Introduction to Probability I: Expectations, Bayes Theorem, Gaussians, and the Poisson Distribution. 1
|
|
- Deirdre Henderson
- 5 years ago
- Views:
Transcription
1 Itroductio to Probability I: Expectatios, Bayes Theorem, Gaussias, ad the Poisso Distributio. 1 Pakaj Mehta February 25, Read: This will itroduce some elemetary ideas i probability theory that we will make use of repeatedly. Stochasticity plays a major role i biology. I these ext few lectures, we will develop the mathematical tools to treat stochasticity i biological systems. I this lecture, we will review some basic cocepts i probability theory. We will start with some basic ideas about probability ad defie various momets ad cumulats. We will the discuss how momets ad cumulats ca be calculated usig geeratig fuctios ad cumulat fuctios. We will the focus o two widely occurrig uiversal distributios : the Gaussia ad the Poisso distributio.
2 distributio. 2 Probability distributios of oe variable Cosider a probability distributio p(x) of a sigle variable where X ca take discrete values i some set x A or cotiuous values over the real umbers 2. We ca defie the expectatio value of a fuctio of this radom variable f (X) as f (X) = p(x) (1) x A 2 We will try to deote radom variables by capital letters ad the values they ca take by the correspodig lower case letter. for the discrete case ad f (X) = dx f (x)p(x), (2) i the cotiuous case. For simplicity, we will write thigs assumig x is discrete but it is straight forward to geeralize to the cotiuous case by replacig by itegrals. We ca defie the -th momet of x as X = p(x)x. (3) x The first momet is ofte called the mea ad we will deote it as x = µ x. We ca also defie the -th cetral of x as (X X ) = p(x)(x x ). (4) x The secod cetral momets is ofte called the variace of the distributio ad i may cases characterizes the typical width of the distributio. We will ofte deote the variace of a distributio by σ 2 x = (X X ) 2. Exercise: Show that the secod cetral momet ca also be writte as X 2 X 2. Example: Biomial Distributio Cosider the Biomial distributio which describes the probability of gettig heads if oe tosses a coi N times. Deote this radom variable by X. Note that X ca take o N + 1 values = 0, 1,..., N. If probability of gettig a head o a coi toss is p, the it is clear that ( ) N p() = p (1 p) N (5)
3 distributio. 3 Let us calculate the mea of this distributio. To do so, it is useful to defie q = (1 p). X = N p() =0 ( N = ) p q N Now we will use a very clever trick. We ca formally treat the above expressio as fuctio of two idepedet variables p ad q. Let us defie G(p, q) = ( ) N p q ( N ) = (p + q) N (6) ad take a partial derivative with respect to p. Formally, we kow the above is the same as X = ( ) N p q N = p p G(p, q) = p p (p + q) N = pn(p + q) N 1 = pn, (7) where i the last lie we have substituted q = 1 p. This is exactly what we expect 3. The mea umber of heads is exactly equal to the umber of times we toss the coi times the umber probability we get a heads. Exercise: Show that the variace of the Biomial distributio is just σ 2 = Np(1 p). 3 Whe oe first ecouters this kid of formal trick, it seems like magic ad cheatig. However, as log as the probability distributios coverge so that we ca iterchage sums/itegrals with derivatives there is othig wrog with this as strage as that seems. Probability distributio of multiple variables I geeral, we will also cosider probability distributios of M variables p(x 1,..., X M ). We ca get the margial distributios of a sigle variable by itegratig (summig over) all other variables 4 We have that p(x j ) = dx 1... dx j 1 dx j+1... dx M p(x 1, x 2,..., x M ). (8) 4 I this sectio, we will use cotiuous otatio as it is easier ad more compact. Discrete variables ca be treated by replacig itegrals with sums. Similarly, we ca defie the margial distributio over ay subset of variables by itegratig over the remaiig variables.
