Dr. Ji Son

Introduction to Probability Distributions

Slide Duration:

Section 1: Introduction
Descriptive Statistics vs. Inferential Statistics

25m 31s

Intro
0:00
0:10
0:11
Statistics
0:35
Statistics
0:36
Let's Think About High School Science
1:12
Measurement and Find Patterns (Mathematical Formula)
1:13
Statistics = Math of Distributions
4:58
Distributions
4:59
Problematic… but also GREAT
5:58
Statistics
7:33
How is It Different from Other Specializations in Mathematics?
7:34
Statistics is Fundamental in Natural and Social Sciences
7:53
Two Skills of Statistics
8:20
Description (Exploration)
8:21
Inference
9:13
Descriptive Statistics vs. Inferential Statistics: Apply to Distributions
9:58
Descriptive Statistics
9:59
Inferential Statistics
11:05
Populations vs. Samples
12:19
Populations vs. Samples: Is it the Truth?
12:20
Populations vs. Samples: Pros & Cons
13:36
Populations vs. Samples: Descriptive Values
16:12
Putting Together Descriptive/Inferential Stats & Populations/Samples
17:10
Putting Together Descriptive/Inferential Stats & Populations/Samples
17:11
Example 1: Descriptive Statistics vs. Inferential Statistics
19:09
Example 2: Descriptive Statistics vs. Inferential Statistics
20:47
Example 3: Sample, Parameter, Population, and Statistic
21:40
Example 4: Sample, Parameter, Population, and Statistic
23:28
Section 2: About Samples: Cases, Variables, Measurements

32m 14s

Intro
0:00
Data
0:09
Data, Cases, Variables, and Values
0:10
Rows, Columns, and Cells
2:03
Example: Aircrafts
3:52
How Do We Get Data?
5:38
Research: Question and Hypothesis
5:39
Research Design
7:11
Measurement
7:29
Research Analysis
8:33
Research Conclusion
9:30
Types of Variables
10:03
Discrete Variables
10:04
Continuous Variables
12:07
Types of Measurements
14:17
Types of Measurements
14:18
Types of Measurements (Scales)
17:22
Nominal
17:23
Ordinal
19:11
Interval
21:33
Ratio
24:24
Example 1: Cases, Variables, Measurements
25:20
Example 2: Which Scale of Measurement is Used?
26:55
Example 3: What Kind of a Scale of Measurement is This?
27:26
Example 4: Discrete vs. Continuous Variables.
30:31
Section 3: Visualizing Distributions
Introduction to Excel

8m 9s

Intro
0:00
Before Visualizing Distribution
0:10
Excel
0:11
Excel: Organization
0:45
Workbook
0:46
Column x Rows
1:50
Tools: Menu Bar, Standard Toolbar, and Formula Bar
3:00
Excel + Data
6:07
Exce and Data
6:08
Frequency Distributions in Excel

39m 10s

Intro
0:00
0:08
Data in Excel and Frequency Distributions
0:09
Raw Data to Frequency Tables
0:42
Raw Data to Frequency Tables
0:43
Frequency Tables: Using Formulas and Pivot Tables
1:28
Example 1: Number of Births
7:17
Example 2: Age Distribution
20:41
Example 3: Height Distribution
27:45
Example 4: Height Distribution of Males
32:19
Frequency Distributions and Features

25m 29s

Intro
0:00
0:10
Data in Excel, Frequency Distributions, and Features of Frequency Distributions
0:11
Example #1
1:35
Uniform
1:36
Example #2
2:58
Unimodal, Skewed Right, and Asymmetric
2:59
Example #3
6:29
Bimodal
6:30
Example #4a
8:29
Symmetric, Unimodal, and Normal
8:30
Point of Inflection and Standard Deviation
11:13
Example #4b
12:43
Normal Distribution
12:44
Summary
13:56
Uniform, Skewed, Bimodal, and Normal
13:57
17:34
Sketch Problem 2: Life Expectancy
20:01
Sketch Problem 3: Telephone Numbers
22:01
Sketch Problem 4: Length of Time Used to Complete a Final Exam
23:43
Dotplots and Histograms in Excel

42m 42s

Intro
0:00
0:06
0:07
Previously
1:02
Data, Frequency Table, and visualization
1:03
Dotplots
1:22
Dotplots Excel Example
1:23
Dotplots: Pros and Cons
7:22
Pros and Cons of Dotplots
7:23
Dotplots Excel Example Cont.
9:07
Histograms
12:47
Histograms Overview
12:48
Example of Histograms
15:29
Histograms: Pros and Cons
31:39
Pros
31:40
Cons
32:31
Frequency vs. Relative Frequency
32:53
Frequency
32:54
Relative Frequency
33:36
Example 1: Dotplots vs. Histograms
34:36
Example 2: Age of Pennies Dotplot
36:21
Example 3: Histogram of Mammal Speeds
38:27
Example 4: Histogram of Life Expectancy
40:30
Stemplots

12m 23s

Intro
0:00
0:05
0:06
What Sets Stemplots Apart?
0:46
Data Sets, Dotplots, Histograms, and Stemplots
0:47
Example 1: What Do Stemplots Look Like?
1:58
Example 2: Back-to-Back Stemplots
5:00
7:46
Example 4: Quiz Grade & Afterschool Tutoring Stemplot
9:56
Bar Graphs

22m 49s

Intro
0:00
0:05
0:08
Review of Frequency Distributions
0:44
Y-axis and X-axis
0:45
Types of Frequency Visualizations Covered so Far
2:16
Introduction to Bar Graphs
4:07
Example 1: Bar Graph
5:32
Example 1: Bar Graph
5:33
Do Shapes, Center, and Spread of Distributions Apply to Bar Graphs?
11:07
Do Shapes, Center, and Spread of Distributions Apply to Bar Graphs?
11:08
Example 2: Create a Frequency Visualization for Gender
14:02
Example 3: Cases, Variables, and Frequency Visualization
16:34
Example 4: What Kind of Graphs are Shown Below?
19:29
Section 4: Summarizing Distributions
Central Tendency: Mean, Median, Mode

38m 50s

Intro
0:00
0:07
0:08
Central Tendency 1
0:56
Way to Summarize a Distribution of Scores
0:57
Mode
1:32
Median
2:02
Mean
2:36
Central Tendency 2
3:47
Mode
3:48
Median
4:20
Mean
5:25
Summation Symbol
6:11
Summation Symbol
6:12
Population vs. Sample
10:46
Population vs. Sample
10:47
Excel Examples
15:08
Finding Mode, Median, and Mean in Excel
15:09
Median vs. Mean
21:45
Effect of Outliers
21:46
Relationship Between Parameter and Statistic
22:44
Type of Measurements
24:00
Which Distributions to Use With
24:55
Example 1: Mean
25:30
Example 2: Using Summation Symbol
29:50
Example 3: Average Calorie Count
32:50
Example 4: Creating an Example Set
35:46
Variability

42m 40s

Intro
0:00
0:05
0:06
0:45
0:46
5:45
5:46
Range, Quartiles and Interquartile Range
6:37
Range
6:38
Interquartile Range
8:42
Interquartile Range Example
10:58
Interquartile Range Example
10:59
Variance and Standard Deviation
12:27
Deviations
12:28
Sum of Squares
14:35
Variance
16:55
Standard Deviation
17:44
Sum of Squares (SS)
18:34
Sum of Squares (SS)
18:35
Population vs. Sample SD
22:00
Population vs. Sample SD
22:01
Population vs. Sample
23:20
Mean
23:21
SD
23:51
Example 1: Find the Mean and Standard Deviation of the Variable Friends in the Excel File
27:21
Example 2: Find the Mean and Standard Deviation of the Tagged Photos in the Excel File
35:25
Example 3: Sum of Squares
38:58
Example 4: Standard Deviation
41:48
Five Number Summary & Boxplots

57m 15s

Intro
0:00
0:06
0:07
Summarizing Distributions
0:37
0:38
5 Number Summary
1:14
Boxplot: Visualizing 5 Number Summary
3:37
Boxplot: Visualizing 5 Number Summary
3:38
Boxplots on Excel
9:01
Using 'Stocks' and Using Stacked Columns
9:02
Boxplots on Excel Example
10:14
When are Boxplots Useful?
32:14
Pros
32:15
Cons
32:59
How to Determine Outlier Status
33:24
Rule of Thumb: Upper Limit
33:25
Rule of Thumb: Lower Limit
34:16
Signal Outliers in an Excel Data File Using Conditional Formatting
34:52
Modified Boxplot
48:38
Modified Boxplot
48:39
Example 1: Percentage Values & Lower and Upper Whisker
49:10
Example 2: Boxplot
50:10
Example 3: Estimating IQR From Boxplot
53:46
Example 4: Boxplot and Missing Whisker
54:35
Shape: Calculating Skewness & Kurtosis

41m 51s

Intro
0:00
0:16
0:17
Skewness Concept
1:09
Skewness Concept
1:10
Calculating Skewness
3:26
Calculating Skewness
3:27
Interpreting Skewness
7:36
Interpreting Skewness
7:37
Excel Example
8:49
Kurtosis Concept
20:29
Kurtosis Concept
20:30
Calculating Kurtosis
24:17
Calculating Kurtosis
24:18
Interpreting Kurtosis
29:01
Leptokurtic
29:35
Mesokurtic
30:10
Platykurtic
31:06
Excel Example
32:04
Example 1: Shape of Distribution
38:28
Example 2: Shape of Distribution
39:29
Example 3: Shape of Distribution
40:14
Example 4: Kurtosis
41:10
Normal Distribution

