STAT Homework 2 - Solutions

Size: px
Start display at page:

Download "STAT Homework 2 - Solutions"

Transcription

1 STAT Homework - Solutios Fall 08 September 4, 08 This cotais solutios for Homework. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better isight. Problem. Suppose we geerate a radom variable X i the followig way. First we flip a fair coi. If the coi is heads, take X to have a N µ, σ distributio. If the coi is tails, take X to have a N µ, σ distributio. Fid: a The mea of X b The stadard deviatio of X Solutio. Let Z Beroulli, the X Z N µ, σ, X Z 0 N µ, σ. Note: Questio asks for stadard deviatio, ot the variace be careful a EX E[[E[X Z]] E[Zµ + Zµ ] µ + µ b VarX Var[E[X Z]] + E[Var[X Z]] Var[Zµ + Zµ ] + E[Zσ + Zσ ] Var[µ µ Z + µ ] + σ + σ µ µ VarZ + σ + σ 4 µ µ + σ + σ sdx VarX 4 µ µ + σ + σ Problem. For a collectio of radom variables prove that: Var i a i X i i j a i a j CovX i, X j. Solutio. Note: It is possible to use iductio to prove this give that we are give the formula. However sometimes just proceedig directly from defiitio i this case of variace ca be easier.

2 stat homework - solutios Var i a i X i E a i X i i i j i j i j [ a i a j EX i X j E i a i X i ] i j a i a j EXi X j EX i EX j a i a j CovX i, X j Problem 3. Let X N µ, σ. Show that M X t expµt + σ t / a i a j EX i EX j Solutio 3. Here we ote that if Z N 0, ad X N µ, σ the 3 X µ + σz Proof. We proceed as follows: M X t : E X e Xt 3 Note: Here we ca fid a affie i.e. scale ad shift represetatio of X i terms of Z. The advatage is that the MGF of Z is much easier to calculate usig Lemma 0. E Z e σz+µt by defiitio of X e µt M Z σt Usig lemma 0. e µt e σ t Usig lemma 0. e µt+ σ t Problem 4. Suppose that X, X..., X are i.i.d. radom variables with E[X i ] µ ad Var[X i ] σ Let ad Show that a E[Y] µ. b Var[Y] σ. c E[S] σ. Y S X i, i i X i Y Solutio 4. We firstly ote that sice the X i s are IID for all it follows that EX i EX µ ad VarX i VarX σ for all.

3 stat homework - solutios 3 a EY µ Proof. We proceed directly 4 : EY E X i i i i µ i µ EX i EX by liearity of expectatio sice X i s are idetically distributed 4 Key poit: We did ot rely o idepedece of X i s here to derive the expectatio of T. Simply usig liearity of Expectatio ad Idetically distributed X i s was eough. Always try ad prove statemets with miimal required assumptios µ b Var[Y] σ. Proof. We firstly ote that 5 VarY Var X i i 5 Key poit: Here we do rely o both idepedece of X i s ad their idetical distributio calculatio to simplify the variace of Y i VarX i by idepedece of X i s i VarX sice X i s are idetically distributed σ i σ σ c E[S] σ Proof. We ote some prelimiary useful idetities to simplify the proof 6. a X i X i i X i X b i X i µ i X i i µ X µ X µ 6 Key poit: This questio shows how to costruct a ubiased estimator of the true variace from the sample data. These are useful idetities to ote dow ad used frequetly i similar proofs

4 stat homework - solutios 4 Approach Proof. Assumig the prelimiary useful idetities, we proceed as follows: E E i X i X X i X i i ix i X i i X i }{{} X + X i X + X Xi X i i E i E Xi E X X E X i X by liearity of expectatio sice X i s are i.i.d VarX + EX Var X + E X σ + µ σ + µ σ from parts a b X i X σ by liearity of expectatio ad rescalig by : ES Approach Assumig the prelimiary useful idetities, we proceed with the followig useful decompositio 7 : 7 Key poit: Decompositios of complicated expressios are really isightful. It may lead to a loger proof i this case but ofte the breakdow ca be more isightful ad iterpretable. I comig weeks we will see the useful bias-variace decompositio which is used to measure squared error loss i machie learig so keep this add-subtract µ approach i mid

