# 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

Save this PDF as:

Size: px
Start display at page:

Download "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"

## Transcription

1 Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp/ Review Questios, Chapters 8, Suppose that Y, Y 2,..., Y deote a radom sample of size from a populatio with a expoetial distributio whose desity is give by (/θ)e y/θ, y > 0 If Y () = mi(y, Y 2,..., Y ) deotes the smallest-order statistic, show that ˆθ = Y () is a ubiased estimator for θ ad fid MSE(ˆθ). Solutio. Let s fid the distributio fuctio of Y : e y/θ, y > 0 F (y) = Now we ca use the formula F Y() (y) = [ F (y) ] or fy() = ( F (y) ) f(y) to fid the the desity fuctio for Y () : for y > 0, f Y() = ( e y/θ) θ e y/θ = θ e y θ. We ca recogize this desity fuctio to be the desity of the expoetial distributio with parameter θ /, Y () Exp ( θ ). Kowig the distributio of Y () allows us to compute the expectatio of ˆθ = Y () : E[ˆθ] = E[Y () ] = θ = θ. So, E[ˆθ] = θ, ad ˆθ is a ubiased estimator of θ. To fid MSE(ˆθ), use the formula MSE(ˆθ) = V [ˆθ] + ( B(ˆθ) ) 2. Sice the estimator is ubiased, its bias B(ˆθ) equals zero. For the variace, remember that Y () is expoetial. We have MSE(ˆθ) = V [ˆθ] + 0 = 2 V [ Y () ] = 2 θ2 2 = θ2.

2 Stat 366 Lab 2 Solutios (September 2, 2006) page Suppose that Y, Y 2,..., Y deote a radom sample of size from a expoetial distributio with desity fuctio give by (/θ)e y/θ, y > 0 I Exercise 8.5 we determied that ˆθ = Y () is a ubiased estimator of θ with MSE(ˆθ)= θ 2. Cosider the estimator ˆθ 2 = Ȳ, ad fid the efficiecy of ˆθ relative to ˆθ 2. Solutio. First compute the variace of ˆθ 2 : [ ] V [ˆθ 2 ] = V [Ȳ ] = V Y + + Y = V [Y Y ] = ( V [Y ] + + V [Y 2 ] ) ) θ 2 = = ( θ θ 2 2 }{{} times 2 = θ2. To fid the relative efficiecy, we eed to fid the ratio of two variaces: eff(ˆθ, ˆθ 2 ) = V (ˆθ 2 ) V (ˆθ ) = θ2 θ 2 =. We coclude that ˆθ 2 is preferable to ˆθ. 9.6 Let Y, Y 2,..., Y deote a radom sample from the probability desity fuctio (θ + )y θ, 0 < y < ; θ > Fid a estimator for θ by the method of momets. Solutio. Let s fid the first momet of this distributio: µ = The method of momets implies 0 (θ + ) y θ+ dy = (θ + ) yθ+2 θ + 2 = θ + 0 θ + 2. Ȳ = ˆθ + ˆθ + 2 ˆθ = 2Ȳ Ȳ.

3 Stat 366 Lab 2 Solutios (September 2, 2006) page Suppose that Y, Y 2,..., Y deote a radom sample from the Poisso distributio with mea λ. (a) Fid the maximum-likelihood estimator ˆλ for λ. (d) What is the MLE for P (Y = 0) = e λ? Solutio. Let s defie the likelihood fuctio L(λ y, y 2,..., y ): p(y i ) = λ y i e λ y i! = λ y i e λ y. i! The problem ow is to fid the maximum value of this fuctio of λ. Let s make a simplifyig trasformatio: ( ) l y i l λ λ Differetiatio with respect to λ yields: l(y i!). d dx l λ y i = 0. Solvig this equatio: The latter is the MLE for λ. λ = y i, or ˆλ = Y i = Ȳ. To aswer (b), recall the ivariace priciple for MLEs: if t(ˆθ) is a oe-to-oe fuctio, the t(θ) = t(ˆθ). I our case t(λ) = e λ, so ê λ = e ˆλ = e Ȳ. 9.75a Suppose that Y, Y 2,..., Y costitute a radom sample from a uiform distributio with probability desity fuctio 2θ +, 0 y 2θ + 0, elsewhere. Obtai the maximum-likelihood estimator of θ.