34m 33s

Intro
0:00
0:13
0:14
What is a Normal Distribution
0:44
The Normal Distribution As a Theoretical Model
0:45
Possible Range of Probabilities
3:05
Possible Range of Probabilities
3:06
What is a Normal Distribution
5:07
Can Be Described By
5:08
Properties
5:49
'Same' Shape: Illusion of Different Shape!
7:35
'Same' Shape: Illusion of Different Shape!
7:36
Types of Problems
13:45
Example: Distribution of SAT Scores
13:46
Shape Analogy
19:48
Shape Analogy
19:49
Example 1: The Standard Normal Distribution and Z-Scores
22:34
Example 2: The Standard Normal Distribution and Z-Scores
25:54
Example 3: Sketching and Normal Distribution
28:55
Example 4: Sketching and Normal Distribution
32:32
Standard Normal Distributions & Z-Scores

41m 44s

Intro
0:00
0:06
0:07
A Family of Distributions
0:28
Infinite Set of Distributions
0:29
Transforming Normal Distributions to 'Standard' Normal Distribution
1:04
Normal Distribution vs. Standard Normal Distribution
2:58
Normal Distribution vs. Standard Normal Distribution
2:59
Z-Score, Raw Score, Mean, & SD
4:08
Z-Score, Raw Score, Mean, & SD
4:09
Weird Z-Scores
9:40
Weird Z-Scores
9:41
Excel
16:45
For Normal Distributions
16:46
For Standard Normal Distributions
19:11
Excel Example
20:24
Types of Problems
25:18
Percentage Problem: P(x)
25:19
Raw Score and Z-Score Problems
26:28
Standard Deviation Problems
27:01
Shape Analogy
27:44
Shape Analogy
27:45
Example 1: Deaths Due to Heart Disease vs. Deaths Due to Cancer
28:24
Example 2: Heights of Male College Students
33:15
Example 3: Mean and Standard Deviation
37:14
Example 4: Finding Percentage of Values in a Standard Normal Distribution
37:49
Normal Distribution: PDF vs. CDF

55m 44s

Intro
0:00
0:15
0:16
Frequency vs. Cumulative Frequency
0:56
Frequency vs. Cumulative Frequency
0:57
Frequency vs. Cumulative Frequency
4:32
Frequency vs. Cumulative Frequency Cont.
4:33
Calculus in Brief
6:21
Derivative-Integral Continuum
6:22
PDF
10:08
PDF for Standard Normal Distribution
10:09
PDF for Normal Distribution
14:32
Integral of PDF = CDF
21:27
Integral of PDF = CDF
21:28
Example 1: Cumulative Frequency Graph
23:31
Example 2: Mean, Standard Deviation, and Probability
24:43
Example 3: Mean and Standard Deviation
35:50
Example 4: Age of Cars
49:32
Section 5: Linear Regression
Scatterplots

47m 19s

Intro
0:00
0:04
0:05
Previous Visualizations
0:30
Frequency Distributions
0:31
Compare & Contrast
2:26
Frequency Distributions Vs. Scatterplots
2:27
Summary Values
4:53
Shape
4:54
Center & Trend
6:41
8:22
Univariate & Bivariate
10:25
Example Scatterplot
10:48
Shape, Trend, and Strength
10:49
Positive and Negative Association
14:05
Positive and Negative Association
14:06
Linearity, Strength, and Consistency
18:30
Linearity
18:31
Strength
19:14
Consistency
20:40
Summarizing a Scatterplot
22:58
Summarizing a Scatterplot
22:59
Example 1: Gapminder.org, Income x Life Expectancy
26:32
Example 2: Gapminder.org, Income x Infant Mortality
36:12
Example 3: Trend and Strength of Variables
40:14
Example 4: Trend, Strength and Shape for Scatterplots
43:27
Regression

32m 2s

Intro
0:00
0:05
0:06
Linear Equations
0:34
Linear Equations: y = mx + b
0:35
Rough Line
5:16
Rough Line
5:17
Regression - A 'Center' Line
7:41
Reasons for Summarizing with a Regression Line
7:42
Predictor and Response Variable
10:04
Goal of Regression
12:29
Goal of Regression
12:30
Prediction
14:50
Example: Servings of Mile Per Year Shown By Age
14:51
Intrapolation
17:06
Extrapolation
17:58
Error in Prediction
20:34
Prediction Error
20:35
Residual
21:40
Example 1: Residual
23:34
Example 2: Large and Negative Residual
26:30
Example 3: Positive Residual
28:13
Example 4: Interpret Regression Line & Extrapolate
29:40
Least Squares Regression

56m 36s

Intro
0:00
0:13
0:14
Best Fit
0:47
Best Fit
0:48
Sum of Squared Errors (SSE)
1:50
Sum of Squared Errors (SSE)
1:51
Why Squared?
3:38
Why Squared?
3:39
Quantitative Properties of Regression Line
4:51
Quantitative Properties of Regression Line
4:52
So How do we Find Such a Line?
6:49
SSEs of Different Line Equations & Lowest SSE
6:50
Carl Gauss' Method
8:01
How Do We Find Slope (b1)
11:00
How Do We Find Slope (b1)
11:01
Hoe Do We Find Intercept
15:11
Hoe Do We Find Intercept
15:12
Example 1: Which of These Equations Fit the Above Data Best?
17:18
Example 2: Find the Regression Line for These Data Points and Interpret It
26:31
Example 3: Summarize the Scatterplot and Find the Regression Line.
34:31
Example 4: Examine the Mean of Residuals
43:52
Correlation

43m 58s

Intro
0:00
0:05
0:06
Summarizing a Scatterplot Quantitatively
0:47
Shape
0:48
Trend
1:11
Strength: Correlation ®
1:45
Correlation Coefficient ( r )
2:30
Correlation Coefficient ( r )
2:31
Trees vs. Forest
11:59
Trees vs. Forest
12:00
Calculating r
15:07
Average Product of z-scores for x and y
15:08
Relationship between Correlation and Slope
21:10
Relationship between Correlation and Slope
21:11
Example 1: Find the Correlation between Grams of Fat and Cost
24:11
Example 2: Relationship between r and b1
30:24
Example 3: Find the Regression Line
33:35
Example 4: Find the Correlation Coefficient for this Set of Data
37:37
Correlation: r vs. r-squared

52m 52s

Intro
0:00
0:07
0:08
R-squared
0:44
What is the Meaning of It? Why Squared?
0:45
Parsing Sum of Squared (Parsing Variability)
2:25
SST = SSR + SSE
2:26
What is SST and SSE?
7:46
What is SST and SSE?
7:47
r-squared
18:33
Coefficient of Determination
18:34
If the Correlation is Strong…
20:25
If the Correlation is Strong…
20:26
If the Correlation is Weak…
22:36
If the Correlation is Weak…
22:37
Example 1: Find r-squared for this Set of Data
23:56
Example 2: What Does it Mean that the Simple Linear Regression is a 'Model' of Variance?
33:54
Example 3: Why Does r-squared Only Range from 0 to 1
37:29
Example 4: Find the r-squared for This Set of Data
39:55
Transformations of Data

27m 8s

Intro
0:00
0:05
0:06
Why Transform?
0:26
Why Transform?
0:27
Shape-preserving vs. Shape-changing Transformations
5:14
Shape-preserving = Linear Transformations
5:15
Shape-changing Transformations = Non-linear Transformations
6:20
Common Shape-Preserving Transformations
7:08
Common Shape-Preserving Transformations
7:09
Common Shape-Changing Transformations
8:59
Powers
9:00
Logarithms
9:39
Change Just One Variable? Both?
10:38
Log-log Transformations
10:39
Log Transformations
14:38
Example 1: Create, Graph, and Transform the Data Set
15:19
Example 2: Create, Graph, and Transform the Data Set
20:08
Example 3: What Kind of Model would You Choose for this Data?
22:44
Example 4: Transformation of Data
25:46
Section 6: Collecting Data in an Experiment
Sampling & Bias

54m 44s

Intro
0:00
0:05
0:06
Descriptive vs. Inferential Statistics
1:04
Descriptive Statistics: Data Exploration
1:05
Example
2:03
To tackle Generalization…
4:31
Generalization
4:32
Sampling
6:06
'Good' Sample
6:40
Defining Samples and Populations
8:55
Population
8:56
Sample
11:16
Why Use Sampling?
13:09
Why Use Sampling?
13:10
Goal of Sampling: Avoiding Bias
15:04
What is Bias?
15:05
Where does Bias Come from: Sampling Bias
17:53
Where does Bias Come from: Response Bias
18:27
Sampling Bias: Bias from Bas Sampling Methods
19:34
Size Bias
19:35
Voluntary Response Bias
21:13
Convenience Sample
22:22
Judgment Sample
23:58
25:40
Response Bias: Bias from 'Bad' Data Collection Methods
28:00
Nonresponse Bias
29:31
Questionnaire Bias
31:10
Incorrect Response or Measurement Bias
37:32
Example 1: What Kind of Biases?
40:29
Example 2: What Biases Might Arise?
44:46
Example 3: What Kind of Biases?
48:34
Example 4: What Kind of Biases?
51:43
Sampling Methods

