Professor Yates

Combining Independent Random Variables

Slide Duration:

Section 1: Introduction
Basic Ideas

17m 34s

Intro
0:00
Basic Definitions
0:09
Element (member, unit)
0:20
Variable
1:01
Observation (measurement)
1:18
Data Set
1:40
Example: Basic Definitions
1:55
Qualitative Variables
4:58
Quantitative Variables
6:16
Discrete Variable
6:33
Continuous Variable
7:36
Cross Section vs Time Series Data
8:58
Summation Notation
10:50
Summation Notation 2
12:59
Summation Notation 3
15:32
Section 2: Exploring Data
Raw Data, Dotplots, Stemplots

27m 24s

Intro
0:00
Raw Data
0:07
Ungrouped Data
0:25
Example: Ages
0:39
Features of Graphical Displays of Distributions
1:28
1:54
Clusters and Gaps
2:04
Outliers (extreme values)
2:12
Symmetric
2:48
Skewed
3:14
Uniform
3:47
Dotplots
4:58
Example: Dotplots
8:51
Stemplot
11:12
Stem and Leaf
11:17
Example: Stemplot
15:18
Extra Example 1
-1
Extra Example 2
-2
Histograms, Cumulative Frequency Plots

10m 21s

Intro
0:00
Features of Graphical Displays of Distributions
0:07
Histogram
3:03
Common Programs
3:09
Example: Histogram
6:14
Cumulative Frequency Plot
7:43
Example: Cumulative Frequency Plot
8:16
Summarizing Distributions, Measuring Center

16m 4s

Intro
0:00
Measures of Central Tendency
0:08
Mean (average)
0:28
Mean for Population Data
0:51
Mean for Sample Data
1:18
Example: Mean
1:57
Example: Mean
2:49
Median
3:53
Example: Median
4:52
Example: Median
6:47
Mode
8:01
Unimodal
8:11
Bimodal
8:19
Multimodal
8:24
Example: Mode
8:34
Example: Mode
9:53
Effect of Changing Units
10:31
Extra Example 1
-1
Extra Example 2
-2
Measuring Spread: Range, IQR, Standard Deviation

18m 4s

Intro
0:00
0:08
Range
1:06
Example
1:16
Example
1:35
Standard Deviation
2:05
Population Standard Deviation
2:14
Sample Standard Deviation
3:13
Example: Standard Deviation
4:11
Example: Standard Deviation
6:05
Interquartile Range (IQR)
8:05
Example: Interquartile Range
9:03
Example: Interquartile Range
10:27
Extra Example 1
-1
Extra Example 2
-2
Measuring Position: Quartiles, Percentiles, Standardized Scores

16m 28s

Intro
0:00
Measure of Position
0:09
Quartile, Percentile, Z-Scores
0:24
Quartiles (Q1, Q2, Q3)
0:32
Example
0:51
Example: Quartiles
1:28
Example: Quartiles
3:27
Percentiles
5:44
Example: Percentiles
6:19
Example: Percentiles
7:24
Standardized Score (Z-Score)
8:27
Example: Standardized Score
9:23
Example: Standardized Score
10:21
Extra Example 1
-1
Extra Example 2
-2
Boxplots

15m 37s

Intro
0:00
What is a Boxplot?
0:05
Five Number Summary
0:15
Example: Boxplot
0:30
Example: Boxplot
4:33
Extra Example 1
-1
Extra Example 2
-2
Comparing Distributions of Univariate Data

24m 19s

Intro
0:00
Comparing Features
0:07
0:11
Compare Clusters & Gaps
0:23
Compare Outliers and Unusual Features
0:33
Compare Shapes
0:55
Symmetric
1:00
Skewed Right
1:20
Skewed Left
1:31
Uniform
1:41
Example: Dotplots
1:56
Example: Back to Back Stemplots
5:16
Example: Parallel Boxplots
10:21
Example: Back to Back Stemplots
15:03
Extra Example 1
-1
Extra Example 2
-2
Exploring Bivariate Data: Scatterplots

