Intro to Probability Instructor: Alexandre Bouchard

Transcription

1 Intro to Probability Instructor: Alexandre Bouchard

2 Plan for today: Waiting times, continued Geometric distribution/pmf Negative binomial distribution/pmf Hypergeometric distribution/pmf Synthesis examples Continuous random variables

3 Reference for distributions See Distributions - Quick Ref under the Tools tab Distributions - Quick reference 07 Oct 2014 Bernoulli: Binomial: Parameter: a success probability Notation in JAGS: X ~ dbern(p) Mathematical notation:, or PMF: Mean: Variance: Relationship(s) to other random variables: Parameters: a success probability, and a number of trials Notation in JAGS: X ~ dbin(p, n) Note: and have a different order in JAGS and the textbook. Mathematical notation:, or PMF:, for Mean: Variance: Relationship(s) to other random variables: If, then can be written as the sum of Bernoulli random variables:

4 Logistics What s new/recent on the website: Webwork: due Friday. Written assignment due now

5 A few more words on joint PMFs

6 Prop 9 (version 2) Expectation when Y=g(X1, X2) You know: P(X1 = x1, X2 = x2) Y = g(x) Definition E[Y] = y * P(Y = y) y Image(Y) This is called the joint PMF, p(x1, x2) Second method: E[Y] = g(x1,x2) * P(X1 = x1, X2 = x2) x2 Image(X2) x1 Image(X1) Def 14

7 One more shortcut If: g(x1, x2) = x1 x2, ie. we want the expectation of the product, E[X1X2] X1 and X2 are independent E[X1X2] = x1 x2 * P(X1 = x1, X2 = x2) x1 x2 = x1 P(X1 = x1) x2 P(X2 = x2) x1 x2 = E[X1]E[X2]

8 Review: geometric distribution

9 Ex 45 Motivating example You need to hire 1 person You can only interview one random person per business day This person will be qualified for the job and hired with independent probability 2/3 What is the probability you succeed in 3 business days or less?

10 1/3 2/3 Decision tree Do not find employee on day 1 (X1 = 0) Find employee on day 1 (X1 = 1) 2/3 1/3 2/3 Do not find employee on day 2 (.., X2 = 0) Find employee on day 2 (..., X2 = 1) (T = 2) 2/9 1/3 2/3 Do not find employee on day 3 (.., X3 = 0) Find employee on day 3 (..., X3 = 1) (T = 3) 1/3 2/3 2/27 (T = 1)

11 k Geometric PMF This gives us a new PMF! What do we need to check? P (X = k) =(1 p) k 1 p k in 1, 2, 3,... Probability Mass Function (PMF) Approximation of P(Y = k)

12 Expectation of geometric Parameter: Probability of success p, a real in (0, 1) Expectation: 1 / p Variance: (1-p) / p 2 See textbook, p.157

13 Probability mass function (PMF) and cumulative distribution function (CDF)

14 Example: Geometric Distribution PMF P (X = k) =(1 p) k 1 p CDF P (X apple k) =1 (1 p) k k in 1, 2, 3,... Probability Mass Function (PMF) Cumulative Distribution Function (CDF) Approximation of P(Y = k) Approximation of P(R <= k) k k

15 Review: negative binomial distribution

16 Ex 46 Motivating example You need to hire 3 people You can only interview one random person per business day This person will be qualified for the job and hired with independent probability 2/3 What is the probability you succeed in exactly 6 business days

17 Negative binomial PMF This gives us a new PMF! P (X = i) = i 1 r 1 p r (1 p) i r i in r, r+1, r+2,... Probability Mass Function (PMF) Approximation of P(Y = k) k

18 Expectation of negative binomial Parameters: Number of required successes r, some positive integer Probability of success p, a real in (0, 1) Expectation: r / p Variance: r(1-p) / p 2 See textbook, p.159

19 Negative binomial: one more example

20 Ex 47 Hiring example, continued The firm is losing $100 for each day of the interview process You have the opportunity to get access to a CV database This would increase the hire probability to 3/4... (from 2/3 without CV database) but it costs $80 (flat rate until you find all 3 candidates)

21 Ex 47 Example continued The firm is losing $100 for each day of the interview process You have the opportunity to get access to a CV database This would increase the hire probability to 3/4... (from 2/3 without CV database) but it costs $80 (flat rate until you find all 3 candidates) A. The database reduces costs B. The database increases costs C. Same cost either way

22 Ex 47 Example continued The firm is losing $100 for each day of the interview process You have the opportunity to get access to a CV database This would increase the hire probability to 3/4... (from 2/3 without CV database) but it costs $80 (flat rate until you find all 3 candidates) A. The database reduces costs B. The database increases costs C. Same cost either way

23 Hypergeometric distribution

24 Ex. 4 Ecology: Estimating animal population sizes Example: finding the number of Sockeye salmon in the Pacific Ocean (!) Very important problem for conservation, setting fishing quotas, etc.

