variance, I can figure out the standard deviation easily and quickly.0577

If you tell me the standard deviation and the variance, you have told me some redundant information.0582

You have not told me anything new.0588

As soon as you tell me the standard deviation, I figure out the variance on my own.0590

Or as soon as you the variance, I figure out the standard deviation on my own.0594

This C here, this is redundant information.0599

On the other hand, if you tell me the mean and the standard deviation,0611

those are independent pieces of information which means you have really told me two useful pieces of information there.0616

You tell me the mean, even after you tell me mean I did not know the standard deviation.0631

Even you told me the standard deviation, I did not know the mean.0635

These are two useful pieces of information, you told me 2 different things about a dataset.0639

The same thing for part B, if you tell me the mean and variance, you have told me 2 pieces of independent information.0644

Those are both useful independently of each other.0654

The mean and variance are both useful to know independently of each other.0663

Even if you tell me one of them, I still do not know what the other one is.0667

When you tell me the other one, I get something useful.0669

The standard deviation and the variance, if you tell me one I can calculate the other.0672

That is something redundant.0678

The question asks which is the least useful in understanding, that would definitely be C then.0680

Because essentially, if you tell me the information in part C, I really only know one thing about my data set.0685

If you tell me any information in part A the mean and standard deviation, I can calculate the variance so I know everything there.0691

If you tell me in part B the mean and variance then I can calculate the standard deviation on my own so I get something a lot more useful there.0699

Let us keep going.0709

In example 3, we are going to look at your probability class again.0710

This is actually very similar to an example we had in the previous lecture0715

where we are calculating your semester average for your probability class.0719

You might want to check that out if you have not just looked through those lectures.0723

In this example, we are going to take 2 midterm exams.0729

They are going to count 25% each of the semester grades.0733

The final exam counts for 30% and the homework counts for 20%.0737

It turns out that the score 79 of both midterms and on the final, and on the homework.0741

We are very consistent here, scoring 79 on everything.0748

We want to calculate the mean, the variance, and the standard deviation of our scores using the weights above.0752

You might be able to guess a lot of the answers to this question ahead of time.0759

In the next question, we are going to mix it up using different numbers.0764

In the next question is an extension of this one.0768

That is why I want to work through this one first and you will see how all the numbers come in.0771

Let me go ahead and start by calculating the mean there.0778

The mean is the expected value of Y and we did calculate one very similar to this.0781

I have different numbers but it was the exact same principle in the previous lectures.0789

You might want to check back in the example in the previous lecture where we calculated one like this.0793

The expected value of Y, just by definition, I gave you the definition in the previous lecture.0800

You add up all the possible values Y can be and then multiply each one by the probability.0805

In this case, you are just scoring a 79 on everything.0812

This Y is 79 for everything, P of Y × 79.0816

There is only one value which is the 79 because that is how you scored on everything.0824

That 79 × the total probability, the total probability of any experiment is 1, that is just 79 is your mean.0829

That was fairly easy to calculate, that is not all surprising.0840

If you score 79 on everything then your average should be 79.0843

The σ² is the variance, what was the mean if we calculated the first.0847

The variance is, the σ² remember is the expected value of Y - μ².0853

I try to find the expected value of the function of our random variable.0864

Let me remind you for the formula for that.0868

The expected value of a function of a random variable, by definition is you add up all the possible values that random variable can take.0872

The probability of each one × not the value itself but G of that value.0881

In this case, you look at all the possible values P of Y.0887

In this case, G of that variable is Y - μ².0894

It is (Y - 79)², the 79 is what I figure out up above.0900

All the possible values, all the values in this case were 79.0911

Y -79², the only values we have here are 79 so 79 -79², of course is 0.0918

The variance here is 0.0929

That should be intuitively clear because you are completely consistent the whole semester, you scored 79 on everything.0931