4 Stat 366 Lab 2 Solutios (September 2, 2006) page 4 Solutio. This is a somewhat differet problem from the previous oe because the support of the desity fuctio depeds o θ. Recall the idicator fuctio I(A). It is equal to oe whe A is true, ad zero if A is false. We ca write the likelihood fuctio i the followig way: f(y i ) = 2θ + I(0 y i 2θ + ) = (2θ + ) I(0 y i 2θ + ). We ca simplify this eve further if we ote that the product of idicator is o-zero oly whe all of the uderlyig coditios fulfill. That is, all y i are less that 2θ + ad positive. Notice that this statemet is equivalet to the followig: 0 y () ad y () 2θ +. (We use order statistics y () = mi(y,..., y ) ad y () = max(y,..., y ).) We have (2θ + ) I(0 y ()) I(y () 2θ + ). Now look at the first part of the likelihood fuctio L, (2θ + ). Notice that this is a decreasig (ad cotiuous) fuctio of θ. If we wat to maximize L, we should choose the value of θ as small as possible. Notice that if 2θ + is smaller tha y (), the the value of L(θ) is zero. So, the miimum of 2θ + is y (). This gives the miimum value for θ ad maximizes the likelihood L(θ). We coclude (provided at least oe observatio i the sample is positive) ( Y () = 2ˆθ + ˆθ = Y() ) Let Y, Y 2,..., Y deote a radom sample from the probability desity fuctio (θ + )y θ, 0 < y < ; θ > Fid the maximum-likelihood estimator for θ. Compare your aswer to the method of momets estimator foud i Exercise 9.6. Solutio. Defie the likelihood fuctio: ( ) θ. (θ + )yi θ = (θ + ) y i Take the logarithms: l l(θ + ) + θ l y i.

5 Stat 366 Lab 2 Solutios (September 2, 2006) page 5 Fid critical poits: so ad fially d dθ l θ + + l y i = 0, θ = l y, i ˆθ = l Y. i This is quite differet from the method of momets estimator foud i Exercise 9.6.

### 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

### 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

### 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

### 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

### Statistical Theory MT 2009 Problems 1: Solution sketches

Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where

### STATISTICAL INFERENCE

STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample

### 5. Likelihood Ratio Tests

1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,

### SOLUTION FOR HOMEWORK 7, STAT np(1 p) (α + β + n) + ( np + α

SOLUTION FOR HOMEWORK 7, STAT 6331 1 Exerc733 Here we just recall that MSE(ˆp B ) = p(1 p) (α + β + ) + ( p + α 2 α + β + p) 2 The you plug i α = β = (/4) 1/2 After simplificatios MSE(ˆp B ) = 4( 1/2 +

### First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >

### Element sampling: Part 2

Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig

### Read carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.

THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each

### Sample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for

Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they

### Properties and Hypothesis Testing

Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

### Lecture 9: September 19

36-700: Probability ad Mathematical Statistics I Fall 206 Lecturer: Siva Balakrisha Lecture 9: September 9 9. Review ad Outlie Last class we discussed: Statistical estimatio broadly Pot estimatio Bias-Variace

### Output 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%

### ECE 901 Lecture 12: Complexity Regularization and the Squared Loss

ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality

### 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

### 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

### 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 ] / (

### 1 Inferential Methods for Correlation and Regression Analysis

1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet

### De Moivre s Theorem - ALL

De Moivre s Theorem - ALL. Let x ad y be real umbers, ad be oe of the complex solutios of the equatio =. Evaluate: (a) + + ; (b) ( x + y)( x + y). [6]. (a) Sice is a complex umber which satisfies = 0,.

### 10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.

0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece

### Statisticians use the word population to refer the total number of (potential) observations under consideration

6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space

### MLE and efficiency 23. P (X = x) = θx Let s try to find the MLE for θ. A random sample drawn from this distribution has the likelihood function

3. Maximum likelihood estimators ad efficiecy 3.1. Maximum likelihood estimators. Let X 1,..., X be a radom sample, draw from a distributio P θ that depeds o a ukow parameter θ. We are lookig for a geeral

### Infinite 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

### 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

### Elementary Statistics

Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of

### Carleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.

Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified

### REVIEW 1, MATH n=1 is convergent. (b) Determine whether a n is convergent.

REVIEW, MATH 00. Let a = +. a) Determie whether the sequece a ) is coverget. b) Determie whether a is coverget.. Determie whether the series is coverget or diverget. If it is coverget, fid its sum. a)

### Seunghee Ye Ma 8: Week 5 Oct 28

Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

### G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan

Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity

### Math 116 Practice for Exam 3

Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur

### STA 4032 Final Exam Formula Sheet

Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace

### Table 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab

Sectio 12 Tests of idepedece ad homogeeity I this lecture we will cosider a situatio whe our observatios are classified by two differet features ad we would like to test if these features are idepedet