5 stat homework - solutios 5 E E i X i X ix i X i i i i X i µ + µ X add ad subtract µ [ X i µ X i µ X µ + X µ ] expad the square [ X i µ ] i [X i µ X µ] + split ito separate sums [ X i µ ] X µ i i remove X µ outside the sum Which gives us the required result i [X i µ] + E X i µ E X µ + } {{ } } {{ } σ σ from part b usig liearity of expectatio σ σ σ + σ i i i [ X µ ] [ X µ ] E X µ }{{} σ from part b X i X σ by liearity of expectatio ad rescalig by : ES Problem 5. Let X, Y have the uiform distributio o [, ] [, ] Fid the probability that X + Y /. Solutio 5. Claim: this is 9 3 Approach Proof. This is the probability that X, Y fall i the upper triagular regio o the box [, ] [, ] give that they are joitly uiform o the box [, ] [, ]. The required probability is simply the ratio of the area of the upper triagle to the area of the box i this case 8 : P X + Y P Y X Key poit: Try ad draw a picture ad exploit the geometry of the problem to fid the quickest solutio. I this case because of the uiform distributio ad thus uiform volume i 3D, we are simply cocered with relative D areas to get our required probability

6 stat homework - solutios X + Y Approach Usig itegratio 9 : Notice that f X,Y x, y 4 o [, ] [, ] so that 9 f X,Y x, y dx dy. Note: itegratio approach is loger tha the geometric approach used i part a but more geeral whe the desity is ot uiform. Agai, drawig a picture helps. P X + Y P Y X / / [ 8 + x / x 8 + x 4 dx ] / dy dx 4 Problem 6. Let X, Y have the uiform distributio o the set {x, y : x + y }. Fid the joit desity fuctio of X, Y. Solutio 6. By defiitio this is the uiform distributio of the closed uit disk circle i R. So we defie the joit desity f X,Y x, y as follows 0 : 0 Key poit: Always remember to write dow the desity for all values x, y R i.e. iclude the 0 desity value for poits outside the uit disk i this case

7 stat homework - solutios 7 π if x + y f X,Y x, y 0 otherwise Problem 7. Let F be a cotiuous, strictly icreasig CDF. Let U be a radom variable uiformly distributed o [0, ]. Show that the radom variable Z F U has CDF F. Remark. This result lets us draw samples from ay distributio usig samples from the uiform distributio o [0, ] Solutio 7. Proof. We deote the CDF of Z : F Z z F Z z : PZ z defiitio of CDF of Z P F U z by defiitio of Z PU Fz by ivertibility of F : F U Fz writig i terms of the CDF of U Fz Usig Lemma 0.3 sice U Uif0, So the radom variable Z F U has CDF F as required. Problem 8. Let X be uiformly distributed o [ 5, ]. Let Y X 4. Fid the CDF ad PDF of Y. Solutio 8. F Y y PY y PX 4 y y [0,] P y 4 X y 4 + y,65] P y 4 X Note: Compare this to the approach take i the similar Q8c of HW y 4 y [0,] y 4 6 dx + y,65] y 4 y [0,] 3 y 4 + y,65] 6 + y 4 This implies that the CDF ad PDF are: 6 dx 0 y 0 F Y y 3 y 4 0 < y 6 + y 4 < y 65 y > 65 y < y f Y y F Y y 4 y 3 4 < y 65 0 otherwise

8 stat homework - solutios 8 Appedix: Supportig Lemmas Here we prove some lemmas from the lecture otes that we use repeatedly i the solutios Lemma 0. Scalig ad Shiftig MGF. For a radom variables X ad Y such that Y ax + b we have the followig relatioship betwee the MGFs: M Y t e bt M X at where the MGFs of X ad Y as deoted as M X t ad M Y t respectively Proof. M Y t : E Y e Yt E Y e ax+bt by defiitio of Y E Y e bt e axt e bt E X e atx e bt M X at From which we obtai the required result. Lemma 0. MGF of stadard ormal. For Z N 0, the M Z t e t Proof. M Z t : E Z e Zt e t e zt f Z zdz e zt π e z dz π e z +zt dz π e z t e t dz π e z t dz }{{} the itegral of N t, over it s etire support e t Which is the required result.