14m 25s

Intro
0:00
0:05
0:06
Biased vs. Unbiased Sampling Methods
0:32
Biased Sampling
0:33
Unbiased Sampling
1:13
Probability Sampling Methods
2:31
Simple Random
2:54
Stratified Random Sampling
4:06
Cluster Sampling
5:24
Two-staged Sampling
6:22
Systematic Sampling
7:25
8:33
Example 2: Describe How to Take a Two-Stage Sample from this Book
10:16
Example 3: Sampling Methods
11:58
Example 4: Cluster Sample Plan
12:48
Research Design

53m 54s

Intro
0:00
0:06
0:07
Descriptive vs. Inferential Statistics
0:51
Descriptive Statistics: Data Exploration
0:52
Inferential Statistics
1:02
Variables and Relationships
1:44
Variables
1:45
Relationships
2:49
Not Every Type of Study is an Experiment…
4:16
Category I - Descriptive Study
4:54
Category II - Correlational Study
5:50
Category III - Experimental, Quasi-experimental, Non-experimental
6:33
Category III
7:42
Experimental, Quasi-experimental, and Non-experimental
7:43
Why CAN'T the Other Strategies Determine Causation?
10:18
Third-variable Problem
10:19
Directionality Problem
15:49
What Makes Experiments Special?
17:54
Manipulation
17:55
Control (and Comparison)
21:58
Methods of Control
26:38
Holding Constant
26:39
Matching
29:11
Random Assignment
31:48
Experiment Terminology
34:09
'true' Experiment vs. Study
34:10
Independent Variable (IV)
35:16
Dependent Variable (DV)
35:45
Factors
36:07
Treatment Conditions
36:23
Levels
37:43
Confounds or Extraneous Variables
38:04
Blind
38:38
Blind Experiments
38:39
Double-blind Experiments
39:29
How Categories Relate to Statistics
41:35
Category I - Descriptive Study
41:36
Category II - Correlational Study
42:05
Category III - Experimental, Quasi-experimental, Non-experimental
42:43
Example 1: Research Design
43:50
Example 2: Research Design
47:37
Example 3: Research Design
50:12
Example 4: Research Design
52:00
Between and Within Treatment Variability

41m 31s

Intro
0:00
0:06
0:07
Experimental Designs
0:51
Experimental Designs: Manipulation & Control
0:52
Two Types of Variability
2:09
Between Treatment Variability
2:10
Within Treatment Variability
3:31
Updated Goal of Experimental Design
5:47
Updated Goal of Experimental Design
5:48
Example: Drugs and Driving
6:56
Example: Drugs and Driving
6:57
Different Types of Random Assignment
11:27
All Experiments
11:28
Completely Random Design
12:02
Randomized Block Design
13:19
Randomized Block Design
15:48
Matched Pairs Design
15:49
Repeated Measures Design
19:47
Between-subject Variable vs. Within-subject Variable
22:43
Completely Randomized Design
22:44
Repeated Measures Design
25:03
Example 1: Design a Completely Random, Matched Pair, and Repeated Measures Experiment
26:16
Example 2: Block Design
31:41
Example 3: Completely Randomized Designs
35:11
Example 4: Completely Random, Matched Pairs, or Repeated Measures Experiments?
39:01
Section 7: Review of Probability Axioms
Sample Spaces

37m 52s

Intro
0:00
0:07
0:08
Why is Probability Involved in Statistics
0:48
Probability
0:49
Can People Tell the Difference between Cheap and Gourmet Coffee?
2:08
Taste Test with Coffee Drinkers
3:37
If No One can Actually Taste the Difference
3:38
If Everyone can Actually Taste the Difference
5:36
Creating a Probability Model
7:09
Creating a Probability Model
7:10
D'Alembert vs. Necker
9:41
D'Alembert vs. Necker
9:42
Problem with D'Alembert's Model
13:29
Problem with D'Alembert's Model
13:30
Covering Entire Sample Space
15:08
Fundamental Principle of Counting
15:09
Where Do Probabilities Come From?
22:54
Observed Data, Symmetry, and Subjective Estimates
22:55
Checking whether Model Matches Real World
24:27
Law of Large Numbers
24:28
Example 1: Law of Large Numbers
27:46
Example 2: Possible Outcomes
30:43
Example 3: Brands of Coffee and Taste
33:25
Example 4: How Many Different Treatments are there?
35:33

20m 29s

Intro
0:00
0:08
0:09
Disjoint Events
0:41
Disjoint Events
0:42
Meaning of 'or'
2:39
In Regular Life
2:40
In Math/Statistics/Computer Science
3:10
3:55
If A and B are Disjoint: P (A and B)
3:56
If A and B are Disjoint: P (A or B)
5:15
5:41
5:42
8:31
If A and B are not Disjoint: P (A or B)
8:32
Example 1: Which of These are Mutually Exclusive?
10:50
Example 2: What is the Probability that You will Have a Combination of One Heads and Two Tails?
12:57
Example 3: Engagement Party
15:17
Example 4: Home Owner's Insurance
18:30
Conditional Probability

57m 19s

Intro
0:00
0:05
0:06
'or' vs. 'and' vs. Conditional Probability
1:07
'or' vs. 'and' vs. Conditional Probability
1:08
'and' vs. Conditional Probability
5:57
P (M or L)
5:58
P (M and L)
8:41
P (M|L)
11:04
P (L|M)
12:24
Tree Diagram
15:02
Tree Diagram
15:03
Defining Conditional Probability
22:42
Defining Conditional Probability
22:43
Common Contexts for Conditional Probability
30:56
Medical Testing: Positive Predictive Value
30:57
Medical Testing: Sensitivity
33:03
Statistical Tests
34:27
Example 1: Drug and Disease
36:41
Example 2: Marbles and Conditional Probability
40:04
Example 3: Cards and Conditional Probability
45:59
Example 4: Votes and Conditional Probability
50:21
Independent Events

24m 27s

Intro
0:00
0:05
0:06
Independent Events & Conditional Probability
0:26
Non-independent Events
0:27
Independent Events
2:00
Non-independent and Independent Events
3:08
Non-independent and Independent Events
3:09
Defining Independent Events
5:52
Defining Independent Events
5:53
Multiplication Rule
7:29
Previously…
7:30
But with Independent Evens
8:53
Example 1: Which of These Pairs of Events are Independent?
11:12
Example 2: Health Insurance and Probability
15:12
Example 3: Independent Events
17:42
Example 4: Independent Events
20:03
Section 8: Probability Distributions
Introduction to Probability Distributions

56m 45s

Intro
0:00
0:08
0:09
Sampling vs. Probability
0:57
Sampling
0:58
Missing
1:30
What is Missing?
3:06
Insight: Probability Distributions
5:26
Insight: Probability Distributions
5:27
What is a Probability Distribution?
7:29
From Sample Spaces to Probability Distributions
8:44
Sample Space
8:45
Probability Distribution of the Sum of Two Die
11:16
The Random Variable
17:43
The Random Variable
17:44
Expected Value
21:52
Expected Value
21:53
Example 1: Probability Distributions
28:45
Example 2: Probability Distributions
35:30
Example 3: Probability Distributions
43:37
Example 4: Probability Distributions
47:20
Expected Value & Variance of Probability Distributions

53m 41s

Intro
0:00
0:06
0:07
Discrete vs. Continuous Random Variables
1:04
Discrete vs. Continuous Random Variables
1:05
Mean and Variance Review
4:44
Mean: Sample, Population, and Probability Distribution
4:45
Variance: Sample, Population, and Probability Distribution
9:12
Example Situation
14:10
Example Situation
14:11
Some Special Cases…
16:13
Some Special Cases…
16:14
Linear Transformations
19:22
Linear Transformations
19:23
What Happens to Mean and Variance of the Probability Distribution?
20:12
n Independent Values of X
25:38
n Independent Values of X
25:39
Compare These Two Situations
30:56
Compare These Two Situations
30:57
Two Random Variables, X and Y
32:02
Two Random Variables, X and Y
32:03
Example 1: Expected Value & Variance of Probability Distributions
35:35
Example 2: Expected Values & Standard Deviation
44:17
Example 3: Expected Winnings and Standard Deviation
48:18
Binomial Distribution

55m 15s

Intro
0:00
0:05
0:06
Discrete Probability Distributions
1:42
Discrete Probability Distributions
1:43
Binomial Distribution
2:36
Binomial Distribution
2:37
Multiplicative Rule Review
6:54
Multiplicative Rule Review
6:55
How Many Outcomes with k 'Successes'
10:23
Adults and Bachelor's Degree: Manual List of Outcomes
10:24
P (X=k)
19:37
Putting Together # of Outcomes with the Multiplicative Rule
19:38
Expected Value and Standard Deviation in a Binomial Distribution
25:22
Expected Value and Standard Deviation in a Binomial Distribution
25:23
Example 1: Coin Toss
33:42
38:03
Example 3: Types of Blood and Probability
45:39
Example 4: Expected Number and Standard Deviation
51:11
Section 9: Sampling Distributions of Statistics
Introduction to Sampling Distributions

