Foundations of Computer Science Lecture 19 Expected Value The - PowerPoint PPT Presentation

Foundations of Computer Science Lecture 19 Expected Value The Average Over Many Runs of an Experiment Mathematical Expectation: A Number that Summarizes a PDF Conditional Expectation Law of Total Expectation

Last Time 1 Random variables. ◮ PDF. ◮ CDF. ◮ Joint-PDF. ◮ Independent random variables. 2 Important random variables. ◮ Bernoulli (indicator). ◮ Uniform (equalizer in strategic games). ◮ Binomial (sum of Bernoullis, e.g. number of heads in n coin tosses). ◮ Exponential Waiting Time Distribution (repeated tries till success). Creator: Malik Magdon-Ismail Expected Value: 2 / 15 Today →

Today: Expected Value Expected value approximates the sample average. 1 Mathematical Expectation 2 Examples 3 Sum of dice. Bernoulli. Uniform. Binomial. Waiting time. Conditional Expectaton 4 Law of Total Expectation 5 Creator: Malik Magdon-Ismail Expected Value: 3 / 15 Sample Average →

Sample Average: Toss Two Coins Many Times Sample Space Ω x ∈ X (Ω) ω HH HT TH TT x 0 1 2 → 1 1 1 1 P ( ω ) 4 4 4 4 1 1 1 P X ( x ) 4 2 4 X ( ω ) 2 1 1 0 ← number of heads Creator: Malik Magdon-Ismail Expected Value: 4 / 15 Mathematical Expectation →

Sample Average: Toss Two Coins Many Times Sample Space Ω x ∈ X (Ω) ω HH HT TH TT x 0 1 2 → 1 1 1 1 P ( ω ) 4 4 4 4 1 1 1 P X ( x ) 4 2 4 X ( ω ) 2 1 1 0 ← number of heads Toss two coins and repeat the experiment n = 24 times: HH TH HT HH HH TH TT TT HH TT HT HT HH HT TT HT TT HT HT TH HH TH TT TH 2 1 1 2 2 1 0 0 2 0 1 1 2 1 0 1 0 1 1 1 2 1 0 1 Creator: Malik Magdon-Ismail Expected Value: 4 / 15 Mathematical Expectation →

Sample Average: Toss Two Coins Many Times Sample Space Ω x ∈ X (Ω) ω HH HT TH TT x 0 1 2 → 1 1 1 1 P ( ω ) 4 4 4 4 1 1 1 P X ( x ) 4 2 4 X ( ω ) 2 1 1 0 ← number of heads Toss two coins and repeat the experiment n = 24 times: HH TH HT HH HH TH TT TT HH TT HT HT HH HT TT HT TT HT HT TH HH TH TT TH 2 1 1 2 2 1 0 0 2 0 1 1 2 1 0 1 0 1 1 1 2 1 0 1 Average value of X : 2 + 1 + 1 + 2 + 2 + 1 + 0 + 0 + 2 + 0 + 1 + 1 + 2 + 1 + 0 + 1 + 0 + 1 + 1 + 1 + 2 + 1 + 0 + 1 = 24 24 = 1 . 24 Creator: Malik Magdon-Ismail Expected Value: 4 / 15 Mathematical Expectation →

Sample Average: Toss Two Coins Many Times Sample Space Ω x ∈ X (Ω) ω HH HT TH TT x 0 1 2 → 1 1 1 1 P ( ω ) 4 4 4 4 1 1 1 P X ( x ) 4 2 4 X ( ω ) 2 1 1 0 ← number of heads Toss two coins and repeat the experiment n = 24 times: HH TH HT HH HH TH TT TT HH TT HT HT HH HT TT HT TT HT HT TH HH TH TT TH 2 1 1 2 2 1 0 0 2 0 1 1 2 1 0 1 0 1 1 1 2 1 0 1 Average value of X : 2 + 1 + 1 + 2 + 2 + 1 + 0 + 0 + 2 + 0 + 1 + 1 + 2 + 1 + 0 + 1 + 0 + 1 + 1 + 1 + 2 + 1 + 0 + 1 = 24 24 = 1 . 24 Re-order outcomes: TT TT TT TT TT TT HT HT HT HT HT HT HT TH TH TH TH TH HH HH HH HH HH HH n 0 = 6 n 1 = 12 n 2 = 6 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 Creator: Malik Magdon-Ismail Expected Value: 4 / 15 Mathematical Expectation →