4 distributio. 4 We ca also defie the coditioal probability of a variable x i which ecodes the probability that variable X i takes o a give value give the values of all the other radom variables. We deote this probability by p(x i X 1,... X i 1, X i+1,..., X M ). Oe importat formula that we will make use of repeatedly is Bayes formula which relates margials ad coditioals to the full probability distributio p(x 1,..., X M ) = p(x i x 1,... X i 1, X i+1,..., X M )p(x 1,... X i 1, X i+1,..., X M ). (9) For the special case of two variables, this reduces to the formula p(x, Y) = p(x Y)p(Y) = p(y X)p(Y). (10) This is oe of the fudametal results i probability ad we will make use of it extesively whe discussig iferece i our discussio of early visio. 5 Exercise: Uderstadig test results. 6. Cosider a mammogram to diagose breast cacer. Cosider the followig facts (the umbers are t exact but reasoable). 1% of wome have breast cacer. 5 Bayes formula is crucial to uderstadig probability ad I hope you have ecoutered it before. If you have ot, please do some exercises that use Bayes formula that are everywhere o the web. 6 This is from the ice website https: //betterexplaied.com/articles/ a-ituitive-ad-short-explaatio-of-bayes-theore Mammograms miss 20% of cacers whe that are there. 10% of mammograms detect cacer eve whe it is ot there. Suppose you get a positive test result, what are the chaces you have cacer? Exercise: Cosider a distributio of two variables p(x, y). Defie a variable Z = X + Y 7 Show that Z = X + Y 7 We will make use of these results repeatedly. Please make sure you ca derive them ad uderstad them thoroughly. Show that σ 2 z = Z 2 Z 2 = σ 2 x + σ 2 y + 2Cov(X, Y) where we have defied the covariace of X ad Y as Cov(x, y) = (X X )(Y Y ).
5 distributio. 5 Show that if x ad y are idepedet variables tha Cov(x, y) = 0 ad σ 2 z = σ 2 x + σ 2 y. Law of Total Variace A importat result we will make use of tryig to uderstad experimets is the law of total variace. Cosider two radom variables X ad Y. We would like to kow how much the variace of Y is explaied by the value of X. I other words, we would like to decompose the variace of Y ito two terms: oe term that captures how much of the variace of Y is explaied by X ad aother term that captures the uexplaied variace. It will be helpful to defie some otatio to make the expectatio value clearer E X [ f (X)] = X x = dxp(x) f (x) (11) Notice that we ca defie the coditioal expectatio of a fuctio Y give that X = x by E Y [ f (Y) X = x] = dyp(y X = x) f (y). (12) For the special case whe f (Y) = Y 2 we ca defie the coditioal variace which measures how much does Y vary is we fix the value of X Var Y [Y X = x] = E Y [Y 2 X = x] E Y [Y X = x] 2 (13) We ca also defie a differet variace that measures the variace of the E Y [Y X] viewed as a radom fuctio of X: Var X [E Y [Y X]] = E X [E Y [Y X] 2 ] E X [E Y [Y X]] 2. (14) This is a well defied probability distributio sice p(y x) is the marigal probability distributio over y. Notice ow, that usig Bayes rule we ca write E Y [ f (y)] = dyp(y) f (y) = dxdyp(y x)p(x) f (y) = dxp(x) dyp(y x) f (y) = E X [E Y [ f (Y) X]] (15)
6 distributio. 6 So ow otice that we ca write σ 2 Y = E Y [Y 2 ] E Y [Y] 2 = E X [E Y [Y 2 X]] E X [E Y [Y X]] 2 = E X [E Y [Y 2 X]] E X [E Y [Y X] 2 ] + E X [E Y [Y X] 2 ] E X [E Y [Y X]] 2 = E X [E Y [Y 2 X] E Y [Y X] 2 ] + Var X [E Y [Y X]] = E X [Var Y [Y X]] + Var X [E Y [Y X]] (16) The first term is ofte called the uexplaied variace ad the secod term is the explaied variace. To see why, cosider the case where the radom variable Y are related by a liear fuctio plus oise Y = ax + ɛ, where ɛ is a third radom variable with mea ɛ = b ad ɛ 2 = σ 2 ɛ + µ 2. Furthermore, let E X [X] = µ x. The otice, that i this case all the radomess of Y is cotaied i ɛ. Notice that E[Y] = aµ x + b (17) ad E ɛ [Y X] = ax + E ɛ [ɛ] = ax + b (18) We ca also write Var X [E Y [Y X]] = E X [E Y [Y X] 2 ] E X [E Y [Y X]] 2 = E X [(ax + b) 2 ] (aµ x + b) 2 = a 2 E X [X 2 ] + 2abµ x + b 2 (aµ x + b) 2 = a 2 σ 2 X (19) This is clearly the variace i Y explaied by the variace i X. Usig the law of total variace, we see that i this case is the uexplaied variace. E X [Var Y [Y X]] = σ 2 Y a2 σ 2 X (20) Geeratig Fuctios ad Cumulat Geeratig Fuctios We are ow i the positio to itroduce some more machiery. These are the momet geeratig fuctios ad cumulat geeratig fuctios. Cosider a probability distributio p(x). We would like a easy ad efficiet way to calculate the momets of this distributio. It turs out that there is a easy way to do this usig ideas that will be familiar from Statistical Mechaics.