48m 17s

Intro
0:00
0:08
0:09
Probability Distributions vs. Sampling Distributions
0:55
Probability Distributions vs. Sampling Distributions
0:56
Same Logic
3:55
Logic of Probability Distribution
3:56
Example: Rolling Two Die
6:56
Simulating Samples
9:53
To Come Up with Probability Distributions
9:54
In Sampling Distributions
11:12
Connecting Sampling and Research Methods with Sampling Distributions
12:11
Connecting Sampling and Research Methods with Sampling Distributions
12:12
Simulating a Sampling Distribution
14:14
Experimental Design: Regular Sleep vs. Less Sleep
14:15
Logic of Sampling Distributions
23:08
Logic of Sampling Distributions
23:09
General Method of Simulating Sampling Distributions
25:38
General Method of Simulating Sampling Distributions
25:39
Questions that Remain
28:45
Questions that Remain
28:46
Example 1: Mean and Standard Error of Sampling Distribution
30:57
Example 2: What is the Best Way to Describe Sampling Distributions?
37:12
Example 3: Matching Sampling Distributions
38:21
Example 4: Mean and Standard Error of Sampling Distribution
41:51
Sampling Distribution of the Mean

1h 8m 48s

Intro
0:00
0:05
0:06
Special Case of General Method for Simulating a Sampling Distribution
1:53
Special Case of General Method for Simulating a Sampling Distribution
1:54
Computer Simulation
3:43
Using Simulations to See Principles behind Shape of SDoM
15:50
Using Simulations to See Principles behind Shape of SDoM
15:51
Conditions
17:38
Using Simulations to See Principles behind Center (Mean) of SDoM
20:15
Using Simulations to See Principles behind Center (Mean) of SDoM
20:16
Conditions: Does n Matter?
21:31
Conditions: Does Number of Simulation Matter?
24:37
Using Simulations to See Principles behind Standard Deviation of SDoM
27:13
Using Simulations to See Principles behind Standard Deviation of SDoM
27:14
Conditions: Does n Matter?
34:45
Conditions: Does Number of Simulation Matter?
36:24
Central Limit Theorem
37:13
SHAPE
38:08
CENTER
39:34
39:52
Comparing Population, Sample, and SDoM
43:10
Comparing Population, Sample, and SDoM
43:11
48:24
What Happens When We Don't Know What the Population Looks Like?
48:25
Can We Have Sampling Distributions for Summary Statistics Other than the Mean?
49:42
How Do We Know whether a Sample is Sufficiently Unlikely?
53:36
Do We Always Have to Simulate a Large Number of Samples in Order to get a Sampling Distribution?
54:40
Example 1: Mean Batting Average
55:25
Example 2: Mean Sampling Distribution and Standard Error
59:07
Example 3: Sampling Distribution of the Mean
1:01:04
Sampling Distribution of Sample Proportions

54m 37s

Intro
0:00
0:06
0:07
Intro to Sampling Distribution of Sample Proportions (SDoSP)
0:51
Categorical Data (Examples)
0:52
Wish to Estimate Proportion of Population from Sample…
2:00
Notation
3:34
Population Proportion and Sample Proportion Notations
3:35
What's the Difference?
9:19
SDoM vs. SDoSP: Type of Data
9:20
SDoM vs. SDoSP: Shape
11:24
SDoM vs. SDoSP: Center
12:30
15:34
Binomial Distribution vs. Sampling Distribution of Sample Proportions
19:14
Binomial Distribution vs. SDoSP: Type of Data
19:17
Binomial Distribution vs. SDoSP: Shape
21:07
Binomial Distribution vs. SDoSP: Center
21:43
24:08
Example 1: Sampling Distribution of Sample Proportions
26:07
Example 2: Sampling Distribution of Sample Proportions
37:58
Example 3: Sampling Distribution of Sample Proportions
44:42
Example 4: Sampling Distribution of Sample Proportions
45:57
Section 10: Inferential Statistics
Introduction to Confidence Intervals

42m 53s

Intro
0:00
0:06
0:07
Inferential Statistics
0:50
Inferential Statistics
0:51
Two Problems with This Picture…
3:20
Two Problems with This Picture…
3:21
Solution: Confidence Intervals (CI)
4:59
Solution: Hypotheiss Testing (HT)
5:49
Which Parameters are Known?
6:45
Which Parameters are Known?
6:46
Confidence Interval - Goal
7:56
When We Don't Know m but know s
7:57
When We Don't Know
18:27
When We Don't Know m nor s
18:28
Example 1: Confidence Intervals
26:18
Example 2: Confidence Intervals
29:46
Example 3: Confidence Intervals
32:18
Example 4: Confidence Intervals
38:31
t Distributions

1h 2m 6s

Intro
0:00
0:04
0:05
When to Use z vs. t?
1:07
When to Use z vs. t?
1:08
What is z and t?
3:02
z-score and t-score: Commonality
3:03
z-score and t-score: Formulas
3:34
z-score and t-score: Difference
5:22
Why not z? (Why t?)
7:24
Why not z? (Why t?)
7:25
But Don't Worry!
15:13
Gossett and t-distributions
15:14
Rules of t Distributions
17:05
t-distributions are More Normal as n Gets Bigger
17:06
t-distributions are a Family of Distributions
18:55
Degrees of Freedom (df)
20:02
Degrees of Freedom (df)
20:03
t Family of Distributions
24:07
t Family of Distributions : df = 2 , 4, and 60
24:08
df = 60
29:16
df = 2
29:59
How to Find It?
31:01
'Student's t-distribution' or 't-distribution'
31:02
Excel Example
33:06
Example 1: Which Distribution Do You Use? Z or t?
45:26
47:41
Example 3: t Distributions
52:15
Example 4: t Distributions , confidence interval, and mean
55:59
Introduction to Hypothesis Testing

1h 6m 33s

Intro
0:00
0:06
0:07
Issues to Overcome in Inferential Statistics
1:35
Issues to Overcome in Inferential Statistics
1:36
What Happens When We Don't Know What the Population Looks Like?
2:57
How Do We Know whether a sample is Sufficiently Unlikely
3:43
Hypothesizing a Population
6:44
Hypothesizing a Population
6:45
Null Hypothesis
8:07
Alternative Hypothesis
8:56
Hypotheses
11:58
Hypotheses
11:59
Errors in Hypothesis Testing
14:22
Errors in Hypothesis Testing
14:23
Steps of Hypothesis Testing
21:15
Steps of Hypothesis Testing
21:16
Single Sample HT ( When Sigma Available)
26:08
26:09
Step1
27:08
Step 2
27:58
Step 3
28:17
Step 4
32:18
Single Sample HT (When Sigma Not Available)
36:33
36:34
Step1: Hypothesis Testing
36:58
Step 2: Significance Level
37:25
Step 3: Decision Stage
37:40
Step 4: Sample
41:36
Sigma and p-value
45:04
Sigma and p-value
45:05
On tailed vs. Two Tailed Hypotheses
45:51
Example 1: Hypothesis Testing
48:37
Example 2: Heights of Women in the US
57:43
Example 3: Select the Best Way to Complete This Sentence
1:03:23
Confidence Intervals for the Difference of Two Independent Means

55m 14s

Intro
0:00
0:14
0:15
One Mean vs. Two Means
1:17
One Mean vs. Two Means
1:18
Notation
2:41
A Sample! A Set!
2:42
Mean of X, Mean of Y, and Difference of Two Means
3:56
SE of X
4:34
SE of Y
6:28
Sampling Distribution of the Difference between Two Means (SDoD)
7:48
Sampling Distribution of the Difference between Two Means (SDoD)
7:49
Rules of the SDoD (similar to CLT!)
15:00
Mean for the SDoD Null Hypothesis
15:01
Standard Error
17:39
When can We Construct a CI for the Difference between Two Means?
21:28
Three Conditions
21:29
Finding CI
23:56
One Mean CI
23:57
Two Means CI
25:45
Finding t
29:16
Finding t
29:17
Interpreting CI
30:25
Interpreting CI
30:26
Better Estimate of s (s pool)
34:15
Better Estimate of s (s pool)
34:16
Example 1: Confidence Intervals
42:32
Example 2: SE of the Difference
52:36
Hypothesis Testing for the Difference of Two Independent Means

50m

Intro
0:00
0:06
0:07
The Goal of Hypothesis Testing
0:56
One Sample and Two Samples
0:57
Sampling Distribution of the Difference between Two Means (SDoD)
3:42
Sampling Distribution of the Difference between Two Means (SDoD)
3:43
Rules of the SDoD (Similar to CLT!)
6:46
Shape
6:47
Mean for the Null Hypothesis
7:26
Standard Error for Independent Samples (When Variance is Homogenous)
8:18
Standard Error for Independent Samples (When Variance is not Homogenous)
9:25
Same Conditions for HT as for CI
10:08
Three Conditions
10:09
Steps of Hypothesis Testing
11:04
Steps of Hypothesis Testing
11:05
Formulas that Go with Steps of Hypothesis Testing
13:21
Step 1
13:25
Step 2
14:18
Step 3
15:00
Step 4
16:57
Example 1: Hypothesis Testing for the Difference of Two Independent Means
18:47
Example 2: Hypothesis Testing for the Difference of Two Independent Means
33:55
Example 3: Hypothesis Testing for the Difference of Two Independent Means
44:22
Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means