### Lecture 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

### MASSACHUSETTS 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

### Chapter 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

### The Sample Variance Formula: A Detailed Study of an Old Controversy

The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace

### Advanced Engineering Mathematics Exercises on Module 4: Probability and Statistics

Advaced Egieerig Mathematics Eercises o Module 4: Probability ad Statistics. A survey of people i give regio showed that 5% drak regularly. The probability of death due to liver disease, give that a perso

### It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.

Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig

### 3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered

### PRACTICE FINAL/STUDY GUIDE SOLUTIONS

Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)

### f(x)dx = 1 and f(x) 0 for all x.

OCR Statistics 2 Module Revisio Sheet The S2 exam is 1 hour 30 miutes log. You are allowed a graphics calculator. Before you go ito the exam make sureyou are fully aware of the cotets of theformula booklet

### Math 181, Solutions to Review for Exam #2 Question 1: True/False. Determine whether the following statements about a series are True or False.

Math 8, Solutios to Review for Exam #2 Questio : True/False. Determie whether the followig statemets about a series are True or False. X. The series a diverges if lim s 5.! False: The series coverges to

### The Random Walk For Dummies

The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

### Basis for simulation techniques

Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios

### 62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

### sin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =

60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece

### Section 14. Simple linear regression.

Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo

### Point Estimation: properties of estimators 1 FINITE-SAMPLE PROPERTIES. finite-sample properties (CB 7.3) large-sample properties (CB 10.

Poit Estimatio: properties of estimators fiite-sample properties CB 7.3) large-sample properties CB 10.1) 1 FINITE-SAMPLE PROPERTIES How a estimator performs for fiite umber of observatios. Estimator:

### Topic 18: Composite Hypotheses

Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:

### KLMED8004 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

### Riemann Sums y = f (x)

Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

### The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that

### 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

### Zeros of Polynomials

Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

### Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract

Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated

### PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)

Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages

### Topic 15: Maximum Likelihood Estimation

Topic 5: Maximum Likelihood Estimatio November ad 3, 20 Itroductio The priciple of maximum likelihood is relatively straightforward. As before, we begi with a sample X (X,..., X of radom variables chose

### 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

### Homework for 2/3. 1. Determine the values of the following quantities: a. t 0.1,15 b. t 0.05,15 c. t 0.1,25 d. t 0.05,40 e. t 0.

Name: ID: Homework for /3. Determie the values of the followig quatities: a. t 0.5 b. t 0.055 c. t 0.5 d. t 0.0540 e. t 0.00540 f. χ 0.0 g. χ 0.0 h. χ 0.00 i. χ 0.0050 j. χ 0.990 a. t 0.5.34 b. t 0.055.753

### Solutions to quizzes Math Spring 2007

to quizzes Math 4- Sprig 7 Name: Sectio:. Quiz a) x + x dx b) l x dx a) x + dx x x / + x / dx (/3)x 3/ + x / + c. b) Set u l x, dv dx. The du /x ad v x. By Itegratio by Parts, x(/x)dx x l x x + c. l x

### NYU Center for Data Science: DS-GA 1003 Machine Learning and Computational Statistics (Spring 2018)

NYU Ceter for Data Sciece: DS-GA 003 Machie Learig ad Computatioal Statistics (Sprig 208) Brett Berstei, David Roseberg, Be Jakubowski Jauary 20, 208 Istructios: Followig most lab ad lecture sectios, we

### 6. Uniform distribution mod 1

6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which

### MATH 10550, EXAM 3 SOLUTIONS

MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,

### Chapter 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

### Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }

UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig

### Name: Math 10550, Final Exam: December 15, 2007

Math 55, Fial Exam: December 5, 7 Name: Be sure that you have all pages of the test. No calculators are to be used. The exam lasts for two hours. Whe told to begi, remove this aswer sheet ad keep it uder

### Large holes in quasi-random graphs

Large holes i quasi-radom graphs Joaa Polcy Departmet of Discrete Mathematics Adam Mickiewicz Uiversity Pozań, Polad joaska@amuedupl Submitted: Nov 23, 2006; Accepted: Apr 10, 2008; Published: Apr 18,

### 7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses

### Algorithms for Clustering

CR2: Statistical Learig & Applicatios Algorithms for Clusterig Lecturer: J. Salmo Scribe: A. Alcolei Settig: give a data set X R p where is the umber of observatio ad p is the umber of features, we wat