7 distributio. 7 Geeratig Fuctios Give a discrete probability distributio, p(x) we ca defie a momet-geeratig fuctio G(t) for the probability distributio as G(t) = e tx = dxp(x)e tx (21) Alteratively, we ca also defie a (ordiary) geeratig fuctio for the distributios G(z) = Z X = p(x)z x (22) x If the probability distributio is cotiuous, we defie the mometgeeratig fuctio as 8 G(t) = e tx = dxp(x)e tx. (23) 8 If x is defied over the reals, this is just the Laplace trasform. Notice that G(t = 0) = G(z = 1) = 1 sice probability distributios are ormalized. These fuctios have some really ice properties. Notice that the th momet of X ca be calculated by takig the -th partial derivative of the momet geeratig fuctio evaluated at t = 0: X N = G(t) t t=0 (24) Alteratively, we ca write i terms of the ordiary geeratig fuctio X N = (z z ) G(z) z=0 (25) where (z z ) meas apply this operator times. Example: Geeratig Fuctio of the Biomial Distributio Let us retur to the Biomial distributio above.the ordiary geeratig fuctio is give by G(z) = ( ) N p q N z = (pz + q) N = (pz + (1 p)) N (26) Let us calculate the mea usig this µ X = z z G(z) = [znp(pz + (1 p)) N 1 ] z=1 = Np (27) We ca also calculate the secod momet X 2 = z z (z z G(z)) = [znp(pz + (1 p)) N 1 ] z=1 + z [Np(pz + (1 p)) N 1 ] z=1 = Np + N(N 1)p 2 = Np(1 p) + N 2 p 2 (28)
8 distributio. 8 ad the variace σ 2 X = X2 X 2 = Np(1 p). (29) Example: Normal Distributio We ow cosider a Normal Radom Variable X whose probability desity is give by 1 p(x) = e (x µ)2 σ 2. (30) 2πσ 2 Let us calculate the momet geeratig fuctio 1 G(t) = dx e (x µ)2 σ 2 e tx. (31) 2πσ 2 It is useful to defie a ew variable u = (x µ)/σ 9. I terms of this variable we have that 1 G(t) = due u2 /2+t(σu+µ) 2π = e µt+t2 σ 2 /2 1 2π due (u σu/2)2 2 9 Notice that u is just the z-score i statistics. = e µt+t2 σ 2 /2 (32) Exercise: Show that the th momet of a Gaussia with mea zero (µ = 0) ca be writte as X p 0 if p is odd = (33) (p 1)!σ p if p is eve This is oe of the most importat properties of the Gaussia/Normal distributio. All higher order momets ca be expressed solely i terms of the mea ad variace! We will make use of this agai. The Cumulat Geeratig Fuctio Aother fuctio we ca defie is the Cumulat geeratig fuctio of a probability distributio K(t) = log e tx = log G(t). (34) The th derivative of K(t) evaluated at t = 0 is called the -th cumulat: κ = K () (0). (35) Let us ot look the first few cumulats. Notice that κ 1 = G (0) G(0) = µ X 1 = µ X. (36)
9 distributio. 9 Figure 1: "Whatever the form of the populatio distributio, the samplig distributio teds to a Gaussia, ad its dispersio is give by the Cetral Limit Theorem" (figure ad captio from Wikipedia) Similarly, it is easy to show that the secod cumulat is just the variace: κ 2 = G (0) G(0) [G (0)] 2 G(0) 2 = X 2 X 2 = σx 2 (37) Notice that the cumulat geeratig fuctio is just the Free Eergy i statistical mechaics ad the momet-geeratig fuctio is the partitio fuctio. Cetral Limit Theorem ad the Normal Distributio Oe of the most ubiquitous distributios that we see i all of physics, statistics, ad biology is the Normal distributio. This is because of the Cetral Limit theorem. Suppose that we draw some radom variables X 1, X 2,..., X N idetically ad idepedetly from a distributio with mea µ ad variace σ 2. Let us defie a variable S N = X 1 + X X