More discrete distributions

Transcription

1 Will Monroe July 14, 217 with materials by Mehran Sahami and Chris Piech More discrete distributions

2 Announcements: Problem Set 3 Posted yesterday on the course website. Due next Wednesday, 7/19, at 12:3pm (before class). (election prediction) (Moby Dick)

3 Announcements: Problem Set 3 Posted yesterday on the course website. Due next Wednesday, 7/19, at 12:3pm (before class). Everybody gets an extra late day! (4 total) (election prediction) (Moby Dick)

4 Review: Bernoulli random variable An indicator variable (a possibly biased coin flip) obeys a Bernoulli distribution. Bernoulli random variables can be or 1. X Ber ( p) p X (1)= p p X ()=1 p ( elsewhere)

5 Review: Bernoulli fact sheet? X Ber ( p) probability of success (heads, ad click,...) PMF: expectation: variance: p X (1)= p p X ()=1 p ( elsewhere) E [ X ]= p Var( X )= p(1 p) image (right): Gabriela Serrano

6 Review: Binomial random variable The number of heads on n (possibly biased) coin flips obeys a binomial distribution. X Bin (n, p) n p k (1 p)n k if k ℕ, k n p X (k)= k otherwise { ()

7 Review: Binomial fact sheet number of trials (flips, program runs,...) X Bin (n, p) probability of success (heads, crash,...) PMF: expectation: variance: n p k (1 p)n k if k ℕ, k n p X (k )= k otherwise { () E [ X ]=np Var( X )=np(1 p) note: Ber ( p)=bin (1, p)

8 Review: Poisson random variable The number of occurrences of an event that occurs with constant rate λ (per unit time), in 1 unit of time, obeys a Poisson distribution. X Poi (λ) p X (k)= { λ e k λ k! if k ℤ, k otherwise

9 Review: Poisson fact sheet X Poi (λ) rate of events (requests, earthquakes, chocolate chips, ) per unit time (hour, year, cookie,...) PMF: expectation: variance: p X (k )= { e k λ if k ℤ, k k! otherwise λ E [ X ]=λ Var( X )=λ

10 Geometric random variable The number of trials it takes to get one success, if successes occur independently with probability p, obeys a geometric distribution. X Geo( p) k 1 (1 p) p if k ℤ, k 1 p X (k )= otherwise {

11 Catching Pokémon Wild Pokémon are captured by throwing Poké Balls at them. Each ball has probability p of capturing the Pokémon. How many are needed on average for a successful capture? X: number of Poké Balls until (and including) capture Ci: event that Pokémon is captured on the i-th throw C 1 C 1 C C k 1 P ( X =k )=P (C C 2 C Ck) C C =P(C ) P (C 2 ) P (C k 1 ) P (C k ) k 1 =(1 p) p

12 Geometric: Fact sheet X Geo( p) probability of success (catch, heads, crash,...) PMF: k 1 (1 p) p if k ℤ, k 1 p X (k )= otherwise {

13 Catching Pokémon X: number of Poké Balls until (and including) capture k 1 P ( X =k )=(1 p) p k 1 E [ X ]= k (1 p) p k =1 E [ X ]=(1 p) E [ X ]+1 (1 (1 p)) E [ X ]=1 p E [ X ]=1 1 E [ X ]= p k 1 = (k 1+1) (1 p) p k=1 k 1 = (k 1) (1 p) p+ (1 p) k=1 k =1 j j j= j 1 p 1 x = 1 x j= = j (1 p) p+ (1 p) p j= k 1 =(1 p) j (1 p) p+ p (1 p) j= 1 =(1 p) E [ X ]+ p 1 (1 p) =(1 p) E [ X ]+1 j= j j

14 Geometric: Fact sheet X Geo( p) probability of success (catch, heads, crash,...) PMF: expectation: k 1 (1 p) p if k ℤ, k 1 p X (k )= otherwise { 1 E[ X ]= p

15 Catching Pokémon Wild Pokémon are captured by throwing Poké Balls at them. Each ball has probability p =.1 of capturing the Pokémon. How many are needed so that probability of successful capture is at least 99%? X: number of Poké Balls until (and including) capture Ci: event that Pokémon is captured on the i-th throw P ( X k )=1 P ( X >k ) C C C =1 P(C 1 C 2 C k ) C C C =1 P(C 1 ) P (C 2 ) P(C k ) k =1 (1 p)

16 Cumulative distribution function The cumulative distribution function (CDF) of a random variable is a function giving the probability that the random variable is less than or equal to a value. F Y (k )=P(Y k ) P(Y k ) (CDF of the sum of two dice) k

17 Geometric: Fact sheet X Geo( p) probability of success (catch, heads, crash,...) PMF: CDF: expectation: k 1 (1 p) p if k ℤ, k 1 p X (k )= otherwise { { k 1 (1 p) F X (k )= 1 E[ X ]= p if k ℤ, k 1 otherwise