13m 45s

Intro
0:00
Bivariate Data
0:08
Example: Student Scores
0:31
Example: Scatterplot
1:08
Example: Scatterplot
2:36
Correlation and Linearity
3:49
Example: Correlation
5:30
Example: Correlation
6:55
Extra Example 1
-1
Extra Example 2
-2
Least Squares Regression

17m 32s

Intro
0:00
Least Squares Regression Line
0:06
Why Least Squares?
0:25
Equations
1:21
Example 1: Age and Price
2:02
Example 2: Weld Diameter
5:47
Diagnostics
8:39
Residuals
8:58
Normal Probability Plot
10:09
Studentized Residuals (Hat Matrix)
10:29
Transformations
10:48
Logarithmic Transformation
11:04
Square Root Transformation
11:44
Extra Example 1
-1
Extra Example 2
-2
Exploring Categorical Data

17m

Intro
0:00
Frequency Tables
0:05
Example: Student Age
0:16
Relative Frequency
0:55
Bar Graphs
1:59
Marginal and Joint Probabilities
3:54
Example 1: Gender and Beer
6:52
Conditional Probabilities
8:47
Example 2: Gender and Beer
11:41
Extra Example 1
-1
Extra Example 2
-2
Section 3: Sampling and Experimentation
Methods of Data Collection

12m 4s

Intro
0:00
Purpose
0:05
Census
1:22
Example: US Census
1:36
Example: Fireworks Factory
2:34
Sample Survey
3:41
Experiment
6:12
Example: Coke vs Pepsi
7:09
Observational Study
8:19
Observational or Experiment
9:30
Example 1
9:53
Example 2
10:24
Example 3
11:17
Planning and Conducting Surveys

13m 51s

Intro
0:00
Ideal Surveys
0:06
Random Selection
0:16
Characteristics of Surveys
0:42
Chance
0:50
Random Samples
1:02
No Source of Bias
1:32
Populations, Samples, Random Selection
2:21
Population
2:28
Sample
2:51
Sources of Bias
4:14
Example
4:33
Sampling Methods
7:27
Simple Random Sampling (SRS)
7:40
Example
8:33
Stratified Random Sampling (Strata)
10:03
Example
11:06
Cluster Sampling
12:19
Example
13:06
Planning and Conducting Experiments

19m 32s

Intro
0:00
Purpose
0:06
Characteristics
1:00
Basic Terms
2:00
Treatment
2:12
Control Group
2:30
Experimental Units
3:17
Random Assignment
3:38
Replication
4:19
Sources of Bias and Confounding
4:48
Counfounding
5:00
Example
5:29
Placebo Effect
6:41
Example
7:08
Blinding
7:56
Example
8:24
Completely Randomized Design
9:12
Randomized Block Design
12:44
Block
12:55
Matched Pairs
13:22
Example
13:41
Randomized Block Design
15:09
Example
15:30
Studies and Surveys vs Experiments
17:03
Section 4: Probability
Experiment, Outcomes, and Sample Space

14m 54s

Intro
0:00
Basic Definitions
0:29
Experiment
0:35
Outcomes
0:55
Sample Space
1:04
Examples
1:34
Roll a Die
1:39
Flip a Coin
2:33
Simple and Compound Events
3:30
Event
3:43
Simple Event
3:58
Compound Event
4:27
Example
5:14
Extra Example 1
-1
Extra Example 2
-2
Calculating Probability

14m 13s

Intro
0:00
What is Probability
0:27
Range
0:53
Sum of Probabilities
1:26
Example: Football Game
2:05
Classical Probability
2:53
Equally Likely Outcomes
3:05
Example: Coin Flip
4:08
Example: Die Roll
5:12
Relative Frequency
6:44
Example
7:22
Subjective Probability
9:38
Example
10:06
Extra Example 1
-1
Extra Example 2
-2
Probability and Events