25 Ex. 4 Insight: the capturerecapture trick Population Capture and tag Recapture and count Examples:

26 Notation Population Capture and tag Recapture and count p = population size [= 14] t = number of tagged animals [= 4] c = number that are recaptured [=5] X = number out of the c recaptured that have the tag [=2] P(X = x)?

27 Exercises

28 Ex 48 Bird migration Consider the following model for a bird colony migrating North from the equator (a 10,000 km journey). At each day, the colony stays where it is with probability 1/3, else it move 500 km North What it the approximate probability that the migration takes exactly 25 days? A. 5% B. 10% C. 11% D. 17%

29 Ex 49 Smartphones Suppose 70% of UBC students have a smartphone What is the probability that exactly 7 people in a class of 10 have a smartphone? A. 98% B. 51% C. 47% D. 27%

30 Ex 50 Without Survey replacement! (a) A. approx. 4.5% B. approx 9% C. approx 18% D. approx 36% E. None of the above (b, c, d) : Left as a study problem

31 Problems Ex 49 Smartphones Ex 50 Survey Without replacement! Suppose 70% of UBC students have a smartphone What is the probability that exactly 7 people in a class of 10 have a smartphone? A. 98% B. 51% C. 47% D. 27% (a) A. approx. 4.5% B. approx 9% C. approx 18% D. approx 36% E. None of the above (b, c, d) : Left as a study problem

32 Ex 48 Bird migration Consider the following model for a bird colony migrating North from the equator (a 10,000 km journey). At each day, the colony stays where it is with probability 1/3, else it move 500 km North What it the approximate probability that the migration takes exactly 25 days? A. 5% B. 10% C. 11% D. 17%

33 Ex 49 Smartphones Suppose 70% of UBC students have a smartphone What is the probability that exactly 7 people in a class of 10 have a smartphone? A. 98% B. 51% C. 47% D. 27%

34 Ex 50 Without Survey replacement! (a) A. approx. 4.5% B. approx 9% C. approx 18% D. approx 36% E. None of the above (b, c, d) : Left as a study problem

35 Continuous distributions

36 The need to go beyond discrete random variables Many aspects of the world are (basically) continuous:

37 The need to go beyond discrete random variables Even when a problem is discrete, it can be simpler to think about it as a continuous quantity Money Population of a country Positions on a genome...

38 How to represent (describe) non-discrete random variables? Probability mass function? (PMF) p(x) = P(X = x) PMF might not sum to one! Example: pmf of roulette angle Better: CDF F(x) = P(X x)

39 Continuous random variables CDF is great for representation (all real valued r.v. have a CDF), but not always convenient for computing expectations). Another description: density function

40 Def 15 Density The function f is a density for X if for any interval A Example: A = [a, b] area = probability height = density

41 Informal interpretation

42 Def 16 Example: the continuous, uniform distribution Parameters: a, b a < b; the boundaries of an interval Roulette angle example : a = 0, b = 360 Density: proportional to an indicator on the interval [a, b] Fact: all real random variables (and more) can be constructed from uniform distributions (more later)

43 Def 16 The uniform distribution Density CDF

44 Important remarks R

45 Ex 52 Exercise If X has density f(x) = { 0 otherwise C exp(-2x) for x 0 What is P(X < 1)? A B C D. 1.0

46 Properties of densities

47 Relation between density and CDF Density CDF c Go from density to CDF by integrating: c The CDF at x is the integral (area under the curve) of the density, from - to x

48 Relation between density and CDF Density CDF c Go from CDF to density by differentiating: The density at x is the derivative (slope) of the CDF c At points where a CDF is differentiable

49 Prop 13 Properties of density By calculus: (at points of F where it is differentiable) By axioms of pr: R R