You know it did not vary at all from your mean of 79.0939

Finally, let me calculate the standard deviation.0943

That is very easy once you figure out the variance.0950

It is just the square root of the variance, V of Y.0953

Our V of Y that is the same as σ², that is √0 which of course is still 0.0959

Your standard deviation is also 0.0967

Again, that is not a surprise if you scored 79 on everything all semester0969

then you have not deviated at all from your mean which means your standard deviation is 0.0975

I will emphasize it in the next example, we are going to mix up the numbers so0980

we will have some more tricky numbers to calculate that would not be so obvious, I hope.0986

Let me remind you where all of these numbers came from.0992

To calculate the mean, we calculated the weights on each possible score.0995

What every possible score here was 79, it was just all the weight went on 79,1001

total probability of 1 on 79, that was your mean of 79.1007

Calculate the σ², by definition, σ² is the expected value of Y - μ².1012

I’m using this formula that we learned in the previous lecture, in the previous video,1020

which was the expected value of a function of a random variable is just the sum1025

of the probabilities × the function of each value.1031

That was what I plugged in here, Y - μ² here, that is the G of y.1034

In this case, the μ was 79 so I just plug in 79 here.1044

Since all the Y’s are 79, we are just adding up 79 – 79², of course that comes out to be 0.1050

The standard deviation is just the square root of the variance, that is √0 which just comes up to be 0.1059

In example 4, we are going to revisit this example except, I’m going to mix up the numbers on you,1066

so you should get a more interesting answer and less predictable answer.1072

In example 4, again, we are in this probability class, this is very similar to the previous example.1076

Same weight, we got two midterm exams with a 25% each and a final exam counts for 30%,1081

and the homework is going to count for 20%.1090

This time, instead of scoring 79 on everything, you are going to score 60 on the first midterm,1092

80 on the second, 80 on the final, and 100 on the homework.1098

I want to calculate the mean, variance, and the standard deviation of all of the scores using these weights.1103

This one should come out a little more interesting.1109

In fact, these are the same numbers that I used in an example on the previous lecture,1112

when you are first learning to calculate expected value.1117

You might want to go back and check on that example from previous lecture1120

because I may be doing the same calculation to get the expected value here.1124

Μ is the unexpected value of Y.1128

By definition, the way you calculate expected value is you look at all the Y values and1137

the weighting or the probability on each one × that value.1143

Here, our Y values are 60 because we scored 60 on something,1147

we scored 80 on a couple different things, and we scored 100 on something.1152

Let us figure out the probabilities or the weights on each one.1159

We scored 60 on the first midterm and that was 25%, 25/100.1162

We scored 80 on the second midterms, 25/100, and also on the final.1171

Let me put a 30/100, fill that in there as well.1177

Both of those get multiplied by 80 because there is a two different test on which we scored 80.1182

We scored 100% on the homework.1189

The homework was 20% of the grade, 20/100 × 100.1192

If you just run the numbers through there, I’m not going to do labor that on my own calculator because I did this ahead of time.1200

You get 79 as your average for the semester, that actually comes out to be the same average that we got in the previous examples.1206

The μ is the same as in the previous example but the standard deviation will definitely not be the same, and the variance will not be the same.1216

Let us calculate those.1225

I’m going to calculate the variance using two different formulas.1227

I’m going to use the original definition of variance then I’m going to use a clever formula that we found out later on.1230

Let me calculate the variance.1238

The first way I’m going to use is the original definition of variance which is the expected value of Y - μ².1241

Now, the expected value of a function of a random variable, that means you calculate it just like the expected value of Y ×1252

the probability of each value except that instead of calculating Y itself, you calculate that function of Y.1261

Y - μ², μ was 79, I will go ahead and fill that in.1269

That is coming from up above, μ is 79.1276

I have to calculate all my possible values -79².1282

I'm going to calculate the same probabilities as above 25/100 except instead of 60, I will have 60 -79².1291

Just like above, I have 25/100 + 30/100, that simplifies to 55/100 × 80 -79² + 20/100 × 100 -79².1307