N. (38) N The as N, the distributio of S N is well described by a Gaussia/Normal distributio with mea µ ad variace σ 2 /N. We will deote such a ormal distributio by N (µ, σ 2 /N). This is true regardless of the origial distributio (see Figure 1). The mai thig we should take away is that the variace decreases as 1/N with the umber of samples N that we take! Example: The Biomial Distributio We ca ask how the sample mea chages with the umber of samples for a Beroulli variable with probability p = 1/2 for various N. This Figure 2 from Wikipedia. The Poisso Distributio There is a secod uiversal distributio that occurs ofte i Biology. This is the distributio that describes the umber of rare evets
10 distributio. 10 Figure 2: Meas from various N samples draw from Beroulli distributio with p = 1/2. Notice it coverges more ad more to a Gaussia. (figure ad captio from Wikipedia) we expect i some time iterval T. The Poisso distributio is applicable if the followig assumptios hold: The umber of evets is discrete k = 0, 1, 2,.... The evets are idepedet, caot occur simultaeously, occur at a costat rate per uit time r. Geerally the umber of trials is large ad probability of success is small. Examples of thigs that ca be described by the Poisso distributio iclude: Photos arrivig i a microscope (especially at low itesity) The umber of mutatios per uit legth of DNA The umber of uclear decays i a time iterval The umber of soldiers killed by horse-kicks each year i each corps i the Prussia cavalry. This example was made famous by a book of Ladislaus Josephovich Bortkiewicz (1868Ð1931) (from Wikipedia) "The targetig of V-1 flyig bombs o Lodo durig World War II ivestigated by R. D. Clarke i 1946". Let us deote the mea umber of evets that occur i a time t by λ = rt. The, the probability that k evets occurs is give by p(k) = e λ λk k!. (39)
11 distributio. 11 Figure 3: Examples of Poisso distributio for differet meas.. (figure from Wikipedia) It is easy to check that k p(k) = 1 usig the Taylor series of the expoetial. We ca also calculate the mea ad variace. Notice we ca defie the geeratig fuctio of the Poisso as G(t) = e λ etk (λ) k z k! The cumulat-geeratig fuctio is just = e λ(1+et). (40) K(t) = log G(t) = λ(1 + e t ) (41) From this we ca easily get the all cumulats sice this is just differetiatig the expressio above times κ = K () (0) = λ. (42) I other words, all the higher order cumulats of the Poisso distributio are the same ad equal to the mea. Aother defiig feature of the Poisso distributio is that it has o memory. Sice thigs happe at a costat rate, there is o memory. We will see that i geeral to create o-poisso distributios (with memory), we will have to bur eergy! More o this cryptic statemet later.
Approximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More informationAMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
More informationExponential Families and Bayesian Inference
Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationEE 4TM4: Digital Communications II Probability Theory
1 EE 4TM4: Digital Commuicatios II Probability Theory I. RANDOM VARIABLES A radom variable is a real-valued fuctio defied o the sample space. Example: Suppose that our experimet cosists of tossig two fair
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationIE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.
IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationMath 525: Lecture 5. January 18, 2018
Math 525: Lecture 5 Jauary 18, 2018 1 Series (review) Defiitio 1.1. A sequece (a ) R coverges to a poit L R (writte a L or lim a = L) if for each ǫ > 0, we ca fid N such that a L < ǫ for all N. If the
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationAdvanced Stochastic Processes.
Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 15
CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationCS284A: Representations and Algorithms in Molecular Biology
CS284A: Represetatios ad Algorithms i Molecular Biology Scribe Notes o Lectures 3 & 4: Motif Discovery via Eumeratio & Motif Represetatio Usig Positio Weight Matrix Joshua Gervi Based o presetatios by
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationDistribution of Random Samples & Limit theorems
STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to
More information6. Sufficient, Complete, and Ancillary Statistics
Sufficiet, Complete ad Acillary Statistics http://www.math.uah.edu/stat/poit/sufficiet.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 6. Sufficiet, Complete, ad Acillary
More informationNOTES ON DISTRIBUTIONS
NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which
More informationThis section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
More informationVariance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationChapter 5. Inequalities. 5.1 The Markov and Chebyshev inequalities
Chapter 5 Iequalities 5.1 The Markov ad Chebyshev iequalities As you have probably see o today s frot page: every perso i the upper teth percetile ears at least 1 times more tha the average salary. I other
More informationSTAT Homework 1 - Solutions
STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
More informationLecture 2: Probability, Random Variables and Probability Distributions. GENOME 560, Spring 2017 Doug Fowler, GS
Lecture 2: Probability, Radom Variables ad Probability Distributios GENOME 560, Sprig 2017 Doug Fowler, GS (dfowler@uw.edu) 1 Course Aoucemets Problem Set 1 will be posted Due ext Thursday before class
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationLast time: Moments of the Poisson distribution from its generating function. Example: Using telescope to measure intensity of an object
6.3 Stochastic Estimatio ad Cotrol, Fall 004 Lecture 7 Last time: Momets of the Poisso distributio from its geeratig fuctio. Gs () e dg µ e ds dg µ ( s) µ ( s) µ ( s) µ e ds dg X µ ds X s dg dg + ds ds
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More information0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =
PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several well-ow discrete probability distributios ad study some of their properties. Some of these distributios, lie
More informationDiscrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions
CS 70 Discrete Mathematics for CS Sprig 2005 Clacy/Wager Notes 21 Some Importat Distributios Questio: A biased coi with Heads probability p is tossed repeatedly util the first Head appears. What is the
More informationChapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p
Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE Part 3: Summary of CI for µ Cofidece Iterval for a Populatio Proportio p Sectio 8-4 Summary for creatig a 100(1-α)% CI for µ: Whe σ 2 is kow ad paret
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationLecture 12: September 27
36-705: Itermediate Statistics Fall 207 Lecturer: Siva Balakrisha Lecture 2: September 27 Today we will discuss sufficiecy i more detail ad the begi to discuss some geeral strategies for costructig estimators.
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationf X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probablity that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c} Pr(X c) = Pr({s S X(s)
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Math 36Q Fall 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2 to
More informationMathematics 170B Selected HW Solutions.