1h 14m 11s

Intro
0:00
0:09
0:10
The Goal of Hypothesis Testing
1:27
One Sample and Two Samples
1:28
Independent Samples vs. Paired Samples
3:16
Independent Samples vs. Paired Samples
3:17
Which is Which?
5:20
Independent SAMPLES vs. Independent VARIABLES
7:43
independent SAMPLES vs. Independent VARIABLES
7:44
T-tests Always…
10:48
T-tests Always…
10:49
Notation for Paired Samples
12:59
Notation for Paired Samples
13:00
Steps of Hypothesis Testing for Paired Samples
16:13
Steps of Hypothesis Testing for Paired Samples
16:14
Rules of the SDoD (Adding on Paired Samples)
18:03
Shape
18:04
Mean for the Null Hypothesis
18:31
Standard Error for Independent Samples (When Variance is Homogenous)
19:25
Standard Error for Paired Samples
20:39
Formulas that go with Steps of Hypothesis Testing
22:59
Formulas that go with Steps of Hypothesis Testing
23:00
Confidence Intervals for Paired Samples
30:32
Confidence Intervals for Paired Samples
30:33
Example 1: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
32:28
Example 2: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
44:02
Example 3: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means
52:23
Type I and Type II Errors

31m 27s

Intro
0:00
0:18
0:19
Errors and Relationship to HT and the Sample Statistic?
1:11
Errors and Relationship to HT and the Sample Statistic?
1:12
7:00
One Sample t-test: Friends on Facebook
7:01
Two Sample t-test: Friends on Facebook
13:46
Usually, Lots of Overlap between Null and Alternative Distributions
16:59
Overlap between Null and Alternative Distributions
17:00
How Distributions and 'Box' Fit Together
22:45
How Distributions and 'Box' Fit Together
22:46
Example 1: Types of Errors
25:54
Example 2: Types of Errors
27:30
Example 3: What is the Danger of the Type I Error?
29:38
Effect Size & Power

44m 41s

Intro
0:00
0:05
0:06
Distance between Distributions: Sample t
0:49
Distance between Distributions: Sample t
0:50
Problem with Distance in Terms of Standard Error
2:56
Problem with Distance in Terms of Standard Error
2:57
Test Statistic (t) vs. Effect Size (d or g)
4:38
Test Statistic (t) vs. Effect Size (d or g)
4:39
Rules of Effect Size
6:09
Rules of Effect Size
6:10
Why Do We Need Effect Size?
8:21
Tells You the Practical Significance
8:22
HT can be Deceiving…
10:25
Important Note
10:42
What is Power?
11:20
What is Power?
11:21
Why Do We Need Power?
14:19
Conditional Probability and Power
14:20
Power is:
16:27
Can We Calculate Power?
19:00
Can We Calculate Power?
19:01
How Does Alpha Affect Power?
20:36
How Does Alpha Affect Power?
20:37
How Does Effect Size Affect Power?
25:38
How Does Effect Size Affect Power?
25:39
How Does Variability and Sample Size Affect Power?
27:56
How Does Variability and Sample Size Affect Power?
27:57
How Do We Increase Power?
32:47
Increasing Power
32:48
Example 1: Effect Size & Power
35:40
Example 2: Effect Size & Power
37:38
Example 3: Effect Size & Power
40:55
Section 11: Analysis of Variance
F-distributions

24m 46s

Intro
0:00
0:04
0:05
Z- & T-statistic and Their Distribution
0:34
Z- & T-statistic and Their Distribution
0:35
F-statistic
4:55
The F Ration ( the Variance Ratio)
4:56
F-distribution
12:29
F-distribution
12:30
s and p-value
15:00
s and p-value
15:01
Example 1: Why Does F-distribution Stop At 0 But Go On Until Infinity?
18:33
Example 2: F-distributions
19:29
Example 3: F-distributions and Heights
21:29
ANOVA with Independent Samples

1h 9m 25s

Intro
0:00
0:05
0:06
The Limitations of t-tests
1:12
The Limitations of t-tests
1:13
Two Major Limitations of Many t-tests
3:26
Two Major Limitations of Many t-tests
3:27
Ronald Fisher's Solution… F-test! New Null Hypothesis
4:43
Ronald Fisher's Solution… F-test! New Null Hypothesis (Omnibus Test - One Test to Rule Them All!)
4:44
Analysis of Variance (ANoVA) Notation
7:47
Analysis of Variance (ANoVA) Notation
7:48
Partitioning (Analyzing) Variance
9:58
Total Variance
9:59
Within-group Variation
14:00
Between-group Variation
16:22
Time out: Review Variance & SS
17:05
Time out: Review Variance & SS
17:06
F-statistic
19:22
The F Ratio (the Variance Ratio)
19:23
S²bet = SSbet / dfbet
22:13
What is This?
22:14
How Many Means?
23:20
So What is the dfbet?
23:38
So What is SSbet?
24:15
S²w = SSw / dfw
26:05
What is This?
26:06
How Many Means?
27:20
So What is the dfw?
27:36
So What is SSw?
28:18
Chart of Independent Samples ANOVA
29:25
Chart of Independent Samples ANOVA
29:26
Example 1: Who Uploads More Photos: Unknown Ethnicity, Latino, Asian, Black, or White Facebook Users?
35:52
Hypotheses
35:53
Significance Level
39:40
Decision Stage
40:05
Calculate Samples' Statistic and p-Value
44:10
Reject or Fail to Reject H0
55:54
Example 2: ANOVA with Independent Samples
58:21
Repeated Measures ANOVA

1h 15m 13s

Intro
0:00
0:05
0:06
The Limitations of t-tests
0:36
Who Uploads more Pictures and Which Photo-Type is Most Frequently Used on Facebook?
0:37
ANOVA (F-test) to the Rescue!
5:49
Omnibus Hypothesis
5:50
Analyze Variance
7:27
Independent Samples vs. Repeated Measures
9:12
Same Start
9:13
Independent Samples ANOVA
10:43
Repeated Measures ANOVA
12:00
Independent Samples ANOVA
16:00
Same Start: All the Variance Around Grand Mean
16:01
Independent Samples
16:23
Repeated Measures ANOVA
18:18
Same Start: All the Variance Around Grand Mean
18:19
Repeated Measures
18:33
Repeated Measures F-statistic
21:22
The F Ratio (The Variance Ratio)
21:23
S²bet = SSbet / dfbet
23:07
What is This?
23:08
How Many Means?
23:39
So What is the dfbet?
23:54
So What is SSbet?
24:32
S² resid = SS resid / df resid
25:46
What is This?
25:47
So What is SS resid?
26:44
So What is the df resid?
27:36
SS subj and df subj
28:11
What is This?
28:12
How Many Subject Means?
29:43
So What is df subj?
30:01
So What is SS subj?
30:09
SS total and df total
31:42
What is This?
31:43
What is the Total Number of Data Points?
32:02
So What is df total?
32:34
so What is SS total?
32:47
Chart of Repeated Measures ANOVA
33:19
Chart of Repeated Measures ANOVA: F and Between-samples Variability
33:20
Chart of Repeated Measures ANOVA: Total Variability, Within-subject (case) Variability, Residual Variability
35:50
Example 1: Which is More Prevalent on Facebook: Tagged, Uploaded, Mobile, or Profile Photos?
40:25
Hypotheses
40:26
Significance Level
41:46
Decision Stage
42:09
Calculate Samples' Statistic and p-Value
46:18
Reject or Fail to Reject H0
57:55
Example 2: Repeated Measures ANOVA
58:57
Example 3: What's the Problem with a Bunch of Tiny t-tests?
1:13:59
Section 12: Chi-square Test
Chi-Square Goodness-of-Fit Test

58m 23s

Intro
0:00
0:05
0:06
Where Does the Chi-Square Test Belong?
0:50
Where Does the Chi-Square Test Belong?
0:51
A New Twist on HT: Goodness-of-Fit
7:23
HT in General
7:24
Goodness-of-Fit HT
8:26
12:17
Null Hypothesis
12:18
Alternative Hypothesis
13:23
Example
14:38
Chi-Square Statistic
17:52
Chi-Square Statistic
17:53
Chi-Square Distributions
24:31
Chi-Square Distributions
24:32
Conditions for Chi-Square
28:58
Condition 1
28:59
Condition 2
30:20
Condition 3
30:32
Condition 4
31:47
Example 1: Chi-Square Goodness-of-Fit Test
32:23
Example 2: Chi-Square Goodness-of-Fit Test
44:34
Example 3: Which of These Statements Describe Properties of the Chi-Square Goodness-of-Fit Test?
56:06
Chi-Square Test of Homogeneity

51m 36s

Intro
0:00
0:09
0:10
Goodness-of-Fit vs. Homogeneity
1:13
Goodness-of-Fit HT
1:14
Homogeneity
2:00
Analogy
2:38
5:00
Null Hypothesis
5:01
Alternative Hypothesis
6:11
Example
6:33
Chi-Square Statistic
10:12
Same as Goodness-of-Fit Test
10:13
Set Up Data
12:28
Setting Up Data Example
12:29
Expected Frequency
16:53
Expected Frequency
16:54
Chi-Square Distributions & df
19:26
Chi-Square Distributions & df
19:27
Conditions for Test of Homogeneity
20:54
Condition 1
20:55
Condition 2
21:39
Condition 3
22:05
Condition 4
22:23
Example 1: Chi-Square Test of Homogeneity
22:52
Example 2: Chi-Square Test of Homogeneity
32:10
Section 13: Overview of Statistics
Overview of Statistics

18m 11s

Intro
0:00
0:07
0:08
The Statistical Tests (HT) We've Covered
0:28
The Statistical Tests (HT) We've Covered
0:29
Organizing the Tests We've Covered…
1:08
One Sample: Continuous DV and Categorical DV
1:09
Two Samples: Continuous DV and Categorical DV
5:41
More Than Two Samples: Continuous DV and Categorical DV
8:21
The Following Data: OK Cupid
10:10
The Following Data: OK Cupid
10:11
Example 1: Weird-MySpace-Angle Profile Photo
10:38
Example 2: Geniuses
12:30
Example 3: Promiscuous iPhone Users
13:37
Example 4: Women, Aging, and Messaging
16:07
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 0 answersPost by Azhar Rahman on June 16, 2013My lecturer gave the formula for Expected values as E(X)=Âµ=âˆ‘x.f(x) with an x under the sum of symbol which i couldnt enter. Why is there a difference? 0 answersPost by James Ulatowski on December 30, 2011On The Random Variable lesson of Intro to Probability Distributions you said for the greater of two die that 6 is 11/36 "because you don't count 6 twice". As you know, you don't count any of the doubles twice, resulting in the odd number. Considering the sample space:if you point out that these are simply the result of adding the rows and columns of the occurrence of each number -without double counting the double-it is a lot clearer as to how the resulting probabilities are obtained.