If I simplify that down, I get 25/100.1328

60 -79 is -19, since I’m squaring, I will make it a positive, 19².1333

Let me make that bigger there so you can read it.1341

+ 55/100, 80 -79 is just 1 × 1² + 20/100.1346

100 -79 is 21².1357

If you work this out, I did this on my calculator.1364

I get 79/100 and that simplifies down to 179, that is the variance.1367

That was using the original definition of variance but we have another way to calculate variance.1382

Let me remind you what that was.1387

The original definition was the expected value of (Y - μ)², that was our original definition.1390

But we also have a clever formula that I want to remind you of.1401

It was the expected value of Y² - the expected value of Y, the whole thing².1408

Let me calculate that one because that was fun to workout here.1418

Of course, we should get the same answer if I do the arithmetic right.1422

Let me calculate V of Y using that new formula.1427

It is E of Y² - the expected value of Y.1431

Of course, we already figured that out,².1438

E of Y² is the sum of Y × P of Y × Y².1441

It is all those same probabilities and I'm going to fill in the values of Y² everywhere, 25/100.1449

My first value of Y was 60, I will put 60² + 55/100.1458

That was corresponding to the value of Y being 80, so 80².1466

20/100 × 100².1473

Now, I still have to subtract off E of Y².1479

This E of Y, that is the same as the μ, that is the mean that we calculated earlier.1482

Let me subtract off 79² because that was the mean that we calculated earlier.1487

If you on run these numbers, I did run them on my calculator, we got 6420 -79², that simplifies down to 179.1495

It is the same thing we got above but it is a slightly different way of calculating.1514

I think it might be a little bit easier but you can make your own decision there, and it does check.1517

We have the mean right there, we have the variance right there.1525

Finally, we have to find the standard deviation.1534

That is a very quick because once you know the variance, the standard deviation is just the square root of the variance.1541

Σ is just by definition, the square root of the variance.1549

In this case, we already calculated the variance that is 179.1559

179, the² root of that is let us se, 13² is 169, 14² is 196.1563

It is a little closer to 13.1571

It is approximately equal to 13, there is definitely some decimals there that I'm not bothering to work out there.1574

If you want to be really accurate, say the standard deviation is the √ 179 and that will certainly be accurate.1582

If you want to get a quick estimation, I estimate that at about 13.1590

Let me remind you how we calculate each one of these.1595

The mean was actually something we calculated before in the previous lecture.1599

I will just do that quickly, looked all the values of Y, took a probability of each one, and then add them all up.1605

The tricky one was this value of Y being 80 because there were 2 different things we scored 80 on.1613

80 on the second midterm and 80 on the final.1620

That is why I took 25% for the second midterm and 30% for the final, that is where that came from.1623

If you run your arithmetic on that, you get a mean of 79.1631

The variance, there is 2 different ways you can calculate that, using the original definition here or using this formula.1634

I calculated it both ways here because I want to check and make sure that we got the same thing both ways.1642

It is also a good practice to see both methods.1648

If you want to calculate the expected value of Y - μ², it is just like the expected value of Y except that you replace Y with Y -79².1652

Same probabilities here, that 25, 55, and 20.1664

Instead of 60, 80, 100, we do each one of those numbers (-79)².1669

And then, if you work the arithmetic out there, it comes out to be 179 for the variance.1678

That is one way of calculating the variance.1683

The other way of calculating the variance is using this formula.1685

The expected value of (Y²) - the expected value of (Y²).1689

It seems like those things are the same but they are not, do not be confused by that.1697

The expected value of Y² is that is the same as μ.1703

That is the 79² that I filled in right there.1707

The expected value of what (Y²) is that same formula with all the probabilities 25, 55, and 20.1712

All the values of Y except we are squaring each one, 60², 80², 100².1720

I ran that through my calculator, simplify down to 179.1727

The standard deviation is very easy to calculate once you know the variance because by definition,1735

it is just the square root of the variance.1741

We just take √179 and I did not run that through my calculators, I just estimated that to be about 13.1743

