Binomial Approximation and Joint Distributions Chris Piech CS109, Stanford University

Transcription

1 Binomial Approximation and Joint Distributions Chris Piech CS109, Stanford University

2 Four Prototypical Trajectories Review

3 The Normal Distribution X is a Normal Random Variable: X ~ N(µ, s 2 ) Probability Density Function (PDF): 1 f ( x) = e s 2p E[X ] = µ 2 Var( X ) = s 2 2 -( x-µ ) / s Also called Gaussian Note: f(x) is symmetric about µ 2 where - < x f (x) < µ x

4 Simplicity is Humble µ Simple. Will generalize probability σ 2 value * A Gaussian maximizes entropy for a given mean and variance

5 Density vs Cumulative 1 CDF of a Normal F(x) PDF of a Normal f(x) f(x) = derivative of probability F(x) = P(X < x)

6 Probability Density Function N (µ, 2 ) exponential the distance to the mean f(x) = 1 p 2 e (x µ) probability density at x a constant sigma shows up twice

7 Cumulative Density Function N (µ, 2 ) CDF of Standard Normal: A function that has been solved for numerically F (x) = x µ The cumulative density function (CDF) of any normal Table of F(z) values in textbook, p. 201 and handout

8 Four Prototypical Trajectories Great questions!

9 Four Prototypical Trajectories 68% rule only for Gaussians?

10 68% Rule? What is the probability that a normal variable X N(µ, has a value within one standard deviation of its mean? µ µ X µ µ + µ P (µ <X<µ+ )=P < < = P ( 1 <Z<1) = (1) ( 1) = (1) [1 (1)] =2 (1) 1 = 2[0.8413] 1 = Only applies to normal 2 )

11 68% Rule? Counter example: Uniform X Uni(, ) Var(X) = ( )2 12 = p Var(X) = p P (µ <X<µ+ ) = 1 apple 2( ) p 12 = p 2 apple p 12 =0.58

12 Four Prototypical Trajectories How does python sample from a Gaussian?

13 from random import * for i in range(10): mean = 5 std = 1 sample = gauss(mean, std) print sample How does this work?

14 How Does a Computer Sample Normal? Inverse Transform Sampling 1 CDF of the Standard Normal (x)

15 How Does a Computer Sample Normal? Step 1: pick a uniform number y between 0,1-5 Inverse Transform Sampling 1 0 Step 2: Find the x such that (x) =y x = 1 (y) CDF of the Standard Normal (x) Further reading: Box Muller transform 5

16 Continuous RV Relative Probability f(x) X = time to finish pset 3 X ~ N(10, 2) Time to finish pset 3 x How much more likely are you to complete in 10 hours than in 5? P (X = 10) P (X = 5) = = = = "f(x = 10) "f(x = 5) f(x = 10) f(x = 5) 1 p 2 2 p p 4 e p 1 4 e 0 = e0 e 25 4 (10 µ)2 2 e 2 2 (5 µ)2 e 2 2 (10 10)2 4 (5 10)2 4 = 518

17 Four Prototypical Trajectories Imagine you are sitting a test

18 Website Testing 100 people are given a new website design X = # people whose time on site increases CEO will endorse new design if X 65 What is P(CEO endorses change it has no effect)? X ~ Bin(100, 0.5). Want to calculate P(X 65) Give a numerical answer X P (X 65) = (0.5) i (1 0.5) 100 i i i=65

19 Normal Approximates Binomial There is a deep reason for the Binomial/Normal similarity

20 Normal Approximates Binomial

21 Four Prototypical Trajectories Let s invent an approximation!

22 Website Testing 100 people are given a new website design X = # people whose time on site increases CEO will endorse new design if X 65 What is P(CEO endorses change it has no effect)? X ~ Bin(100, 0.5). Want to calculate P(X 65) P(X = x) What is variance? E[X] = 50 x

23 Website Testing 100 people are given a new website design X = # people whose time on site increases CEO will endorse new design if X 65 What is P(CEO endorses change it has no effect)? X ~ Bin(100, 0.5). Want to calculate P(X 65) np = 50 np( 1 p) = 25 np( 1 p) = 5 Use Normal approximation: Y ~ N(50, 25) P (Y 65) = P Y 50 5 > = P (Z >3) = 1 (3) Using Binomial: P( X ³ 65)»

24 Website Testing P(X = x) E[X] = 50 x

25 Continuity Correction If Y (normal) approximates X (binomial) P (X 65) What about 64.9? P (Y 64.5) p(x) or f(x) Bin(100, 0.5) Normal(50, 25) x

26 Continuity Correction If Y (normal) approximates X (binomial) Discrete (eg Binomial) probability question Continuous (Normal) probability question X = < Y < 6.5 X >= 6 Y > 5.5 X > 6 Y > 6.5 X < 6 Y < 5.5 X <= 6 Y < 6.5 * Note: Binomial is always defined in units of 1

27 Comparison when n = 100, p = 0.5 P(X = k) k

28 Who Gets to Approximate? X ~ Bin(n, p) Poisson approx. n large (> 20), p small (< 0.05) Normal approx. n large (> 20), p is mid-ranged np(1-p) > 10 If there is a choice, go with the normal approximation

29 Stanford Admissions Stanford accepts 2050 students this year Each accepted student has 84% chance of attending X = # students who will attend. X ~ Bin(2050, 0.84) What is P(X > 1745)? np = 1722 np(1 p) = 276 p np(1 p) = 16.6 Use Normal approximation: Y ~ N(1722, 276) P (X >1745) P (Y > ) P (Y ) = P Y > = P (Z >1.4) 0.08

30 Changes in Stanford Admissions Class of 2021 Admit Rates Lowest in University History Fewer students were admitted to the Class of 2021 than the Class of 2019, due to the increase in Stanford s yield rate which has increased over 5 percent in the past four years, according to Colleen Lim M.A. 80, Director of Undergraduate Admission. 68% 10 years ago 84% last year

31 Continuous Random Variables Uniform Random Variable X Uni(, ) All values of x between alpha and beta are equally likely. Normal Random Variable X N (µ, Aka Gaussian. Defined by mean and variance. Goldilocks distribution. 2 ) Exponential Random Variable X Exp( ) Time until an event happens. Parameterized by lambda (same as Poisson). Beta Random Variable How mysterious and curious. You must wait a few classes J.