Sample Average: Toss Two Coins Many Times Sample Space Ω x ∈ X (Ω) ω HH HT TH TT x 0 1 2 → 1 1 1 1 P ( ω ) 4 4 4 4 1 1 1 P X ( x ) 4 2 4 X ( ω ) 2 1 1 0 ← number of heads Toss two coins and repeat the experiment n = 24 times: HH TH HT HH HH TH TT TT HH TT HT HT HH HT TT HT TT HT HT TH HH TH TT TH 2 1 1 2 2 1 0 0 2 0 1 1 2 1 0 1 0 1 1 1 2 1 0 1 Average value of X : 2 + 1 + 1 + 2 + 2 + 1 + 0 + 0 + 2 + 0 + 1 + 1 + 2 + 1 + 0 + 1 + 0 + 1 + 1 + 1 + 2 + 1 + 0 + 1 = 24 24 = 1 . 24 Re-order outcomes: TT TT TT TT TT TT HT HT HT HT HT HT HT TH TH TH TH TH HH HH HH HH HH HH n 0 = 6 n 1 = 12 n 2 = 6 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 Average value of X : 6 × 0 + 12 × 1 + 6 × 2 = 24 n 24 Creator: Malik Magdon-Ismail Expected Value: 4 / 15 Mathematical Expectation →

Mathematical Expectation of a Random Variable X TT TT TT TT TT TT HT HT HT HT HT HT HT TH TH TH TH TH HH HH HH HH HH HH n 0 = 6 n 1 = 12 n 2 = 6 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 Creator: Malik Magdon-Ismail Expected Value: 5 / 15 Expected Number of Heads →

Mathematical Expectation of a Random Variable X TT TT TT TT TT TT HT HT HT HT HT HT HT TH TH TH TH TH HH HH HH HH HH HH n 0 = 6 n 1 = 12 n 2 = 6 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 Average value of X : n 0 × 0 + n 1 × 1 + n 2 × 2 + n 3 × 3 n Creator: Malik Magdon-Ismail Expected Value: 5 / 15 Expected Number of Heads →

Mathematical Expectation of a Random Variable X TT TT TT TT TT TT HT HT HT HT HT HT HT TH TH TH TH TH HH HH HH HH HH HH n 0 = 6 n 1 = 12 n 2 = 6 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 Average value of X : n 0 × 0 + n 1 × 1 + n 2 × 2 + n 3 × 3 n 0 × 0 + n 1 × 1 + n 2 = × 2 n n n n ↑ ↑ ↑ ≈ P X (0) ≈ P X (1) ≈ P X (2) Creator: Malik Magdon-Ismail Expected Value: 5 / 15 Expected Number of Heads →

Mathematical Expectation of a Random Variable X TT TT TT TT TT TT HT HT HT HT HT HT HT TH TH TH TH TH HH HH HH HH HH HH n 0 = 6 n 1 = 12 n 2 = 6 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 Average value of X : n 0 × 0 + n 1 × 1 + n 2 × 2 + n 3 × 3 n 0 × 0 + n 1 × 1 + n 2 = × 2 n n n n ↑ ↑ ↑ ≈ P X (0) ≈ P X (1) ≈ P X (2) ≈ P X (0) × 0 + P X (1) × 1 + P X (2) × 2 Creator: Malik Magdon-Ismail Expected Value: 5 / 15 Expected Number of Heads →

Mathematical Expectation of a Random Variable X TT TT TT TT TT TT HT HT HT HT HT HT HT TH TH TH TH TH HH HH HH HH HH HH n 0 = 6 n 1 = 12 n 2 = 6 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 Average value of X : n 0 × 0 + n 1 × 1 + n 2 × 2 + n 3 × 3 n 0 × 0 + n 1 × 1 + n 2 = × 2 n n n n ↑ ↑ ↑ ≈ P X (0) ≈ P X (1) ≈ P X (2) ≈ P X (0) × 0 + P X (1) × 1 + P X (2) × 2 = x ∈ X (Ω) x · P X ( x ) � For two coins the expected value is 0 × 1 4 + 1 × 1 2 + 2 × 1 4 = 1. Add the possible values x weighted by their probabilities P X ( x ) , E [ X ] = x ∈ X (Ω) x · P X ( x ) . � Synonyms: Expectation; Expected Value; Mean; Average. � Important Exercise. Show that E [ X ] = X ( ω ) P ( ω ). ω ∈ Ω Creator: Malik Magdon-Ismail Expected Value: 5 / 15 Expected Number of Heads →

