Foundations of Computer Science Lecture 21 Deviations from the Mean - PowerPoint PPT Presentation

Foundations of Computer Science Lecture 21 Deviations from the Mean How Good is the Expectation as a Sumary of a Random Variable? Variance: Uniform; Bernoulli; Binomial; Waiting Times. Variance of a Sum Law of Large Numbers: The 3- σ Rule

Last Time 1 Expected value of a Sum. ◮ Sum of dice ◮ Binomial ◮ Waiting time ◮ Coupon collecting. 2 Build-up expectation. 3 Expected value of a product. 4 Sum of Indicators. ◮ Random arrangement of hats on heads. Creator: Malik Magdon-Ismail Deviations from the Mean: 2 / 13 Today →

Today: Deviations from the Mean How well does the expected value (mean) summarize a random variable? 1 Variance. 2 Variance of a sum. 3 Law of large numbers 4 The 3- σ rule. Creator: Malik Magdon-Ismail Deviations from the Mean: 3 / 13 Up to Now →

Probability For Analyzing a Random Experiment. Experiment (random)

Probability For Analyzing a Random Experiment. Experiment Outcomes (random) (complex)

Probability For Analyzing a Random Experiment. Measurement X Experiment Outcomes (random) (complex) (random variable)

Probability For Analyzing a Random Experiment. Measurement X Summary E [ X ] Experiment Outcomes (random) (complex) (random variable) (expectation)

Probability For Analyzing a Random Experiment. Measurement X Summary E [ X ] Experiment Outcomes How good (random) (complex) (random variable) (expectation) is E [ X ]? Creator: Malik Magdon-Ismail Deviations from the Mean: 4 / 13 Average of n Dice →

Probability For Analyzing a Random Experiment. Measurement X Summary E [ X ] Experiment Outcomes How good (random) (complex) (random variable) (expectation) is E [ X ]? Experiment. Roll n dice and compute X , the average of the rolls. Creator: Malik Magdon-Ismail Deviations from the Mean: 4 / 13 Average of n Dice →

Probability For Analyzing a Random Experiment. Measurement X Summary E [ X ] Experiment Outcomes How good (random) (complex) (random variable) (expectation) is E [ X ]? Experiment. Roll n dice and compute X , the average of the rolls. E [ average ] Creator: Malik Magdon-Ismail Deviations from the Mean: 4 / 13 Average of n Dice →

Probability For Analyzing a Random Experiment. Measurement X Summary E [ X ] Experiment Outcomes How good (random) (complex) (random variable) (expectation) is E [ X ]? Experiment. Roll n dice and compute X , the average of the rolls. � 1 � E [ average ] = E n · sum Creator: Malik Magdon-Ismail Deviations from the Mean: 4 / 13 Average of n Dice →

Probability For Analyzing a Random Experiment. Measurement X Summary E [ X ] Experiment Outcomes How good (random) (complex) (random variable) (expectation) is E [ X ]? Experiment. Roll n dice and compute X , the average of the rolls. � 1 � 1 E [ average ] = E n · sum = n · E [ sum ] Creator: Malik Magdon-Ismail Deviations from the Mean: 4 / 13 Average of n Dice →

Probability For Analyzing a Random Experiment. Measurement X Summary E [ X ] Experiment Outcomes How good (random) (complex) (random variable) (expectation) is E [ X ]? Experiment. Roll n dice and compute X , the average of the rolls. � 1 � 1 n × n × 3 1 1 E [ average ] = E n · sum = n · E [ sum ] = 2 Creator: Malik Magdon-Ismail Deviations from the Mean: 4 / 13 Average of n Dice →

Probability For Analyzing a Random Experiment. Measurement X Summary E [ X ] Experiment Outcomes How good (random) (complex) (random variable) (expectation) is E [ X ]? Experiment. Roll n dice and compute X , the average of the rolls. � 1 � 1 n × n × 3 1 1 2 = 3 1 E [ average ] = E n · sum = n · E [ sum ] = 2 . Creator: Malik Magdon-Ismail Deviations from the Mean: 4 / 13 Average of n Dice →