22m 8s

Intro
0:00
Mutually Exclusive Events
0:17
Example: Coin Flip
0:27
Example: Die Roll
3:03
Independent Events
5:13
Notation
3:31
Example: Coin
6:01
Independent Events, cont.
9:19
Example: Coffee Drinkers
9:23
Mutually Exclusive vs Independent
13:03
Complementary Events
14:08
Example: Coffee Drinkers
15:37
Extra Example 1
-1
Extra Example 2
-2
Intersection of Events and the Multiplication Rule

19m 58s

Intro
0:00
Intersection of Events
0:08
Venn Diagram
1:20
Multiplication Rule
2:22
Joint Probability
2:23
Example
3:23
Example
6:30
Multiplication Rule for Independent Events
10:30
Example
11:39
Joint Probability of Mutually Exclusive Events
15:24
Extra Example 1
-1
Extra Example 2
-2
Union of Events and the Addition Rule

18m 28s

Intro
0:00
Union of Events
0:06
Venn Diagram
0:52
2:01
Example: Coffee Drinkers
3:25
Example
6:26
Addition Rule for Mutually Exclusive Events
9:11
Example
10:27
Extra Example 1
-1
Extra Example 2
-2
Bayes' Rule

16m 59s

Intro
0:00
Partition of Events
0:07
Venn Diagram
0:17
Law of Total Probability
3:12
Bayes' Rule
6:11
Example
9:09
Extra Example 1
-1
Section 5: Discrete Random Variables and Probability Distribution
Random Variables

7m 52s

Intro
0:00
Definition
0:06
Example
0:24
Discrete Random Variables
1:22
Example
1:56
Continuous Random Variable
3:53
Example
4:12
Extra Example 1
-1
Probability Distribution of a Discrete Random Variable

15m 55s

Intro
0:00
Definition
0:09
Example
0:24
Rules of a Probability Distribution
3:27
Rule 1
3:33
Rule 2
4:30
Example 1
4:59
Example 2
6:00
Example 3
6:38
Example: Defective DVDs
7:19
Extra Example 1
-1
Extra Example 2
-2
Mean and Standard Deviation of a Discrete Random Variable

17m 37s

Intro
0:00
Mean of a Discrete Random Variable
0:10
Example
1:17
Example: Numbers Game
3:09
Standard Deviation of a Discrete Random Variable
6:02
Example
7:46
Example: Cars in a Town
10:12
Extra Example 1
-1
Extra Example 2
-2
Factorials, Combinations, Permutations

15m 43s

Intro
0:00
Counting Rule
0:08
Example: Coin Toss
0:56
Example: Football Team
1:45
Factorials
2:54
Example
3:39
Zero Factorial
4:03
Example
4:20
Combinations
5:16
Example
6:23
Permutations
8:16
Example
9:01
Extra Example 1
-1
Extra Example 2
-2
Binomial Probability Distribution

21m 38s

Intro
0:00
Binomial Experiment
0:07
Discrete Random Variable
0:34
Trial
1:01
Bernoulli Trials
1:26
Example: Roll Die
2:37
Binomial Probability Distribution
4:36
Example: Winter Holiday Stress
6:58
Example: MRI
9:51
Probability of Success and Shape
12:42
Symmetric
12:54
Skewed Right
13:23
Skewed Left
14:13
Mean/Standard Deviation of Binomial Distribution
15:03
Example: Stress
16:06
Example: MRI
17:07
Extra Example 1
-1
Extra Example 2
-2
Poisson Probability Distribution

13m 40s

Intro
0:00
Poisson Probability Distribution
0:06
Conditions
0:43
Example: Complaints
3:18
5:01
Mean/Standard Deviation of Poisson Distribution
7:52
Example: Complaints
8:53
9:46
Extra Example 1
-1
Extra Example 2
-2
Geometric and Hypergeometric Probability Distributions