9 stat homework - solutios 9 Lemma 0.3 CDF of Uif0,. Let U Uif0, ad let F U u deote the CDF of U. The F U u u u [0, ] Proof. F U u : PU u u 0 [t] u 0 u dt by defiitio of Y From which we obtai the required result. Refereces

STAT Homework 1 - Solutions

STAT 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 information

Lecture 7: Properties of Random Samples

Lecture 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 information

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 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 information

Lecture 19: Convergence

Lecture 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 information

AMS570 Lecture Notes #2

AMS570 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 information

January 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS

January 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 information

HOMEWORK I: PREREQUISITES FROM MATH 727

HOMEWORK I: PREREQUISITES FROM MATH 727 HOMEWORK I: PREREQUISITES FROM MATH 727 Questio. Let X, X 2,... be idepedet expoetial radom variables with mea µ. (a) Show that for Z +, we have EX µ!. (b) Show that almost surely, X + + X (c) Fid the

More information

Chapter 2 The Monte Carlo Method

Chapter 2 The Monte Carlo Method Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful

More information

EE 4TM4: Digital Communications II Probability Theory

EE 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 information

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.

This 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 information

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:

Problems 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 information

Lecture 12: September 27

Lecture 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 information

Lecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables

Lecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables CSCI-B609: A Theorist s Toolkit, Fall 06 Aug 3 Lecture 0: the Cetral Limit Theorem Lecturer: Yua Zhou Scribe: Yua Xie & Yua Zhou Cetral Limit Theorem for iid radom variables Let us say that we wat to aalyze

More information

Chapter 5. Inequalities. 5.1 The Markov and Chebyshev inequalities

Chapter 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 information

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13 BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the

More information

Expectation and Variance of a random variable

Expectation 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 information

Chapter 6 Principles of Data Reduction

Chapter 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 information

Mathematical Statistics - MS

Mathematical Statistics - MS Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios

More information

The standard deviation of the mean

The 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 information

AMS 216 Stochastic Differential Equations Lecture 02 Copyright by Hongyun Wang, UCSC ( ( )) 2 = E X 2 ( ( )) 2

AMS 216 Stochastic Differential Equations Lecture 02 Copyright by Hongyun Wang, UCSC ( ( )) 2 = E X 2 ( ( )) 2 AMS 216 Stochastic Differetial Equatios Lecture 02 Copyright by Hogyu Wag, UCSC Review of probability theory (Cotiued) Variace: var X We obtai: = E X E( X ) 2 = E( X 2 ) 2E ( X )E X var( X ) = E X 2 Stadard

More information

Machine Learning Brett Bernstein

Machine Learning Brett Bernstein Machie Learig Brett Berstei Week 2 Lecture: Cocept Check Exercises Starred problems are optioal. Excess Risk Decompositio 1. Let X = Y = {1, 2,..., 10}, A = {1,..., 10, 11} ad suppose the data distributio

More information

Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].

Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)]. Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x

More information

7.1 Convergence of sequences of random variables

7.1 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 information

Variance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Variance 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 information

Randomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)

Randomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018) Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black

More information

This section is optional.

This 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 information

IE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.

IE 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 information

Final Review for MATH 3510

Final 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 information

Lecture 2: Concentration Bounds

Lecture 2: Concentration Bounds CSE 52: Desig ad Aalysis of Algorithms I Sprig 206 Lecture 2: Cocetratio Bouds Lecturer: Shaya Oveis Ghara March 30th Scribe: Syuzaa Sargsya Disclaimer: These otes have ot bee subjected to the usual scrutiy

More information

Quick Review of Probability

Quick 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 information

Quick Review of Probability

Quick 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 information

STAT Homework 7 - Solutions

STAT Homework 7 - Solutions STAT-36700 Homework 7 - Solutios Fall 208 October 28, 208 This cotais solutios for Homework 7. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better

More information

Lecture 6 Simple alternatives and the Neyman-Pearson lemma

Lecture 6 Simple alternatives and the Neyman-Pearson lemma STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull

More information

Mathematics 170B Selected HW Solutions.

Mathematics 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 information

Distribution of Random Samples & Limit theorems

Distribution 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 information

Exercise 4.3 Use the Continuity Theorem to prove the Cramér-Wold Theorem, Theorem. (1) φ a X(1).