### Introduction to Probability Distributions

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
• Sampling vs. Probability 0:57
• Sampling
• Missing
• What is Missing?
• Insight: Probability Distributions 5:26
• Insight: Probability Distributions
• What is a Probability Distribution?
• From Sample Spaces to Probability Distributions 8:44
• Sample Space
• Probability Distribution of the Sum of Two Die
• The Random Variable 17:43
• The Random Variable
• Expected Value 21:52
• Expected Value
• Example 1: Probability Distributions 28:45
• Example 2: Probability Distributions 35:30
• Example 3: Probability Distributions 43:37
• Example 4: Probability Distributions 47:20

### Transcription: Introduction to Probability Distributions

Hi and welcome to www.educator .com.0000

Today we are going to be talking about probability distributions.0002

We are just going to start talking about them.0006

So far we have covered sampling method as well as a little bit about basics of fundamental of probability.0012

In probability distribution we are going to be playing those two ideas together and work on that so we can get what we want out of them.0019

Because of that we need to talk about what sampling and probability cannot do.0030

They cannot tell us something and because of that we need probability distribution and0035

that is going to be the solution to what sampling and probability alone cannot accomplish.0040

With the probability distribution we are going to be talking about random variable and what that means and the expected value is.0047

We know a little bit about sampling and a little bit about probability, basically in sampling we know that0061

what we are trying to do is estimate some variable by taking the sample of the population.0065

Here is the population of all the people we are interested in and we take a sample out and0071

we look at those individuals to try and estimate what that variable is like in the population.0079

We know a little bit about the probability now in a particular population, we know the chance of getting a specific sample.0092

What is the probability of getting this particular sample.0105

You might want to ask yourself what is the relationship between these two things.0111

It is where we are getting the sample and we are looking at the mean and the standard deviation of some particular variable in order to estimate that variable.0116

In probability, what we are doing is it is not that we are actually taking a sample but we have a population.0130

We are trying to think about all the different kinds of samples, the sample space and then calculate the probability of the specific sample.0139

In one we are actually taking the sample and the other we are not actually taking a sample.0153

Here there are some limitations because even if we try to take a random sample, one that is not biased in some way.0159

Even though we try to take a sample we do not necessarily know how close that sample is to the population.0169

Here we have taken a sample but we know the probability of getting that particular sample.0176

We are still missing one piece of the puzzle, neither sampling nor probability helps us know the actual characteristics,0186

Things like shape and spread of the population.0194

We can know how likely this sample is given the population.0199

We could take our sample out and summarize that sample.0203

But we do not know how to understand the population if we do not already know what it is.0209

What do we do?0216

Here is where the probability distributions are going to come in.0219

In probability distributions, what we are going to do is take this population and look at the distribution of all the different possible samples and get their probabilities.0232

This way it is not that we are calculating one probability for one particular sample but we are calculating all the possible sample and all the probability that go along with it.0259

That is the probability distribution.0271

We are no longer dealing with single probability or single samples but we are looking at the universe of all the sample0274

and the probability that go with them that might happen from a low population.0282

It is not necessary to know or we definitely know that this is a population but it could be a theorize population like a fair sighted point.0292

Those could be theoretical model.0309

That is the known population, we have sense of the population and from that we win from it,0311

all the possible sample and all the possibilities that go with the sample.0321

Here is why we want to do that.0329

First we use probability, the fundamental principles of probability to figure out what samples from real populations look like.0332

We would want to do it from one sample at a time now we are going to do it for all the possible samples.0341

We figure out what the distribution of samples looks like then we take a sample from a different unknown population.0347

Here we have known population and now we get all samples and corresponding probability of those samples.0357

Then we have this unknown one, unknown population, and we take a sample.0377

What we do is we compare this sample to the results of the samples from the known population.0387

Why this sample like this is our theorize population?0394

Why is this sample unlikely in this universe of all possible samples from such a know population.0399

If this sample is highly like given that we know that this is probability then we could say these two guys are similar.0407

But if this sample is very unlikely then maybe we could say perhaps these two population are not similar.0417

This one is different from this known population.0428

That is the insight, we are going to use these probability distributions in order to help us figure out samples0432

that come from population to try and figure out what the unknown population is like.0440

It helps us to find what a probability distribution is.0450

These are all the samples that can be drawn from an a known population and their corresponding probabilities.0456

When you think about probability distributions, I also want you to think of what it is called a sampling distribution.0489

What it is really is, it is not a distribution of a single entity or people or companies.0502

It is not a distribution of single data point.0511

Instead it is a sampling distribution.0515

It is a distribution made up of a whole bunch of samples.0519

Now that we know what probability distributions are, for and why we need them so desperately.0529

Let us try to smoothen out the road from probability to probability distributions.0535

In probability we learn about sample spaces.0546

Remember when we are talking about sample spaces, we are talking about all the different possible samples that we could have.0550

When we talk about flipping two coins and we look at what is the probability of getting 2 heads in a row?0557

We created these sample spaces of all the possible different outcomes.0565

For instance, if we look at the sample space for 2 dice we might look at something like this, dice 1, 2.0579

We might have 1, 2, 3, 4, 5, 6 and 1, 2, 3, 4, 5.0597

If we look at the sample space that would be 1-1, 1-2, 1-3, 1-4, 1-5, 1-6.0610

Or it might be 2-6, 3-6, 4-6, 5-6, 6-6.0622

If you fill in all the different possible outcomes there are 36 possible outcomes.0632

If you fill in the rest of this table you could see the entire sample space.0642

Each of these roll of a dice presuming that this dice are fair, each of these we have a probability of 1 out of 36.0647

The likelihood of this out of all the possible outcomes is 36.0659

That would be looking at the sample space.0668

That is the probabilities of all the different outcomes.0672

When we talk about probability distributions, often we are talking about just one number and the distribution around that one variable, whatever that variable is.0677

Here notice that we have 2 numbers but we may want something like the sum of 2 dice.0690

Here the sum would be 2 and the sum would be 3 and the sum would be 4.0699

Maybe we want the probability distribution that looks something like this.0706

Here I will put all the different possible sums.0713

I may not have room for this so I might draw them separately.0717

For instance, can I have a sum of 0?0720

No, I cannot have a sum of 1 either.0724

If I roll 2 dice the lowest possible sum I could have is 2, a 1 and a 1.0727

I could have a sum of 2, 3, 4, 5, 6, 7 but I could also have, I will continue this on this side.0733

It will also have sums of 8, 9, 10, 11, 12.0747

I cannot have any sum higher than 12 because the 6 and 6 on each dice is the highest possible sum I could have.0752

My probability distribution would be what is the probability of the sum of 2?0764

That would be only one outcome out of all 36 outcomes would give you that sum of 2.0773

That would be 1 out of 36.0783

The same thing with 12, the only possible combination that I could get a 12 is this 6 and 6 right there.0792

That would be also 1 out of 36.0800

Now let us think about 3, how many rolls of the dice could possibly add up to 3?0803

1 and 2 but also 2 and 1.0813

Any numbers higher than that is not going to work.0818

Here we would write 2/36 because 2 different rolls could give us the same sum.0823

Notice that not all of the probabilities are equal.0833

This one is less probable and this is more probable.0836

For 4, now we are starting to get into some bigger numbers here.0844

1 and 3 will work but so is 2 and 2, as well as 3 and 1.0849

That is going to be 3/36.0859

Notice the pattern here so far?0863

Also notice the pattern here is going like this and like this.0866

Let us continue that pattern and see if it is correct.0872

Our prediction would be that the sum for 5 would be 4/36 and let us see if that is true.0876

If the sums combine and so thus 4 and 1.0883

Let us see if these diagonals also do.0891

These diagonals would be 2 and 3, as well as 3 and 2.0893

Again, we know we could see our rule generalizing here.0899

That is going to be 4/36.0904

Actually this ends up working all the way up to 7.0910

As 7, this would be going to be the longest diagonal.0917

I will that in right here, 6 and 1, but also 5 and 2, 4 and 3, 3 and 4, 2 and 5, and also 1 and 6.0922

All 6 of these here on the diagonal will add up to a sum of 7.0949

That is why in games like stocks 7 is the highest probability rule.0957

Is another game where 7 is the highest probability roll that is why it is not allowed.0964

Here, you will see that what happens after these diagonal is that it becomes smaller again but in a perfectly symmetrical way.0976