19m 8s

Intro
0:00
Geometric Probability Distribution
0:08
Example: Engine Malfunction
3:00
Example: Interviews
5:45
Hypergeometric Probability Distribution
7:36
Example: Engineers
10:16
Example: Marbles
12:55
Extra Example 1
-1
Extra Example 2
-2
Combining Independent Random Variables

20m 26s

Intro
0:00
Independence vs Dependence
0:09
Mean of Sums for Independent Random Variables
2:32
Example
4:02
Example
5:58
Variance for Sums of Independent Random Variables
8:49
Example
10:30
Example
12:26
Extra Example 1
-1
Extra Example 2
-2
Section 6: Continuous Random Variables and the Normal Distribution
Continuous Probability Distribution

6m 19s

Intro
0:00
Continuous Random Variable
0:07
Probability Density Function
0:54
3:07
4:06
Normal Distribution

6m 42s

Intro
0:00
Normal Distribution
0:05
Bell Shaped Curve
0:09
Properties of the Normal Distribution
1:02
Area Under the Curve (Density Curve)
1:07
1:40
Two Tails
2:21
Normal Distribution
3:07
Different Means
3:10
Different Standard Deviations
4:32
Standard Normal Distribution

13m 25s

Intro
0:00
Standard Normal Distribution
0:06
Z-Scores
1:08
Examples
1:57
More Examples
4:43
More Examples
7:12
Extra Example 1
-1
Extra Example 2
-2
Standardizing a Normal Distribution

12m 22s

Intro
0:00
Standardizing a Normal Distribution
0:07
Mean and Standard Deviation of X
1:13
Examples
1:39
More Examples
3:22
More Examples
6:17
Extra Example 1
-1
Extra Example 2
-2
Applications of the Normal Distribution

12m 20s

Intro
0:00
Standardizing a Normal Distribution
0:08
Example: US Debt
0:59
Example: Toy Assembly
3:19
Example: Soda
5:04
Example: Calculator
7:27
Extra Example 1
-1
Extra Example 2
-2
Finding Values When the Probability is Known

12m 44s

Intro
0:00
Example 1
0:10
Example 2
1:32
Example 3
3:12
Example 4: Battery Life
4:18
Example 5: SAT Scores
6:33
Extra Example 1
-1
Extra Example 2
-2
Section 7: Sampling Distributions
Population and Sampling Distributions

12m 2s

Intro
0:00
Population Distribution
0:06
Example: Teaching Experience
0:14
Sampling Distribution
1:31
Example: Teaching Experience
2:16
Sampling Error
5:29
Random and No Non-Sampling Error
6:00
Example
6:10
Non-Sampling Error
7:22
Example
7:38
Example: Six Numbers
9:17
Mean, Standard Deviation, and the Shape of the Sampling Distribution of the Sampling Mean

4m 57s

Intro
0:00
Mean/Standard Deviation of Sample Mean
0:10
Estimator
0:57
Unbiased Estimator
1:15
Sampling Distribution of Sample Mean
1:50
1:53
Standard Deviation
2:18
Consistent Estimator
2:40
Shape of Sampling Distribution
2:51
Normal
3:21
Shape of Sampling Distribution, cont.
3:50
Central Limit Theorem
4:15
Applications of the Sampling Distribution of the Sample Mean

14m 50s

Intro
0:00
Example 1: Speed Limit
0:08
Example 2: Speed Limit
2:50
Example 3: Speed Limit
4:20
Example 4: Study Times
6:20
Example 5: Study Times
9:02
Extra Example 1
-1
Extra Example 2
-2
Mean, Standard Deviation, and the Shape of the Sampling Distribution of the Sample Proportion

3m 58s

Intro
0:00
Population vs Sample Proportions
0:10
Population Proportion
0:16
Sample Proportion
0:23
Sample: Eye Color
0:36
Mean/Standard Deviation of Sample Proportion
1:47
Mean
1:51
Unbiased Estimator
2:07
Standard Deviation
2:28
Shape of the Distribution
3:07
Applications of the Sampling Distribution of the Sample Proportion