Mathematics 17B Selected HW Solutios. F 4. Suppose X is B(,p). (a)fidthemometgeeratigfuctiom (s)of(x p)/ p(1 p). Write q = 1 p. The MGF of X is (pe s + q), sice X ca be writte as the sum of idepedet Beroulli
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationLecture 12: November 13, 2018
Mathematical Toolkit Autum 2018 Lecturer: Madhur Tulsiai Lecture 12: November 13, 2018 1 Radomized polyomial idetity testig We will use our kowledge of coditioal probability to prove the followig lemma,
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationLECTURE 8: ASYMPTOTICS I
LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece
More informationFinal Review for MATH 3510
Fial Review for MATH 50 Calculatio 5 Give a fairly simple probability mass fuctio or probability desity fuctio of a radom variable, you should be able to compute the expected value ad variace of the variable
More informationDiscrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15
CS 70 Discrete Mathematics ad Probability Theory Sprig 2012 Alistair Siclair Note 15 Some Importat Distributios The first importat distributio we leared about i the last Lecture Note is the biomial distributio
More informationLecture 2: April 3, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 2: April 3, 203 Scribe: Shubhedu Trivedi Coi tosses cotiued We retur to the coi tossig example from the last lecture agai: Example. Give,
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationLecture 2: Probability, Random Variables and Probability Distributions. GENOME 560, Spring 2015 Doug Fowler, GS
Lecture 2: Probability, Radom Variables ad Probability Distributios GENOME 560, Sprig 2015 Doug Fowler, GS (dfowler@uw.edu) 1 Course Aoucemets Problem Set 1 will be posted Due ext Thursday before class
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationCS 330 Discussion - Probability
CS 330 Discussio - Probability March 24 2017 1 Fudametals of Probability 11 Radom Variables ad Evets A radom variable X is oe whose value is o-determiistic For example, suppose we flip a coi ad set X =
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationElements of Statistical Methods Lots of Data or Large Samples (Ch 8)
Elemets of Statistical Methods Lots of Data or Large Samples (Ch 8) Fritz Scholz Sprig Quarter 2010 February 26, 2010 x ad X We itroduced the sample mea x as the average of the observed sample values x
More information10/31/2018 CentralLimitTheorem
10/31/2018 CetralLimitTheorem http://127.0.0.1:8888/bcovert/html/cs237/web/homeworks%2c%20labs%2c%20ad%20code/cetrallimittheorem.ipyb?dowload=false 1/10 10/31/2018 CetralLimitTheorem Cetral Limit Theorem
More informationJanuary 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationTests of Hypotheses Based on a Single Sample (Devore Chapter Eight)
Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........
More informationLecture 3. Properties of Summary Statistics: Sampling Distribution
Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary
More informationECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002
ECE 330:541, Stochastic Sigals ad Systems Lecture Notes o Limit Theorems from robability Fall 00 I practice, there are two ways we ca costruct a ew sequece of radom variables from a old sequece of radom
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter & Teachig Material.
More informationUnderstanding Samples
1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We
More informationSome Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables
Some Basic Probability Cocepts 2. Experimets, Outcomes ad Radom Variables A radom variable is a variable whose value is ukow util it is observed. The value of a radom variable results from a experimet;
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Math 36Q Sprig 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter 2 & Teachig
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationOutput Analysis and Run-Length Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More informationPoisson approximations
The Bi, p) ca be thought of as the distributio of a sum of idepedet idicator radom variables X +...+ X, with {X i = } deotig a head o the ith toss of a coi. The ormal approximatio to the Biomial works
More informationDiscrete probability distributions
Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop
More informationUNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY
UNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY Structure 2.1 Itroductio Objectives 2.2 Relative Frequecy Approach ad Statistical Probability 2. Problems Based o Relative Frequecy 2.4 Subjective Approach
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationLimit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1-p).
Limit Theorems Covergece i Probability Let X be the umber of heads observed i tosses. The, E[X] = p ad Var[X] = p(-p). L O This P x p NM QP P x p should be close to uity for large if our ituitio is correct.
More information4. Basic probability theory
Cotets Basic cocepts Discrete radom variables Discrete distributios (br distributios) Cotiuous radom variables Cotiuous distributios (time distributios) Other radom variables Lect04.ppt S-38.45 - Itroductio
More informationCSE 527, Additional notes on MLE & EM
CSE 57 Lecture Notes: MLE & EM CSE 57, Additioal otes o MLE & EM Based o earlier otes by C. Grat & M. Narasimha Itroductio Last lecture we bega a examiatio of model based clusterig. This lecture will be
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
More informationLecture 4. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za
More informationDISTRIBUTION LAW Okunev I.V.
1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated
More informationLearning Theory: Lecture Notes
Learig Theory: Lecture Notes Kamalika Chaudhuri October 4, 0 Cocetratio of Averages Cocetratio of measure is very useful i showig bouds o the errors of machie-learig algorithms. We will begi with a basic
More information