What goes up? 1, 2, 3, 4, 5, 6 comes down 5, 4, 3, 2, 1.0985

That is because you are just finding those diagonals but now it is going from big to smaller and smaller then you will end up here the 1/36.1000

This is probability distributions for us to start with because it is easy for us to see how we got these probabilities from these sums.1013

This is a relatively small sample space and a small probability distribution we could fill in.1026

There are going to be bigger spaces lined up each knot necessarily calculated by looking at the different combination and1033

we will look at some of the algorithms and theorems that have been developed in order that we could have shortcuts so that we do not have to look at the entire sample space.1041

But I want you to know where this comes from.1054

It all comes from the sample space and looking at the probabilities.1056

In the previous probability distribution we looked at the sum of the probability distributions of two dice.1065

In that case, the random variable of one single variable that we are interested in on this probability distribution is the sum of the two dice.1075

That is one example of a random variable.1085

It is the thinking here, the thing that you are finding the probability of, the one variable.1088

It does not have to be sum, it could be something else.1099

For instance, in other games you might be interested in the probability distribution of the greater of the 2 dice.1103

Here we have another probability distribution where instead of the sum I am looking for the probabilities of those sums.1110

We choose the greater number and the probability of the greater number.1119

Here I have just rewritten what I have written on the previous page except nice and neatly.1130

One thing I want you to notice is if you add up all of these probabilities they add up to 1.1135

That is good because this shows me that we have covered the entire scan of the different outcomes as well as the different probabilities.1141

There is no part of the sample space that have not has been untouched.1155

You have touched all of it.1159

Here we see once again as the sums go up 7 will be the highest probability roll of the highest probability combination.1162

It starts going down after 7.1174

2 and 1 is perfectly symmetrical so that these sides are just the same and 7 is the mirror point.1178

If we are looking at we roll some dice and we are looking at what is the probability that the greatest number there out of the two is 1?1189

There is only one case of that, when it is 1 and 1 the greater number is 1.1198

That is 1/36.1207

But notice that this does not goes up and down because here what is the probability that when you roll a dice that the greatest number there will be 2?1210

That is only going to happen in case of 1 and 2, 2 and 1, 2 and 2.1221

That is 3/36.1230

Let us get down to 6.1233

6 is the highest number so it is going to be the greatest number at the time where all of the combinations that are like 6 and 1, 6 and 2, 6 and 3, 6 and 4.1235

As well as 1 and 6, 2 and 6, 3 and 6.1248

6 is frequently the highest number but we do not count 6 and 6 twice which is why it is 11/36.1255

Even so we add all of the probabilities we have a total of 1 and that shows me we have covered the entire space.1262

These are samples of probability distributions of two different random variables.1271

One is the random variable of sum and the other is the random variable of greater number.1276

We could have a random variable like which is the product, two numbers multiplied together.1284

We could have all sorts of different random variables that we are interested in.1298

The fact is that we think upon which the probability distribution is being decided.1301

That is the thing you want to find the probabilities of.1308

Now we know the variables and now we know the probability distributions, what is expected value?1315

Here is the thing, once you a probability distribution like this, this is that rolling two dice and what is the probability of the sums?1321

Here once you have this probability distribution of the sums, that is our random variable, and we often call our random variable X.1332

We could also call it I or M or Y but we call it big letters.1342

It would be handy to know the mean of this probability distribution because every distribution has a mean.1351

It would be nice to know what the mean of this distribution is.1360

That is often called the expected value, the mean of the probability distribution or the mean of the sampling distributions.1365

The reason it is called the expected value is this.1375

Over time if you keep rolling two dice over and over again, this probability should emerge, the law of large numbers.1378

Over 10,000,000 rolls of two dice, we should see the frequencies corresponding to this probability.1389

It would be nice to know what we should expect even without rolling a dice 10,000,000 times.1401

It would be nice to know what the mean would be on average.1410

In order to find the expected value, what we are doing here is what we did average because these two is not quite as frequent as the 7.1419

These two should not count as much as the 7.1431

This 12 although it is a bigger number it should not count as much as 7 because it is only going to have a small portion of a time.1433

What we are doing is we are going to not only pick the sum and put it in our estimate but we are going to weight it by how likely we need it.1441

If it is very unlikely it only contributes a little bit but it is very likely then it contributes a lot to the expected value.1455

Out of those contributions you should be able to see what is the expected value over time?1475

What is going to be the most likely value of this random variable?1483

In order to find the expected value, you will often see it notated like this, the expected value of x, this random variable.1492

Another notation for this is mu sub X.1505

The reason why we think of it as a mu is that its probability distributions are theoretical distributions.1512

Remember, theoretical distributions are more like populations than samples.1518

It is mu sub X, it is not just mu.1524

If it is only mu it would be the population.1528

When you see mu sub X this means this is the expected value.1531

This can be easily found by taking the sum from i(whatever your x are).1538

In this case this is my x because the sum is my random variable × how likely it is.1551

We need to do this for however many x we have.1569

Here what we see is that if x contributes a lot because the probability is very high then this is going to be a bigger number.1573

If x only contributes a little bit its value is going to be a little bit diminished.1596

When we add all of those up, we hear the course of who is the loudest?1602

That is our expected value.1609

Some people contribute a lot and some people contribute a little bit when we add all of them up what we find is the true story.1612

In order to find this, it is going to be helpful to use Excel.1622

If you want to download the available Excel file I have put in all the sums as well as the probability.1625

I have just put in 1/36 but Excel will put it in decimal form for you.1638

What we want to do is you want to think of this multiply together as the contribution of how much is it contributing to the expected value?1645

In order to do that, we would take this x and multiply it by the probability of the x.1654

This 2 contributes a little bit but the 7 contributes quite a bit.1664

It is much bigger than this 2.1675

It goes down again where the 12 only contributes a little bit.1681

In order to find expected value we add all of these up.1686

We add all of these contributions up.1690

Let us see what is left.1693

Our expected value or the mu sub X is 7.1696

Over many rolls of the dice what we will see is on average the expected value is 7, that is the mean of this probability distribution.1701

That expected value.1723

Now let us move on to some examples.1727

You have all the tools of the game.1731

In example 1, at the state fair you could play a fish for cash, a game of chance that costs $1 to play.1733 You blind the fish out of cart that has a dollar amount that you have won from a giant fish bowl.1739 The games have these probabilities of winning posted on the wall.1745 Is it worth playing this game?1749 Here we the winnings that you could potentially earn.1751 You could win from$1 all the way up to $900.1755 Here are your chances of winning.1759 You have 1 out of 10 chance of earning back your$1 and you have 1/120,000 of winning $900.1761 You are getting a lot of bang for your buck.1774 Is it worth playing this game?1777 I have put all of this information on example 1 tab, these are the winnings and I just put them in here and Excel has changed it in decimal form.1781 The first thing you want to do is check whether all of this probability actually add up to 1.1796 Remember these are posted on the wall.1803 Even out there they are telling you the whole story.1806 Let us sum these probabilities up and make sure that we cover the entire space of winnings and probability of those winnings.1810 These probabilities only cover 22% of that probability space and there is the probability of 1 a 100% total.1824 Let us think.1835 What must be missing?1837 I think the game probably does not advertise that you count the probability of winning that game.1842 There is that probability and they are probably not just telling you.1848 Let us see.1852 I am going to take all of that and add that over here.1859 In this row I am going to put 0 as the potential winning and the probability of getting 0 should be 1 – the sum of all the rest of the possible outcomes.1871 That is almost 78%.1894 You have likely to win nothing but once we do this, if we add up all of these including the 0 now we should have a total of 1.1898 That is good and we want that.1911 This shows us this is now a complete probability distribution of the random variable winnings.1914 What is the contribution of each of these winnings to this expected value?1925 Over many plays of this game, what is the average winning?1931 Let us look at each one contribution and then multiply that out and we sum this up.1941 When we look at this sum here, you will see$.60 as the expected value.1959

Even though there is this chance of winning $900 as well as$.60.1965

It turned out that on average if you play the game over and over again, the average winning is going to be about $.60.1970 You can win$.60 on average so the expected value of winnings or you can write it as mu (w) is $.60.1982 Is it worth it playing this game?2001 That costs you a buck to play the game so if you play the game over and over again, let us say a 100 times that is going to cost you$100.2005

But over time if you multiply this by a 100 you are going to win $60 for every$100 you spend.2016

That is not worth it.2027

Over the long call you are going to be losing money.2030

It does not matter that much if you are just going to play the game once.2033

It is not going to tell you whether that part that you picked is going to be $.60 because remember there is no possible way you could earn$.60.2043

This is not what that means, probably it is going to be 0 because it is 79%.2053

4 out of 5 times you are going to be drawing 0.2060

It is not about any 1 particular turn.2064

You might think that is useless.2069

Actually it is not.2073

Move yourself around and put yourself in the seat of the guy who owns this game, fish for catch.2075

You want to know the expected value.2083

You are the owner of the game.2085

You want to know on average, all of these different people are going to play are you going to be losing money at the end of the day or you are going to be grateful you did not?2088