10m 45s

Intro
0:00
Example 1: Retirement Plan
0:07
Example 2: Retirement Plan
3:04
Example 3: Voters
4:35
Extra Example 1
-1
Extra Example 2
-2
Section 8: Estimation of the Mean and Proportion
Introduction to Estimation

12m 52s

Intro
0:00
Estimation
0:06
Parameter
0:29
Estimate
1:02
Estimator
1:10
Example
1:20
Steps for Estimation
2:21
Example: Dartboard
3:08
Consistent/Bias
3:41
Inconsistent/Unbiased
4:09
Consistent/Unbiased
4:44
Point Estimate
5:33
Example
5:50
Interval Estimate
6:35
Margin of Error
7:15
Confidence Interval
7:35
Confidence Level
7:55
Example
8:10
More on Confidence Intervals
10:18
Confidence Level Increase
11:41
Sample Size Increase
12:25
Estimation of a Population Mean: Standard Deviation Known

17m 3s

Intro
0:00
Population is Normal, n<30
0:10
Confidence Interval
0:28
Example 1
2:34
Example 2
5:54
When n>30, Any Distribution
7:58
Confidence Interval
8:48
Example 3
9:14
Example 4
11:16
Extra Example 1
-1
Extra Example 2
-2
Sample Size for Estimation of a Population Mean

10m 39s

Intro
0:00
Determining Sample Size
0:07
Finding n
0:30
Origin of Equation
0:56
Example 1
2:16
Example 2
4:42
Extra Example 1
-1
Extra Example 2
-2
Estimation of Population Mean: Sigma Not Known

19m 25s

Intro
0:00
t-Distribution
0:10
Examples: t-Distribution
0:38
Using the t-Distribution
4:25
Confidence Interval
5:03
Example 1: Waiting Time
5:54
Example 2: MPG
9:35
Extra Example 1
-1
Extra Example 2
-2
Estimation of Population Proportion: Large Sample

17m 26s

Intro
0:00
Population vs Sample Proportion
0:10
Confidence Intervals for p
1:50
Example 1: Credit
2:18
Example 2: Time
4:59
Sample Size for the Estimation of p
7:31
Margin of Error
7:55
Conservative Estimate
8:17
Example 3: Gambling
8:40
Example 4: Clocks
10:53
Extra Example 1
-1
Extra Example 2
-2
Large Sample Confidence Intervals for Difference in Population Proportion

16m 16s

Intro
0:00
Sampling Distribution for Difference in Sample Proportion
0:08
Large and Independent Samples
0:11
Confidence Intervals for p1-p2
1:28
Example 1: Toothpaste
2:04
Example 2: Seat Belts
6:20
Extra Example 1
-1
Extra Example 2
-2
Confidence Intervals for a Difference in Means

27m 58s

Intro
0:00
Independent Samples: Standard Deviations Known
0:07
Confidence Interval for Difference of Means
1:12
Example 1: Starting Salary
1:35
Example 2: Fill
5:36
Independent Samples: Standard Deviations Not Known
7:54
Pooled Standard Deviation for Two Samples
8:46
Confidence Interval for Difference of Means
9:32
Example 3: Caffeine
10:35
Example 4: Test Scores
15:20
Inference about Difference of Means for Paired Samples
19:05
Paired or Matched Sample
19:21
Inference about Difference of Means for Paired Samples
20:58
Extra Example 1
-1
Extra Example 2
-2
Confidence Intervals for the Slope of a Least Squares Regression Line

18m 47s

Intro
0:00
Sampling Distribution of b
0:08
Calculating the Estimator of Standard Deviation of b
1:03
Confidence Interval for Beta
1:31
Example 1: Age and Price
2:24
Example 2: Weld Diameter
6:41
Extra Example 1
-1
Extra Example 2
-2
Section 9: Tests of Significance
Introduction: Hypothesis Tests