32 Four Prototypical Trajectories Joint Distributions

33 CS109 Joint Go to this URL:

34 Four Prototypical Trajectories Events occur with other events

35 Probability Table for Discrete States all possible outcomes with several discrete variables A probability table is not parametric If #variables is > 2, you can have a probability table, but you can t draw it on a slide All values of A All values of B P(A = 1, B = 1) Here, means and Every outcome falls into a bucket

36 Discrete Joint Mass Function For two discrete random variables X and Y, the Joint Probability Mass Function is: p X Y = Marginal distributions:, ( a, b) = P( X = a, Y b) p = = = å X ( a) P( X a) px, Y ( a, y) Example: X = value of die D 1, Y = value of die D 2 y =å p b) = P( Y = b) p ( x, b) Y ( X, Y x 6 6 p X, Y (1, y) = y= 1 y= 1 = å å 1 36 P( X 1) = = 1 6

37 A Computer (or Three) In Every House Consider households in Silicon Valley A household has X Macs and Y PCs Can t have more than 3 Macs or 3 PCs Y X p Y (y) ? p X (x)

38 A Computer (or Three) In Every House Consider households in Silicon Valley A household has X Macs and Y PCs Can t have more than 3 Macs or 3 PCs Y X p Y (y) p X (x)

39 A Computer (or Three) In Every House Consider households in Silicon Valley A household has X Macs and Y PCs Can t have more than 3 Macs or 3 PCs Y X p Y (y) p X (x) Marginal distributions

40 CS109 Joint Results Go to this URL:

41 Four Prototypical Trajectories Way Back

42 Permutations How many ways are there to order n distinct objects? n!

43 Binomial How many ways are there to make an unordered selection of r objects from n objects? How many ways are there to order n objects such that: r are the same (indistinguishable) (n r) are the same (indistinguishable)? r!(n n! r)! = n r Called the binomial because of something from Algebra

44 Multinomial How many ways are there to order n objects such that: n 1 are the same (indistinguishable) n 2 are the same (indistinguishable) n r are the same (indistinguishable)? n! n 1!n 2!...n r! = n n 1,n 2,...,n r Note: Multinomial > Binomial

45 Binomial Distribution Consider n independent trials of Ber(p) rand. var. X is number of successes in n trials X is a Binomial Random Variable: X ~ Bin(n, p) Probability of exactly i successes Binomial # ways of ordering the successes ænö i n-i P( X = i) = p( i) = ç p (1 - p) i = è i ø 0,1,..., n Probability of each ordering of i successes is equal + mutually exclusive

46 Four Prototypical Trajectories End Way Back

47 The Multinomial Multinomial distribution n independent trials of experiment performed Each trial results in one of m outcomes, with respective probabilities: p 1, p 2,, p m where X i = number of trials with outcome i m å i = 1 p i = 1 P c1 c2 ( X = c1, X 2 = c2,..., X m = cm) = ç p1 p2 c1, c2,..., cm where æ n 1 ö ç è ø Joint distribution m åci = i=1 n and Multinomial # ways of ordering the successes æ ç èc, c n,..., c ö ø = c! c n!! c 1 2 m 1 2 m!... p c m m Probabilities of each ordering are equal and mutually exclusive

48 6-sided die is rolled 7 times Roll results: 1 one, 1 two, 0 three, 2 four, 0 five, 3 six This is generalization of Binomial distribution Binomial: each trial had 2 possible outcomes Multinomial: each trial has m possible outcomes !1!0!2!0!3! 7! 3) 0, 2, 0, 1, 1, ( ø ö ç è æ = ø ö ç è æ ø ö ç è æ ø ö ç è æ ø ö ç è æ ø ö ç è æ ø ö ç è æ = = = = = = = X X X X X X P Hello Die Rolls, My Old Friends

49 Probabilistic Text Analysis Ignoring order of words, what is probability of any given word you write in English? P(word = the ) > P(word = transatlantic ) P(word = Stanford ) > P(word = Cal ) Probability of each word is just multinomial distribution What about probability of those same words in someone else s writing? P(word = probability writer = you) > P(word = probability writer = non-cs109 student) After estimating P(word writer) from known writings, use Bayes Theorem to determine P(writer word) for new writings!

50 A Document is a Large Multinomial According to the Global Language Monitor there are 988,968 words in the english language used on the internet. The

51 Text is a Multinomial P Example document: Pay for Viagra with a credit-card. Viagra is great. So are credit-cards. Risk free Viagra. Click for free. n = 18 Viagra = 2 Free = 2 Risk = 1 Credit-card: 2 For = 2 Probability of seeing this document spam spam = It s a Multinomial! n! 2!2!...2! p2 viagrap 2 free...p 2 for The probability of a word in spam being viagra

52 Four Prototypical Trajectories Who wrote the federalist papers?

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54 Old and New Analysis Authorship of Federalist Papers 85 essays advocating ratification of US constitution Written under pseudonym Publius o Really, Alexander Hamilton, James Madison and John Jay Who wrote which essays? o Analyzed probability of words in each essay versus word distributions from known writings of three authors

55 Four Prototypical Trajectories Let s write a program!