What the order of this game would say is this is a good game because if people play even though I will lose $.60 for every$1 roughly on average, I will be gaining $.40.2097 That is my profit.2113 This is not really for just single people.2116 We are not trying to predict single events but we are trying to predict events over time and over many examples.2122 Here is example 2.2131 According to our recent government report only 16% of occupants in trucks wear seat belts, supposed you randomly sample 3 occupants of trucks2134 what is the random variable and which doubt that this 16% estimate if not of the 3 is wearing seat belts?2146 Let us see.2155 In order to create a probability distribution we want to decide on what the random variable should be.2157 We eventually want to know what if none of the 3 wear seat belts?2164 Maybe we want to know how many seat belts and passengers?2170 Seat belt and passengers.2173 We have 0 seat belts and passengers that is our none of the 3.2179 We could have 1, 2, or 3.2183 We also want to know what is the probability of these seat belted passengers?2185 Now we want to figure this out.2198 If this was like head or tails, all of these combinations we have equal probability but2201 this all have equal probability because only 16% of occupants of trucks say they wear seat belts.2209 Although it is useful to have the sample space this is not going to be enough.2216 First let us just look at the sample space.2227 Here is 1, 2, and 3, and I will put in s for seat belt and n for no seat belts.2230 Half of these are seat belted and half are not.2245 Here is our sample space and we see that the 3 people all wear seat belts and that is 1/8 of the sample space.2261 This is also 1/8 of the entire sample space.2273 The story is not that simple because that would only be if wearing a seat belt and not wearing a seat belt or equally likely.2278 That is not the case.2286 These are independent events so we could use the multiplication rule.2289 Let us say we have these 3 spaces what is the probability that this guy here is not wearing a seat belt?2296 That is going to be 84%.2302 16% wearing a seat belt and the other side of that point is 84%.2304 That is the probability for each of these seat belts because think of them as independent events.2316 For number 3 where all 3 people are nice seat belt wearing, well abiding citizens in this truck that is going to be 163.2324 These are a little more complicated.2344 Here let us think about this, what is the probability that one person is wearing a seat belt?2346 This got one of the seats but these people are not.2352 This is not much more likely also in the sample space.2364 There are 3 of these.2370 Here is one, here is another one, there is another one.2372 As we are using the addition rule, we would add these 3 times but we could also just multiply it by 3 to make it easier.2380 What about if 2 people are wearing seat belts and 1 person is not?2391 That would be .16 for the 2 people wearing seat belt that is the probability that2396 2 people are wearing seat belts × .84 the probability that they would not be wearing seat belts.2407 Here we see that there are also 3 cases.2416 We could get the addition rule and add these 3 times but we could just multiply it by 3.2423 Once we do that then we could see the actual probability.2431 Here we do not need to multiply by anything because there is only 1.2436 It is like multiplying by 1.2440 I am just going to calculate this in my Excel file so I could put in .843 so that is .84 × .84 × .84.2446 That is 59.27%.2466 This should be .5927.2479 We could do the other one just to finish it up 2 × .1, 3 × .16 × .84 × .84.2488 The next one is 3 × .16 × .16 × .84.2510 The last one is .163.2523 We see that it is very likely where all 3 of them are not wearing a seat belt.2528 It is less likely than this but still pretty likely about 34% of the time that your sample have 1% wearing seat belts.2538 Less likely that two people would be wearing seat belt that is only like 6%.2551 If there is less than 1% chance that all 3 people will be wearing a seat belt.2555 Would you tend to doubt the 16% estimate if none of the 3 were wearing a seat belt?2581 We know that if we took a sample 59% at the time that sample will have 0% wearing seat belts.2587 It is likely that sample is consistent with this estimate of 16% of occupants in trucks do not wear seat belts.2599 We would not necessarily doubt that 16% occupants.2609 We might still be wrong but we do not have reason to doubt it.2612 Here is example 3, Apple is going to get you to buy the Apple care warranty for$250 for your laptop.2620

If you buy this Apple care thing you will get unlimited number of free repairs but if you do not you must pay $150 per repair.2627 You find this information below on the web.2642 What is the random variable here and according to this data is this worth getting the warranty?2646 Here you want to predict how likely it is going to be that I am going to need a bunch of repairs?2653 If I need only 1 repair or 0 repair the I probably should not buy this warranty thing but if I am going to need to get repair frequently then I probably want to buy the warranty.2660 Here we could see the probability of the repairs listed here and here are the repairs.2675 This probability distribution has the random variable of repairs.2682 According to this data is it worth getting this warranty?2688 One thing we want to do is to look and see what the expected value is.2693 On average, over million people are buying these laptops what is likely the value of repairs.2699 How many repairs are we going to need on average?2710 I have put this data on this table here in example 3, it is the same table.2714 It would be nice to find the expected value of x.2725 In this case x is repairs.2731 In order to find that, we want to find the contribution of each value of this random variable.2735 It is this times how likely it is.2749 0 contributes more than 4 because it is much more frequent that people need only 0 repairs.2753 Then we add these all up.2769 Remember our expected value is repairs.2773 We on average will expect about 1.1 repairs.2777 No one person will need 1.1 repairs they might need 1 repair or 2.2784 On average, it will average up on 1.1.2790 Remember this is on repairs, we do not know how much it is in dollar amounts.2794 If we want it to know in dollar amount, we know that each repair if we identify the warranty will cost$150.2798

That will be just this times $150.2807 On average we would expect to spend$155 is that going to make Apple care worth it?2812

I will say according to this data it is highly likely that you will need an average 1 repair.2822

It is probably not worth getting Apple care.2832

Here is example 4, there are 5 television shows you want to watch before your class that will begin 74 minutes later.2842

If you randomly pick 2 shows to watch out of this 5, what is the probability that you will have finish watching both shows before your class?2849

It is helpful here to keep in line the questions even though it is not asked here.2859

What is the random variable?2863

Do not be tricked because here it shows the shows A, B, C, D, and E, and it shows the minutes how long each shows is.2866

It does not show you the probability distribution at all.2881

In fact it is not like that there is a probability distribution for A.2886

How like is A?2890

Equally like as B, C, and D if we just pick randomly.2892

What we want to know is how many minutes 2 shows will be when added together?2896

Picking 2 shows and getting the sum of those 2 shows.2905

Eventually what we want to have is sum of the length of 2 shows.2912

We want all the different combinations likely AB, AC, AD, and all their sums and then the probabilities of those sums.2926

That is going to take more room than what we have here so if you pull out this Excel file and go to example 4.2939

I have put this information on here.2952

What you want to do is first start off by just looking at the sample space.2955

The sample space might look something like this.2964

First we want all of the combinations with A as the first show we picked.2967

B will be the second show.2976

C will be the second show.2980

Here we have D as the second show.2983

Maybe it is helpful if we label these.2985

First show and then first show × second show.2986

We have all these different things.2999

Later what we can do is get the sum by adding this and this.3002

We could do that later.3011

Let us create the sample space.3013

The next combination should be all the combinations with B as the first show and A will be the second show.3015

C, D, E as the second show.3032

Next we have all the combinations where C as the first show and then A and B are the second show D and E are the second show.3036

Let us go down to D being the first show and you could have A or B as the first show.3055

C as the first show, A and B as the second show, C as the second show, or E as the second show.3066

Presumably you do not want to watch A after you just watched A.3074

Finally, we have E as the first show, A, B, C, D as the second show.3082

We have all these different shows.3094

All the combinations and we could quite easily calculate the sums.3097

These are all the sum total minutes.3107

What we want to do is sort this by sum because eventually we want a probability distribution that will use sum.3111

Here I wanted to sort by sum because eventually we want to have probability distribution that have just these sum on it.3135

We do not care if 63 was created by watching B and D or D and B.3145

We just care that it adds up to 63 minutes.3151

I am going to go over here and start my probability distribution.3158

Here I will put the sum and I have a couple of different sums here.3163

For instance I have 63 that is the sum that come up, 65 comes up, 68 comes up, 70, 71, 73, 75, 78.3168

I want to know how likely it is 63 given all of these potential different combinations.3190

These are all equally probable because each show has 1 out of 5 chance of being picked as the first show and 1 out of 4 chance being picked as the second show.3198

Here what we will do is we need a formula that we could put countif and we could say in this range we could lock it down and it is not going to change for us.3220

In this range countif is it is 63.3235

Let us put all of that over however many different combinations there are.3240

That is all of this.3247

I think there is actually 20 combinations.3251

In this way we can have Excel do the work for us.3253

I am locking that down and once I do that it says that 2 out of 20 that ends up being 1 out of 10 which is .1.3258

What we can do is just copy and paste this all the way down.3271

We find that most of these are equally probable like 63, 65 minutes as the sum but 68 minutes is twice as likely because it is 4 out of 20.3275

Twice as likely as this 70 minutes or 71 minutes.3291

Here I will put probability of these particular sums.3297

What ends up being helpful is to look back at the questions and say now that we have what we want to know here?3303

If you randomly pick 2 shows to watch what is the probability that you will have finish watching both shows before your class.3311

Let us say you just arrive at your class you could use all 74 minutes to watch.3318

The cut off would be around here.3324

We could watch all these different combinations and still be under 74 minutes.3326

These are the only combinations that are not under 75 minutes.3334

The way that we would write this in probable notation is p where x is the sum in this case is less than 74 minutes.3339

You could just add these probabilities up.3365

In order to get that and we could put the sums and that should give us 80% because3369

the only 2 we are leaving out are these and each of those have a probability of 10%.3382

If you randomly pick shows to watch there is 80% probability that you will finish watching both of those shows before you go on to your class.3387

Nice and helpful of probability to help this figure out our TV watching schedule.3397

Thanks for using www.educator.com.3403

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