Exercise 4.3 Use the Continuity Theorem to prove the Cramér-Wold Theorem, Theorem. (1) φ a X(1). Assigmet 7 Exercise 4.3 Use the Cotiuity Theorem to prove the Cramér-Wold Theorem, Theorem 4.12. Hit: a X d a X implies that φ a X (1) φ a X(1). Sketch of solutio: As we poited out i class, the oly tricky

More information

Lecture 20: Multivariate convergence and the Central Limit Theorem

Lecture 20: Multivariate convergence and the Central Limit Theorem Lecture 20: Multivariate covergece ad the Cetral Limit Theorem Covergece i distributio for radom vectors Let Z,Z 1,Z 2,... be radom vectors o R k. If the cdf of Z is cotiuous, the we ca defie covergece

More information

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment HW5 Solution

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment HW5 Solution Departmet of Civil Egieerig-I.I.T. Delhi CEL 899: Evirometal Risk Assessmet HW5 Solutio Note: Assume missig data (if ay) ad metio the same. Q. Suppose X has a ormal distributio defied as N (mea=5, variace=

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete 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 information

7.1 Convergence of sequences of random variables

7.1 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 information

Massachusetts Institute of Technology

Massachusetts Institute of Technology Solutios to Quiz : Sprig 006 Problem : Each of the followig statemets is either True or False. There will be o partial credit give for the True False questios, thus ay explaatios will ot be graded. Please

More information

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6) STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated

More information

An Introduction to Randomized Algorithms

An Introduction to Randomized Algorithms A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis

More information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

PRACTICE PROBLEMS FOR THE FINAL

PRACTICE 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 information

Understanding Samples

Understanding 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 information

1 Convergence in Probability and the Weak Law of Large Numbers

1 Convergence in Probability and the Weak Law of Large Numbers 36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec

More information

Stat410 Probability and Statistics II (F16)

Stat410 Probability and Statistics II (F16) Some Basic Cocepts of Statistical Iferece (Sec 5.) Suppose we have a rv X that has a pdf/pmf deoted by f(x; θ) or p(x; θ), where θ is called the parameter. I previous lectures, we focus o probability problems

More information

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n, CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe

More information

The 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.

The 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 information

Learning Theory: Lecture Notes

Learning 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

Random Variables, Sampling and Estimation

Random 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 information

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10

DS 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 information

Advanced Stochastic Processes.

Advanced 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 information

Simulation. Two Rule For Inverting A Distribution Function

Simulation. Two Rule For Inverting A Distribution Function Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump

More information

Solutions: Homework 3

Solutions: Homework 3 Solutios: Homework 3 Suppose that the radom variables Y,...,Y satisfy Y i = x i + " i : i =,..., IID where x,...,x R are fixed values ad ",...," Normal(0, )with R + kow. Fid ˆ = MLE( ). IND Solutio: Observe

More information

Machine Learning Brett Bernstein

Machine Learning Brett Bernstein Machie Learig Brett Berstei Week Lecture: Cocept Check Exercises Starred problems are optioal. Statistical Learig Theory. Suppose A = Y = R ad X is some other set. Furthermore, assume P X Y is a discrete

More information

IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 2006 SECTION A IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim

More information

17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15

17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15 17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig

More information

Lecture 18: Sampling distributions

Lecture 18: Sampling distributions Lecture 18: Samplig distributios I may applicatios, the populatio is oe or several ormal distributios (or approximately). We ow study properties of some importat statistics based o a radom sample from

More information

0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =

0, 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 information

Statistical and Mathematical Methods DS-GA 1002 December 8, Sample Final Problems Solutions

Statistical and Mathematical Methods DS-GA 1002 December 8, Sample Final Problems Solutions Statistical ad Mathematical Methods DS-GA 00 December 8, 05. Short questios Sample Fial Problems Solutios a. Ax b has a solutio if b is i the rage of A. The dimesio of the rage of A is because A has liearly-idepedet

More information

Notes 5 : More on the a.s. convergence of sums

Notes 5 : More on the a.s. convergence of sums Notes 5 : More o the a.s. covergece of sums Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: Dur0, Sectios.5; Wil9, Sectio 4.7, Shi96, Sectio IV.4, Dur0, Sectio.. Radom series. Three-series

More information

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1 EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum

More information

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample. Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized

More information

Lecture 3 : Random variables and their distributions

Lecture 3 : Random variables and their distributions Lecture 3 : Radom variables ad their distributios 3.1 Radom variables Let (Ω, F) ad (S, S) be two measurable spaces. A map X : Ω S is measurable or a radom variable (deoted r.v.) if X 1 (A) {ω : X(ω) A}

More information

Lecture 7: Density Estimation: k-nearest Neighbor and Basis Approach

Lecture 7: Density Estimation: k-nearest Neighbor and Basis Approach STAT 425: Itroductio to Noparametric Statistics Witer 28 Lecture 7: Desity Estimatio: k-nearest Neighbor ad Basis Approach Istructor: Ye-Chi Che Referece: Sectio 8.4 of All of Noparametric Statistics.