18 Catching Pokémon Wild Pokémon are captured by throwing Poké Balls at them. Each ball has probability p =.1 of capturing the Pokémon. How many are needed so that probability of successful capture is at least 99%? X: number of Poké Balls until (and including) capture k P ( X k )=1 (1 p).99 k.1 (1 p) log.1 k log (1 p) log k log (1 p)

19 Geometric: Fact sheet X Geo( p) probability of success (catch, heads, crash,...) PMF: CDF: expectation: variance: k 1 (1 p) p if k ℤ, k 1 p X (k )= otherwise { { k 1 (1 p) F X (k )= 1 E[ X ]= p 1 p Var( X )= 2 p if k ℤ, k 1 otherwise

20 Break time!

21 Negative binomial random variable The number of trials it takes to get r successes, if successes occur independently with probability p, obeys a negative binomial distribution. X NegBin (r, p) n 1 pr (1 p)n r if n ℤ, n r p X (n)= r 1 otherwise { ( )

22 Getting that degree A conference accepts papers (independently and randomly?) with probability p =.25. A hypothetical grad student needs 3 accepted papers to graduate. What is the probability this takes exactly 1 submissions? X: number of tries to get 3 accepts Y: number of accepts in first 9 tries accept on 1th try P( X =1)=P(Y =2) p = (1 p) p p.75 2 ()

23 Getting that degree A conference accepts papers (independently and randomly?) with probability p. A hypothetical grad student needs r accepted papers to graduate. What is the probability this takes exactly n submissions? X: number of tries to get r accepts Y: number of accepts in first n 1 tries accept on nth try P( X =1)=P(Y =r 1) p n r r 1 n 1 = (1 p) p p r 1 ( )

24 Negative binomial: Fact sheet number of sucesses (heads, crash,...) X NegBin (r, p) number of trials (flips, program runs,...) PMF: probability of success n 1 pr (1 p)n r if n ℤ, n r p X (n)= r 1 otherwise { ( )

25 Getting that degree A conference accepts papers (independently and randomly?) with probability p =.25. A hypothetical grad student needs 3 accepted papers to graduate. How many submissions will be necessary on average? X: number of tries to get 3 accepts E [ X ]=? A) 3.25 B) 3.25 C) D) Room: CS19SUMMER17

26 Getting that degree A conference accepts papers (independently and randomly?) with probability p. A hypothetical grad student needs r accepted papers to graduate. How many submissions will be necessary on average? X: number of tries to get r accepts r E [ X ]= p Room: CS19SUMMER17

27 Negative binomial: Fact sheet number of sucesses (heads, crash,...) X NegBin (r, p) number of trials (flips, program runs,...) PMF: expectation: variance: probability of success n 1 pr (1 p)n r if n ℤ, n r p X (n)= r 1 otherwise r E[ X ]= p r (1 p) note: Var( X )= 2 p Geo( p)=negbin (1, p) { ( )

28 A few optional (but hopefully interesting) distributions (these won t be on tests or problem sets)

29 Hypergeometric distribution balls to draw number of red balls X HypG (n, N, m) number of red balls drawn without replacement m N m k n k ( )( ) PMF: p (k)= N (n) X { expectation: E [ X ]=n total number of balls (black + red) if k ℤ, k min (n,m) otherwise m N variance: Var ( X )= n m( N2 n)( N m) N ( N 1)

30 Benford distribution base of number system (e.g. 1) X Benford (b) first digit of naturally occurring number 1 log b (1+ ) if d ℤ, d <b PMF: p X (d)= d otherwise {.35 Benford's Law Physical Constants.3 Frequency First Digit

31 Zipf distribution power law exponent (often close to 1) X Zipf (s, N ) rank of randomly chosen word PMF: p X (k)= { vocabulary size 1/k N s if k ℤ, k N s (1/n ) n=1 otherwise

32 A grid of random variables number of successes One trial X Ber( p) n=1 Several trials Interval of time X Bin(n, p) X Poi(λ) time to get successes X Geo( p) One success r=1 X NegBin (r, p) X Exp(λ) (coming soon!) Several successes One success after interval of time

33 Rapid-fire random variables number of Snapchats you receive today A) Ber ( p) D) Geo( p) B) Bin (n, p) E) NegBin (r, p) C) Poi(λ) Room: CS19SUMMER17

34 Rapid-fire random variables number of children until the first one with brown eyes A) Ber ( p) D) Geo( p) B) Bin (n, p) E) NegBin (r, p) with r = 1 C) Poi(λ) Room: CS19SUMMER17

35 Rapid-fire random variables whether the stock market went up today (1 = up, = down) A) Ber ( p) D) Geo( p) B) Bin (n, p) E) NegBin (r, p) with n = 1 C) Poi(λ) Room: CS19SUMMER17

36 Rapid-fire random variables number of years in some decade with more than 6 Atlantic hurricanes A) Ber ( p) D) Geo( p) B) Bin (n, p) E) NegBin (r, p) C) Poi(λ) Room: CS19SUMMER17