But of course, that is not 100% accurate.1750

The accurate value would be √179.1752

The interesting thing about this example is that we have a very same mean on this example,1758

as we had on the previous example, example 3.1763

Example 3 was also calculating your scores over the semester.1767

But in example 3, you got that mean of 79 by getting a 79 on everything all semester.1771

You had a mean of 79 and no variance at all.1778

Here, we still ended up with a mean of 79 by much more variable set of scores1782

which is why we had a much more interesting variance now, we have a positive variance.1788

Last time, we just had a variance of 0.1793

In example 5, we are going to roll one dice and Y is going to be the number showing on that dice.1798

We are going to calculate the mean, and variance, and the standard deviation of Y.1805

Let us jump right into that.1811

The mean, μ, also known as E of Y, those things mean the same thing.1814

Remember that means you calculate all the possible values of Y that you might see in the probability of each one.1825

When you roll a dice, you can see a 1, 2, 3, 4, 5, 6.1832

The probabilities of each one, not ½, that is for sure, it is 1/6 + 1/6 × 2 + 1/6 × 3 + 1/6 × 4 + 1/6 × 5 + 1/6 × 6.1841

The easy way to do that is to factor out 1/6 and then add up 1 + 2 + 3, up to 6.1859

1 + 2 + 3 up to 6 is 21, 21/6, and that simplifies down to 7/2.1870

That is the mean, that is the expected value if you roll one dice.1878

You are not going to see a 7/2 because no side on a dice has 7/2.1883

But, that means if you roll the dice many times then the average of all the answers that you see will be 7/2.1889

To find the σ², I’m going to use I think the new formula σ² divided by the variance.1899

We have a definition for variance but I’m not going to use the definition, we will use this new formula that we have.1908

The expected value of Y² - (E of Y)².1914

I think that is going to be a little more easy to calculate.1920

Let us calculate first the expected value of Y² which means we do1923

the same calculation as above except that to we are going to look at Y², instead of Y.1929

We actually did this one as an example on the previous lecture.1937

I think it was the last example on the previous lecture.1941

You can go back and look that up, if you want.1943

I will do the same thing as I did above here, I will list out my probabilities 1/6, 1/6, 1/6, 1/6, 1/6, and 1/6.1945

I list out my numbers 1, 2, 3, 4, 5, 6, except I’m supposed to square each one.1960

Let me square each one of those.1968

Again, it is useful to factor out the 1/6 on all of these so we get 1/6 × 1 + 2² is 4, 3² is 9, 4² is 16, 25, 36.1975

If you add up those numbers, we actually did this on the previous lecture which I remember doing.1990

I remember that I got 91/6, that is not the variance yet, that is the expected value of Y².1997

In order find the variance, we have to subtract something off.2006

Σ² or V of Y, this is calculating the variance now, is E of Y² as a quantity, - E of Y by itself².2011

We get 91/6 and then we calculated E of Y up above, there it is, it is7/2 except we have to square that.2032

That is 91/6 - 7/2 is 49², not 47 for sure, 49/4 is 7/2².2043

It looks like a common denominator there will be 12.2059

I have to multiply 91 by 2, I got 182.2064

After multiply 49 by 3 which gives me 147.2068

182 -147 is 35 and I divide that by 12, that is my variance.2073

That is my variance if I roll one dice.2085

Let us figure out the standard deviation.2089

It is easy to find the standard deviation after you find the variance.2096

Find the variance first and then you can find the standard deviation very easily,2100

just by doing the square root of the variance, that is the definition of standard deviation.2104

We just get the, not very enlightening here but √ 35/12 or the standard deviation.2110

Let me recap how we calculated each one of those.2124

For the mean, we just use the standard definition for expected value.2128

That means you list out all the possible values you might see and the probabilities of each one, multiply and add.2133