More information

Solutions to HW Assignment 1

Solutions to HW Assignment 1 Solutios to HW: 1 Course: Theory of Probability II Page: 1 of 6 Uiversity of Texas at Austi Solutios to HW Assigmet 1 Problem 1.1. Let Ω, F, {F } 0, P) be a filtered probability space ad T a stoppig time.

More information

2.2. Central limit theorem.

2.2. Central limit theorem. 36.. Cetral limit theorem. The most ideal case of the CLT is that the radom variables are iid with fiite variace. Although it is a special case of the more geeral Lideberg-Feller CLT, it is most stadard

More information

LECTURE 8: ASYMPTOTICS I

LECTURE 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 information

Stat 421-SP2012 Interval Estimation Section

Stat 421-SP2012 Interval Estimation Section Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible

More information

Monte Carlo Integration

Monte Carlo Integration Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce

More information

Unbiased Estimation. February 7-12, 2008

Unbiased Estimation. February 7-12, 2008 Ubiased Estimatio February 7-2, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom

More information

Econ 325: Introduction to Empirical Economics

Econ 325: Introduction to Empirical Economics Eco 35: Itroductio to Empirical Ecoomics Lecture 3 Discrete Radom Variables ad Probability Distributios Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-1 4.1 Itroductio to Probability

More information

Lecture 3. Properties of Summary Statistics: Sampling Distribution

Lecture 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 information

Topic 8: Expected Values

Topic 8: Expected Values Topic 8: Jue 6, 20 The simplest summary of quatitative data is the sample mea. Give a radom variable, the correspodig cocept is called the distributioal mea, the epectatio or the epected value. We begi

More information

Lecture 12: November 13, 2018

Lecture 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 information

Review Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn

Review Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom

More information

Lecture Chapter 6: Convergence of Random Sequences

Lecture Chapter 6: Convergence of Random Sequences ECE5: Aalysis of Radom Sigals Fall 6 Lecture Chapter 6: Covergece of Radom Sequeces Dr Salim El Rouayheb Scribe: Abhay Ashutosh Doel, Qibo Zhag, Peiwe Tia, Pegzhe Wag, Lu Liu Radom sequece Defiitio A ifiite

More information

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22

Discrete 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 information

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering

CEE 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 information

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio

More information

32 estimating the cumulative distribution function

32 estimating the cumulative distribution function 32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio

More information

Kernel density estimator

Kernel density estimator Jauary, 07 NONPARAMETRIC ERNEL DENSITY ESTIMATION I this lecture, we discuss kerel estimatio of probability desity fuctios PDF Noparametric desity estimatio is oe of the cetral problems i statistics I

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 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 information

Stat 400: Georgios Fellouris Homework 5 Due: Friday 24 th, 2017

Stat 400: Georgios Fellouris Homework 5 Due: Friday 24 th, 2017 Stat 400: Georgios Fellouris Homework 5 Due: Friday 4 th, 017 1. A exam has multiple choice questios ad each of them has 4 possible aswers, oly oe of which is correct. A studet will aswer all questios

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY 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 information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece

More information

Partial match queries: a limit process

Partial match queries: a limit process Partial match queries: a limit process Nicolas Brouti Ralph Neiiger Heig Sulzbach Partial match queries: a limit process 1 / 17 Searchig geometric data ad quadtrees 1 Partial match queries: a limit process

More information

Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions

Discrete 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 information

Lecture 11 and 12: Basic estimation theory

Lecture 11 and 12: Basic estimation theory Lecture ad 2: Basic estimatio theory Sprig 202 - EE 94 Networked estimatio ad cotrol Prof. Kha March 2 202 I. MAXIMUM-LIKELIHOOD ESTIMATORS The maximum likelihood priciple is deceptively simple. Louis

More information

2. The volume of the solid of revolution generated by revolving the area bounded by the

2. The volume of the solid of revolution generated by revolving the area bounded by the IIT JAM Mathematical Statistics (MS) Solved Paper. A eigevector of the matrix M= ( ) is (a) ( ) (b) ( ) (c) ( ) (d) ( ) Solutio: (a) Eigevalue of M = ( ) is. x So, let x = ( y) be the eigevector. z (M

More information

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes. Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (

More information

f X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36

f 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 information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 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 information

Lecture 2: Monte Carlo Simulation

Lecture 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 information

Math 10A final exam, December 16, 2016

Math 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 information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties

More information