Last Time... Sanity Check Let X be a RV that takes on values in A . - PowerPoint PPT Presentation

Last Time... Sanity Check Let X be a RV that takes on values in A . ◮ Expectation describes the weighted Expectation Continued: Tail Let Y be a RV that takes on values in B . average of a RV. Sum, Coupon Collector, and Let c ∈ R be a constant. ◮ For more complicated RVs, use linearity Functions of RVs Both c · X and X + Y are also RVs! Today: ◮ Proof of linearity of expectation CS 70, Summer 2019 ◮ The tail sum formula Lecture 20, 7/29/19 ◮ Expectations of Geometric and Poisson ◮ Expectation of a function of an RV 1 / 26 2 / 26 3 / 26 Proof of Linearity of Expectation I Proof of Linearity of Expectation II The Tail Sum Formula Recall linearity of expectation: Next, we show E [ X + Y ] = E [ X ] + E [ Y ] . Let X be a RV with values in { 0 , 1 , 2 , . . . , n } . We use “tail” to describe P [ X ≥ i ] . E [ X 1 + . . . + X n ] = E [ X 1 ] + . . . + E [ X n ] What does � ∞ i = 1 P [ X ≥ i ] look like? For constant c , E [ cX i ] = c · E [ X i ] Small example: X only takes values { 0 , 1 , 2 } : First, we show E [ cX i ] = c · E [ X i ] : Two variables to n variables? 4 / 26 5 / 26 6 / 26

The Tail Sum Formula Expectation of a Geometric I Expectation of a Geometric II The tail sum formula states that: Let X ∼ Geometric( p ) . Use memorylessness : the fact that the geometric RV “resets” after each trial. ∞ P [ X ≥ i ] = � E [ X ] = P [ X ≥ i ] i = 1 Apply the tail sum formula : Two Cases: Proof: Let p i = P [ X = i ] . 7 / 26 8 / 26 9 / 26 Expectation of a Geometric III Coupon Collector I Coupon Collector II Lastly, an intuitive but non-rigorous idea. (Note 19.) I’m out collecting trading cards. Let X i = There are n types total. I get a random trading Let X i be an indicator variable for success in a card every time I buy a cereal box. What is the dist. of X 1 ? single trial. Recall trials are i.i.d. What is the expected number of boxes I need to What is the dist. of X 2 ? X i ∼ buy in order to get all n trading cards? What is the dist. of X 3 ? E [ X 1 + X 2 + . . . + X k ] = High level picture: In general, what is the dist. of X i ? 10 / 26 11 / 26 12 / 26

Coupon Collector III Aside: (Partial) Harmonic Series Break Harmonic Series: � ∞ 1 Let X = A Bad Harmonic Series Joke... k = 1 k Approximation for � n 1 X = A countably infinite number of mathematicians k in terms of n ? k = 1 walk into a bar. The first one orders a pint of E [ X ] = beer, the second one orders a half pint, the third one orders a third of a pint, the fourth one orders a fourth of a pint, and so on. The bartender says ... 13 / 26 14 / 26 15 / 26 Expectation of a Poisson I Expectation of a Poisson II Rest of Today: Functions of RVs! Recall the Poisson distribution: values 0 , 1 , 2 , . . . , Optional but intuitive / non-rigorous approach: Recall X from Lecture 19:  P [ X = i ] = λ i Think of a Poisson( λ ) as a Bin( n , λ 1 wp 0.4 n ) distribution, i ! e − λ   taken as n → ∞ . 1 X = wp 0.25 2 − 1  wp 0.35  We can use the definition to find E [ X ] ! Let X ∼ Bin( n , λ 2 n ) . Refresh your memory: What is X 2 ? X = 16 / 26 17 / 26 18 / 26

Example: Functions of RVs In General: Functions of RVs Square of a Bernoulli � Let X be a RV with values in A . Let X ∼ Bernoulli( p ) . 1 wp 0.4 X 2 = Distribution of f ( X ) : Write out the distribution of X . 1 wp 0.6 4 What is E [ X 2 ] ? What is X 2 ? E [ X 2 ] ? E [ f ( X )] = What is E [ 3 X 2 − 5 ] ? 19 / 26 20 / 26 21 / 26 Product of RVs Product of Two Bernoullis Square of a Binomial I Let X be a RV with values in A . Let X ∼ Bernoulli( p 1 ) , and Y ∼ Bernoulli( p 2 ) . Let X ∼ Bin( n , p ) . Let Y be a RV with values in B . X and Y are independent . Decompose into X i ∼ Bernoulli( p ) . XY is also a RV! What is its distribution? What is the distribution of XY ? X = (Use the joint distribution! ) E [ X ] = What is E [ XY ] ? 22 / 26 23 / 26 24 / 26

Square of a Binomial II Summary Recall, E [ X 2 i ] = p , and E [ X i X j ] = p 2 . Today: ◮ Proof of linearity of expectation: did not use independence, but did use joint distribution ◮ Tail sum for non-negative int.-valued RVs! ◮ Coupon Collector: break problem down into a sum of geometrics. ◮ Expectation of a function of an RV: can apply definition and linearity of expectation (after expanding) as well!! 25 / 26 26 / 26