Our possible values when you roll a dice are 1, 2, 3, 4, 5, 6.2140

1/6 probability of each one, that is where these 1/6 come from.2144

If you simplify the numbers there, you get 7/2.2148

For the variance, I was going to use this formula.2153

It is not the original definition of the variance but a very useful formula.2157

We find the expected value of Y² - the expected value of Y, all².2161

As a preliminary step here, I calculated the expected value of Y² which looks a lot like the expected value of Y.2169

You list out all those probabilities and all the values of Y except you use Y² for each one.2177

I² each one of those numbers, that is the only difference from the calculation above2184

is we are squaring each one of those numbers, as we go through.2189

We get a different answer, 91/6, that is E of Y².2192

That is easy now to drop into our formula for variance which is E of Y² - the expected value².2200

That is our formula for variance which we can write is V of Y or σ².2211

Those would mean exactly the same thing.2214

91/6 comes from above, the 7/2 comes from up here, that was our mean that we already found.2217

Then it is just a matter of squaring 7/2 and reducing the fractions down to 35/12,2225

that is our variance for that experiment.2231

The standard deviation is easier because once you know the variance, the standard deviation,2234

by definition is the square root of variance.2239

We get that with √ 35/2.2244

We got one more example here.2250

When a flip a coin 3 × and we are going to let Y be the number of heads,2251

and we want to calculate the mean, the variance, and the standard deviation of Y.2257

What I'm going to do here is list all the possible outcomes of this experiment.2263

There are better ways to do this kind of example, after we learned about some probability distributions later on that.2268

That is a couple of lectures down below.2277

Unfortunately, at this point we are reduced to listing all the outcomes.2279

After learn about the binomial probability distribution, we will have a better way of calculating these.2283

But in the meantime, let us just look at all the possible outcomes that can happen with this experiment.2287

I’m going to list the outcomes, I’m going to list Y for each one.2296

Because we are going to need that later when we calculate the variance, I'm also doing it as Y², that would be useful.2300

When you flip a coin 3 ×, you can get a head-head-head, head-head-tail, head-tail-head, head-tail-tail.2307

I’m trying to list these in some kind of a binary counting order.2319

I get tail-head-head, tail-head-tail, tail-tail-head, or tail-tail-tail.2323

I’m going to list the values of Y for each one of those.2334

Y was the number of heads we see each time.2338

There is 3 heads there,2 heads, 2 heads, 1 head, 2 heads, 1 head, 1 head, and 0 heads.2341

I’m going to need Y² later on.2350

Let me fill in the Y², 9, 4, 4, 1, 4, 1, 1, 0.2353

I'm in good shape to calculate my expected value, my variance, and standard deviation.2363

The expected value of Y by definition is the same as the mean.2370

If they ask you the mean, the expected value, that is the exact same thing.2377

You can calculate one for the other, they mean the same thing.2383

It is all the probabilities.2386

The probabilities are something I'm going to have to figure out but the probabilities of each one of these outcomes, the probabilities are all the same.2388

The probability is 1/8 for each of these.2406

Let me just factor that 1/8 and then I will add up all of the Y that I see 3 + 2 + 2 + 1 + 2 + 1 + 1 + 0.2410

And I hope I have not made a mistake there.2424

3 + 2 + 2 is 5 + 1 is 6 + 2 is 8.2426

I think I have left something out there, let me go back and check there.2434

3 + 2 + 2 is not 5, 3 + 2 + 2 is 7 + 1 is 8 + 2 is 10 + 1 is 11 + another 1 is 12 + 0 is 12.2439

12/8 and of course that simplifies down to 3/2.2450

That is the mean or the expected value of Y, in this experiment.2458

That is not surprising, if you flip a coin 3 ×, I'm not saying you expect to see exactly 3/2 heads.2463

I’m saying that the average over many iterations of this experiment will be 3/2 heads.2470

On average, I will see 3/2 heads per time that you run this experiment.2477

Let us find the variance now.2483

In order to find the variance, first I’m going to find the expected value of Y².2487

The reason I'm doing that is because we have this nice formula for the variance2492

which says that σ² which is another way of saying the variance.2499

V of Y is also a way of saying the variance, it is the expected value of (Y²) - the expected value of Y by itself and then².2506