Average of n Dice 6 5 Average roll 4 3 . 5 3 2 1 10 2 10 3 10 4 10 5 1 10 Number of dice n

Average of n Dice 6 5 Average roll 4 3 . 5 3 2 1 10 2 10 3 10 4 10 5 1 10 Number of dice n 0.1 σ 0.1 σ Probability Probability 0 0 1 2 3 3 . 5 4 5 1 2 3 3 . 5 4 5 6 6 Average of 4 dice Average of 100 dice Creator: Malik Magdon-Ismail Deviations from the Mean: 5 / 13 Variance →

Variance: Size of the Deviations From the Mean X = sum of 2 dice. E [ X ] = 7 ← µ ( X ) Creator: Malik Magdon-Ismail Deviations from the Mean: 6 / 13 Risk →

Variance: Size of the Deviations From the Mean X = sum of 2 dice. E [ X ] = 7 ← µ ( X ) X 2 3 4 5 6 7 8 9 10 11 12 ∆ − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 ← X − µ 1 2 3 4 5 6 5 4 3 2 1 P X 36 36 36 36 36 36 36 36 36 36 36 Pop Quiz. What is E [ ∆ ] ? Creator: Malik Magdon-Ismail Deviations from the Mean: 6 / 13 Risk →

Variance: Size of the Deviations From the Mean X = sum of 2 dice. E [ X ] = 7 ← µ ( X ) X 2 3 4 5 6 7 8 9 10 11 12 ∆ − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 ← X − µ 1 2 3 4 5 6 5 4 3 2 1 P X 36 36 36 36 36 36 36 36 36 36 36 Pop Quiz. What is E [ ∆ ] ? Variance, σ 2 , is the expected value of the squared deviations, σ 2 = E [ ∆ 2 ] = E [( X − µ ) 2 ] = E [( X − E [ X ]) 2 ] σ 2 = E [ ∆ 2 ] = 1 36 · 25 + Creator: Malik Magdon-Ismail Deviations from the Mean: 6 / 13 Risk →

Variance: Size of the Deviations From the Mean X = sum of 2 dice. E [ X ] = 7 ← µ ( X ) X 2 3 4 5 6 7 8 9 10 11 12 ∆ − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 ← X − µ 1 2 3 4 5 6 5 4 3 2 1 P X 36 36 36 36 36 36 36 36 36 36 36 Pop Quiz. What is E [ ∆ ] ? Variance, σ 2 , is the expected value of the squared deviations, σ 2 = E [ ∆ 2 ] = E [( X − µ ) 2 ] = E [( X − E [ X ]) 2 ] σ 2 = E [ ∆ 2 ] = 36 · 25 + 2 1 36 · 16 + Creator: Malik Magdon-Ismail Deviations from the Mean: 6 / 13 Risk →

Variance: Size of the Deviations From the Mean X = sum of 2 dice. E [ X ] = 7 ← µ ( X ) X 2 3 4 5 6 7 8 9 10 11 12 ∆ − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 ← X − µ 1 2 3 4 5 6 5 4 3 2 1 P X 36 36 36 36 36 36 36 36 36 36 36 Pop Quiz. What is E [ ∆ ] ? Variance, σ 2 , is the expected value of the squared deviations, σ 2 = E [ ∆ 2 ] = E [( X − µ ) 2 ] = E [( X − E [ X ]) 2 ] σ 2 = E [ ∆ 2 ] = 36 · 25 + 2 1 36 · 16 + 3 36 · 9 + 4 36 · 4 + 5 36 · 1 + 6 36 · 0 + 5 36 · 1 + 4 36 · 4 + 3 36 · 9 + 2 36 · 16 + 1 36 · 25 Creator: Malik Magdon-Ismail Deviations from the Mean: 6 / 13 Risk →