14m 9s

Intro
0:00
Two Hypotheses
0:13
Null Hypothesis
0:21
Alternative Hypothesis
0:36
Example
1:05
Example: Two Hypotheses
1:43
Rejection and Non-Rejection Regions
3:25
Type 1 and Type 2 Errors
5:30
Type 1 Error
6:44
Significance Level
7:08
Type 2 Error
7:42
Power of the Test
8:30
Tails of the Test
9:29
Large Sample Test for a Proportion

14m 30s

Intro
0:00
Test Statistic Z
0:08
Why Z?
0:29
Example 1: TV Violence
1:10
Example 2: Smoking
5:16
Extra Example 1
-1
Extra Example 2
-2
Large Sample Test for a Difference in Two Proportions

19m 14s

Intro
0:00
Pooled Estimate of P1 and P2
0:09
Example 1: Softball Bases
1:34
Example 2: Sleep Problems
6:59
Extra Example 1
-1
Extra Example 2
-2
Test for a Mean

14m 57s

Intro
0:00
Standard Deviation is Known
0:07
Central Limit Theory for n>30
0:32
Example 1: Cheese Weight
0:53
Example 2: Observations
3:53
Standard Deviation Not Known
6:15
t-Distribution Usage
6:24
Degrees of Freedom
6:53
Example 3: Height
7:01
Example 4: Sampling
9:50
Extra Example 1
-1
Extra Example 2
-2
Test for a Difference Between Two Means

23m 13s

Intro
0:00
Standard Deviation Known, Unpaired
0:08
Example 1: Boredom
1:17
Example 2: Smoking
4:15
Population Standard Deviations Unknown, But Equal
7:10
Pooled Standard Deviation for Two Samples
7:49
Example 3: Diet Soda
8:28
Example 4: TV
12:12
Paired Samples
15:50
Example 5: Hormone Level
16:33
Example 6: Hypnotism
19:43
Chi-Square Tests: One Way and Two Way

24m 33s

Intro
0:00
Goodness of Fit Test
0:07
Right-Tailed Test
0:52
Example 1: Die Rolls
1:16
Example 2: Stolen Vehicles
3:31
Test of Independence
7:02
Example 3: Debt
7:51
Example 4: Contraceptive Use
13:14
Test of Homogeneity
16:31
Example 5: New Product
17:09
Example 6: Oil
21:24
Hypothesis Testing for the Slope of a Least Squares Regression Line

17m 48s

Intro
0:00
Sampling Distribution of b
0:08
Calculating the Estimator of Standard Deviation of b
1:18
Hypothesis Testing for Beta
1:50
Example 1: Age
2:25
Example 2: Weld Diameter
6:42
Extra Example 1
-1
Extra Example 2
-2
Bookmark & Share Embed

## Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
×
• - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

• ## Related Books

 1 answerLast reply by: Anil GeorgeThu Aug 16, 2012 11:00 AMPost by Anil George on August 16, 2012For the first Variance example, shouldn't the 3 and 5 be squared, since the sigma is squared in the variance equation? So the numbers should read 1 + 4(9) + 25 = 62?

### Combining Independent Random Variables

• If two random variables are independent, their joint probability is the product of the two marginal probabilities.

• The mean of the sum of independent random variables is the sum of the means of the independent random variables.

• The variance of the sum of independent random variables is the sum of the variances of the independent random variables.

### Combining Independent Random Variables

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

• Intro 0:00
• Independence vs Dependence 0:09
• Mean of Sums for Independent Random Variables 2:32
• Example 4:02
• Example 5:58
• Variance for Sums of Independent Random Variables 8:49
• Example 10:30
• Example 12:26
• Extra Example 1
• Extra Example 2

OR

### Start Learning Now

Our free lessons will get you started (Adobe Flash® required).