Expected Number of Heads from 3 Coin Tosses is 1 1 2 ! E [ X ] = x ∈ X (Ω) x · P X ( x ) . � Creator: Malik Magdon-Ismail Expected Value: 6 / 15 Expected Sum of Two Dice →

Expected Number of Heads from 3 Coin Tosses is 1 1 2 ! E [ X ] = x ∈ X (Ω) x · P X ( x ) . � Sample Space Ω x ∈ X (Ω) ω HHH HHT HTH HTT THH THT TTH TTT x 0 1 2 3 → 1 1 1 1 1 1 1 1 P ( ω ) 1 3 3 1 8 8 8 8 8 8 8 8 P X ( x ) 8 8 8 8 X ( ω ) 3 2 2 1 2 1 1 0 Creator: Malik Magdon-Ismail Expected Value: 6 / 15 Expected Sum of Two Dice →

Expected Number of Heads from 3 Coin Tosses is 1 1 2 ! E [ X ] = x ∈ X (Ω) x · P X ( x ) . � Sample Space Ω x ∈ X (Ω) ω HHH HHT HTH HTT THH THT TTH TTT x 0 1 2 3 → 1 1 1 1 1 1 1 1 P ( ω ) 1 3 3 1 8 8 8 8 8 8 8 8 P X ( x ) 8 8 8 8 X ( ω ) 3 2 2 1 2 1 1 0 E [ number heads ] = 0 × 1 8 + 1 × 3 8 + 2 × 3 8 + 3 × 18 12 = 8 = 1 1 2 . What does this mean?!? Exercise. Let X be the the value of a fair die roll. Show that E [ X ] = 3 1 2 . Creator: Malik Magdon-Ismail Expected Value: 6 / 15 Expected Sum of Two Dice →

Expected Sum of Two Dice Probability Space X = sum 1 1 1 1 1 1 7 8 9 10 11 12 36 36 36 36 36 36 1 1 1 1 1 1 Die 2 Value 6 7 8 9 10 11 36 36 36 36 36 36 1 1 1 1 1 1 5 6 7 8 9 10 36 36 36 36 36 36 x 2 3 4 5 6 7 8 9 10 11 12 1 1 1 1 1 1 4 5 6 7 8 9 1 2 3 4 5 6 5 4 3 4 1 36 36 36 36 36 36 P X ( x ) 1 1 1 1 1 1 36 36 36 36 36 36 36 36 36 36 36 3 4 5 6 7 8 36 36 36 36 36 36 1 1 1 1 1 1 2 3 4 5 6 7 36 36 36 36 36 36 Die 1 Value Die 1 Value Creator: Malik Magdon-Ismail Expected Value: 7 / 15 Bernoulli →

Expected Sum of Two Dice Probability Space X = sum 1 1 1 1 1 1 7 8 9 10 11 12 36 36 36 36 36 36 1 1 1 1 1 1 Die 2 Value 6 7 8 9 10 11 36 36 36 36 36 36 1 1 1 1 1 1 5 6 7 8 9 10 36 36 36 36 36 36 x 2 3 4 5 6 7 8 9 10 11 12 1 1 1 1 1 1 4 5 6 7 8 9 1 2 3 4 5 6 5 4 3 4 1 36 36 36 36 36 36 P X ( x ) 1 1 1 1 1 1 36 36 36 36 36 36 36 36 36 36 36 3 4 5 6 7 8 36 36 36 36 36 36 1 1 1 1 1 1 2 3 4 5 6 7 36 36 36 36 36 36 Die 1 Value Die 1 Value � E [ X ] = x x · P X ( x ) Creator: Malik Magdon-Ismail Expected Value: 7 / 15 Bernoulli →