Variance: Size of the Deviations From the Mean X = sum of 2 dice. E [ X ] = 7 ← µ ( X ) X 2 3 4 5 6 7 8 9 10 11 12 ∆ − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 ← X − µ 1 2 3 4 5 6 5 4 3 2 1 P X 36 36 36 36 36 36 36 36 36 36 36 Pop Quiz. What is E [ ∆ ] ? Variance, σ 2 , is the expected value of the squared deviations, σ 2 = E [ ∆ 2 ] = E [( X − µ ) 2 ] = E [( X − E [ X ]) 2 ] σ 2 = E [ ∆ 2 ] = 36 · 25 + 2 1 36 · 16 + 3 36 · 9 + 4 36 · 4 + 5 36 · 1 + 6 36 · 0 + 5 36 · 1 + 4 36 · 4 + 3 36 · 9 + 2 36 · 16 + 1 36 · 25 = 5 5 6 . Creator: Malik Magdon-Ismail Deviations from the Mean: 6 / 13 Risk →

Variance: Size of the Deviations From the Mean X = sum of 2 dice. E [ X ] = 7 ← µ ( X ) X 2 3 4 5 6 7 8 9 10 11 12 ∆ − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 ← X − µ 1 2 3 4 5 6 5 4 3 2 1 P X 36 36 36 36 36 36 36 36 36 36 36 Pop Quiz. What is E [ ∆ ] ? Variance, σ 2 , is the expected value of the squared deviations, σ 2 = E [ ∆ 2 ] = E [( X − µ ) 2 ] = E [( X − E [ X ]) 2 ] σ 2 = E [ ∆ 2 ] = 36 · 25 + 2 1 36 · 16 + 3 36 · 9 + 4 36 · 4 + 5 36 · 1 + 6 36 · 0 + 5 36 · 1 + 4 36 · 4 + 3 36 · 9 + 2 36 · 16 + 1 36 · 25 = 5 5 6 . Standard Deviation, σ , is the square-root of the variance, � � � σ = E [ ∆ 2 ] = E [( X − µ ) 2 ] = E [( X − E [ X ]) 2 ] � 5 5 σ = 6 ≈ 2 . 52 sum of two dice rolls = 7 ± 2 . 52 . Practice. Exercise 21.2. Creator: Malik Magdon-Ismail Deviations from the Mean: 6 / 13 Risk →

Variance is a Measure of Risk Game 1 Game 2 Creator: Malik Magdon-Ismail Deviations from the Mean: 7 / 13 More Convenient Variance →

Variance is a Measure of Risk Game 1 Game 2 probability = 2 win $2 3 ; X 1 : probability = 1 lose $1 3 . Creator: Malik Magdon-Ismail Deviations from the Mean: 7 / 13 More Convenient Variance →

Variance is a Measure of Risk Game 1 Game 2 probability = 2 probability = 2 win $2 3 ; win $102 3 ; X 1 : X 2 : probability = 1 probability = 1 lose $1 lose $201 3 . 3 . Creator: Malik Magdon-Ismail Deviations from the Mean: 7 / 13 More Convenient Variance →

Variance is a Measure of Risk Game 1 Game 2 probability = 2 probability = 2 win $2 3 ; win $102 3 ; X 1 : X 2 : probability = 1 probability = 1 lose $1 lose $201 3 . 3 . E [ X 1 ] = $1 Creator: Malik Magdon-Ismail Deviations from the Mean: 7 / 13 More Convenient Variance →

Variance is a Measure of Risk Game 1 Game 2 probability = 2 probability = 2 win $2 3 ; win $102 3 ; X 1 : X 2 : probability = 1 probability = 1 lose $1 lose $201 3 . 3 . E [ X 1 ] = $1 E [ X 2 ] = $1 Creator: Malik Magdon-Ismail Deviations from the Mean: 7 / 13 More Convenient Variance →