### 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.

### Probability and Statistics

ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,

### ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig

### MAS111 Convergence and Continuity

MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece

### Lecture 10 October Minimaxity and least favorable prior sequences

STATS 300A: Theory of Statistics Fall 205 Lecture 0 October 22 Lecturer: Lester Mackey Scribe: Brya He, Rahul Makhijai Warig: These otes may cotai factual ad/or typographic errors. 0. Miimaxity ad least

### Fourier Series and the Wave Equation

Fourier Series ad the Wave Equatio We start with the oe-dimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig

### Math 106 Fall 2014 Exam 3.2 December 10, 2014

Math 06 Fall 04 Exam 3 December 0, 04 Determie if the series is coverget or diverget by makig a compariso (DCT or LCT) with a suitable b Fill i the blaks with your aswer For Coverget or Diverget write

### 5.1. The Rayleigh s quotient. Definition 49. Let A = A be a self-adjoint matrix. quotient is the function. R(x) = x,ax, for x = 0.

40 RODICA D. COSTIN 5. The Rayleigh s priciple ad the i priciple for the eigevalues of a self-adjoit matrix Eigevalues of self-adjoit matrices are easy to calculate. This sectio shows how this is doe usig

### Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.

Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios

### Solution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1

Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity

### Closed book and notes. No calculators. 60 minutes, but essentially unlimited time.

IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of

### Math 21B-B - Homework Set 2

Math B-B - Homework Set Sectio 5.:. a) lim P k= c k c k ) x k, where P is a partitio of [, 5. x x ) dx b) lim P k= 4 ck x k, where P is a partitio of [,. 4 x dx c) lim P k= ta c k ) x k, where P is a partitio

### How to Maximize a Function without Really Trying

How to Maximize a Fuctio without Really Tryig MARK FLANAGAN School of Electrical, Electroic ad Commuicatios Egieerig Uiversity College Dubli We will prove a famous elemetary iequality called The Rearragemet

### Estimation of Gumbel Parameters under Ranked Set Sampling

Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 11-2014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com

### Statistical Inference

Statistical Iferece Professor Abolfazl Sakhai School of Social Work Columbia Uiversity Notes by Yiqiao Yi i L A TEX December 18, 2016 Abstract This is the otes for Statistical Iferece at Columbia Uiversity.

### MAT1026 Calculus II Basic Convergence Tests for Series

MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

### Lecture 6 Ecient estimators. Rao-Cramer bound.

Lecture 6 Eciet estimators. Rao-Cramer boud. 1 MSE ad Suciecy Let X (X 1,..., X) be a radom sample from distributio f θ. Let θ ˆ δ(x) be a estimator of θ. Let T (X) be a suciet statistic for θ. As we have

### ARIMA Models. Dan Saunders. y t = φy t 1 + ɛ t

ARIMA Models Da Sauders I will discuss models with a depedet variable y t, a potetially edogeous error term ɛ t, ad a exogeous error term η t, each with a subscript t deotig time. With just these three

Regressio with quadratic loss Maxim Ragisky October 13, 2015 Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X,Y, where, as before,

### MA131 - Analysis 1. Workbook 2 Sequences I

MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

### CHAPTER 5. Theory and Solution Using Matrix Techniques

A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

### λ n e λ n f(x;λ) f(y;λ) = λn e λ = e λ ( n = 2 θ θ 1 2 y2 1 3 y3 ] θ = 1 3 θ. E(Y i ) = 3 1 n n1 3 θ = θ. θ 1 3 y3 1 4 y4 ] θ = 1 6 θ2.

CHAPTER Exercise 1 Suppose that Y (Y 1,,Y ) is a radom sample from a Exp(λ) distributio The we may write f Y (y) λe λy i λ e λ y i }{{} }{{} 1 g λ (T(Y )) h(y) It follows thatt(y ) Y i is a sufficiet statistic

### STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

### Markov Decision Processes

Markov Decisio Processes Defiitios; Statioary policies; Value improvemet algorithm, Policy improvemet algorithm, ad liear programmig for discouted cost ad average cost criteria. Markov Decisio Processes

### MATH1035: Workbook Four M. Daws, 2009

MATH1035: Workbook Four M. Daws, 2009 Roots of uity A importat result which ca be proved by iductio is: De Moivre s theorem atural umber case: Let θ R ad N. The cosθ + i siθ = cosθ + i siθ. Proof: The

### Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.