That is a good way to calculate the variance quickly.2518

But in order to do that, you have to know what E of Y² is.2522

That is the same as the same probabilities but we are looking at Y² instead.2526

That is 1/8 × all the values of Y², 9 + 4 + 4 + 1 + 4 + 1 + 1 + 0.2534

Let me see if I can add those up better than I did for the mean.2548

9 + 4 + 4 is 17 + 1 is 18 + 4 is 22 + 1 is 23 + 1 is 24, and 0 does not change it.2552

24/8 gives me exactly 3 there.2564

I did not box that because that is not the variance yet, that is just E of Y².2571

We have not found the variance yet.2577

In order to find the variance, we are going to drop in this other formula down below.2579

E of Y² is 3 – E of Y by itself was 3/2.2583

We calculated that above, I² that.2589

I get 3 - 9/4, 3 is 12/4 so 3 -9/4 is ¾.2593

That is the variance is ¾.2604

Finally, we have to find the standard deviation but that is easy once you know the variance2610

because the standard deviation is just the square root of the variance.2616

By definition, it is the square root of the variance.2621

That is the V of Y right there and that is the square root of what we just figured out, ¾.2624

I can simplify that a little bit to be √3/2.2636

I hope you watch a few more lectures and you learn about the binomial distribution.2644

We will calculate the mean and variance and standard deviation of the binomial distribution.2650

You might want to come back and check this problem because that is one of the very quick ways to get to these answers.2656

But we have not learned that yet, in the meantime, we are just working things out from scratch.2661

Let me remind you what we did.2665

We listed all the possible things that can happen, all the possible outcomes that can occur when you flip a coin 3 ×,2667

which is just all strings of heads and tails, strings of 3 heads and tails.2675

The probability of each one of those is 1/8 because you got to flip a coin 3 ×,2680

you got a 50% chance of getting the right flip each time.2685

There are 8 of them in total because there are 2 outcomes for each individual flip.2689

2 × 2 × 2 is where that 8 come from.2695

We want to calculate, Y was the number of head.2698

I listed the number of heads that we get from each one of those outcomes here.2702

Because we are going to use it for calculating the variance, I also listed Y².2707

That is what these numbers are here.2713

To find the mean, you add up the probabilities of each one × the value of Y.2715

Since, the probabilities were all equal, 1/8 everywhere, I just added up all these values of Y and I multiply by 1/8.2721

We get 12/8 is 3/2 is the mean.2731

Not surprising, if you flip a coin 3 ×, on average, you expect to get 3/2 heads.2734

Of course, you get a whole number of head that will be exactly 3/2.2740

In the long run, you will average 3/2 heads per experiment.2744

E of Y² is a preliminary step towards finding the variance.2749

It is because the variance formula is this E of Y² - (E of Y)².2755

We calculate E of Y² first, that is the same, you list the same probabilities but instead of looking at Y, you look at Y².2763

Instead of 3, we have 9.2771

Instead of 2, we have 4.2773

Those add up to 24/8 which simplifies down to 3.2776

I will not box on that because that is not our final answer.2779

To calculate the variance, σ² is one notation for the variance.2782

E of Y is another notation, we are calculating the same thing either way there.2787

We have this nice formula to calculate it.2792

V of Y² was the 3 and E of Y is the 3/2 and then if you just simplify that down, you get ¾.2794

Finally the standard deviation is easy, you just take the² root of the variance, assuming we already calculated that.2805

It is √ ¾ reduces down to √3/2.2810

As I mentioned, we have an easier way to calculate all these,2815

after we learn about the binomial distribution and all its important properties.2818

But that is a couple of lectures down the road.2823

In the meantime, we are just calculating things from scratch.2825

I hope that clears things up here, this is the end of our lecture on standard deviation and variance.2829

This is part of the probability lecture series here on www.educator.com and my name is Will Μurray, thank you for watching, bye.2836

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