Probability 2008 Rong-Jaye Chen. p2. Course Resources Webpage:

Slides:

Advertisements

Similar presentations

Practice Midterm Welcome to Pstat5E: Statistics with Economics and Business Applications Yuedong Wang Solution to Practice Midterm Exam.

Advertisements

Let X 1, X 2,..., X n be a set of independent random variables having a common distribution, and let E[ X i ] = . then, with probability 1 Strong law.

Week11 Parameter, Statistic and Random Samples A parameter is a number that describes the population. It is a fixed number, but in practice we do not know.

BINOMIAL AND NORMAL DISTRIBUTIONS BINOMIAL DISTRIBUTION “Bernoulli trials” – experiments satisfying 3 conditions: 1. Experiment has only 2 possible outcomes:

Probability and Statistics Dr. Saeid Moloudzadeh Sample Space and Events 1 Contents Descriptive Statistics Axioms of Probability Combinatorial.

Sampling Distributions

Variance and Standard Deviation The Expected Value of a random variable gives the average value of the distribution The Standard Deviation shows how spread.

CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Multiple random variables Transform methods (Sec , 4.5.7)

Probability theory 2010 Outline  The need for transforms  Probability-generating function  Moment-generating function  Characteristic function  Applications.

2003/02/13 Chapter 4 1頁1頁 Chapter 4 : Multiple Random Variables 4.1 Vector Random Variables.

CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Course outline and schedule Introduction Event Algebra (Sec )

Probability and Statistics Review

CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Course outline and schedule Introduction (Sec )

The moment generating function of random variable X is given by Moment generating function.

Syllabus for Engineering Statistics Probability: rules, cond. probability, independence Summarising and presenting data Discrete and continuous random.

2003/04/24 Chapter 5 1頁1頁 Chapter 5 : Sums of Random Variables & Long-Term Averages 5.1 Sums of Random Variables.

Section 5.6 Important Theorem in the Text: The Central Limit TheoremTheorem (a) Let X 1, X 2, …, X n be a random sample from a U(–2, 3) distribution.

Normal and Sampling Distributions A normal distribution is uniquely determined by its mean, , and variance,  2 The random variable Z = (X-  /  is.

Sampling Distributions  A statistic is random in value … it changes from sample to sample.  The probability distribution of a statistic is called a sampling.

Probability and Statistics Dr. Saeid Moloudzadeh Axioms of Probability/ Basic Theorems 1 Contents Descriptive Statistics Axioms of Probability.

General information CSE : Probabilistic Analysis of Computer Systems

Chapter 6 Random Variables. Make a Sample Space for Tossing a Fair Coin 3 times.

Introduction to Biostatistics and Bioinformatics Estimation II This Lecture By Judy Zhong Assistant Professor Division of Biostatistics Department of Population.

Advanced Higher Statistics Data Analysis and Modelling Hypothesis Testing Statistical Inference AH.

MA-250 Probability and Statistics Nazar Khan PUCIT Lecture 26.

Probability and Statistics Dr. Saeid Moloudzadeh Counting Principle/ Permutations 1 Contents Descriptive Statistics Axioms of Probability.

King Saud University Women Students

Discrete distribution word problems –Probabilities: specific values, >, =, … –Means, variances Computing normal probabilities and “inverse” values: –Pr(X

Week11 Parameter, Statistic and Random Samples A parameter is a number that describes the population. It is a fixed number, but in practice we do not know.

STA347 - week 51 More on Distribution Function The distribution of a random variable X can be determined directly from its cumulative distribution function.

Chapter 7 Sampling and Sampling Distributions ©. Simple Random Sample simple random sample Suppose that we want to select a sample of n objects from a.

1 Since everything is a reflection of our minds, everything can be changed by our minds.

Exam 1 next Thursday (March 7 th ) in class 15% of your grade Covers chapters 1-6 and the central limit theorem I will put practice problems, old exams,

Probability and Statistics Dr. Saeid Moloudzadeh Random Variables/ Distribution Functions/ Discrete Random Variables. 1 Contents Descriptive.

HW solutions are on the web. See website for how to calculate probabilities with minitab, excel, and TI calculators.

Test of Goodness of Fit Lecture 43 Section 14.1 – 14.3 Fri, Apr 8, 2005.

CpSc 881: Machine Learning Evaluating Hypotheses.

Chapter 7 Point Estimation of Parameters. Learning Objectives Explain the general concepts of estimating Explain important properties of point estimators.

The final exam solutions. Part I, #1, Central limit theorem Let X1,X2, …, Xn be a sequence of i.i.d. random variables each having mean μ and variance.

Week 121 Law of Large Numbers Toss a coin n times. Suppose X i ’s are Bernoulli random variables with p = ½ and E(X i ) = ½. The proportion of heads is.

Introduction to Statistics I MATH 1131, Summer I 2008, Department of Math. & Stat., York University.

Probability and Statistics Dr. Saeid Moloudzadeh Joint Distribution of Two Random Variables/ Independent Random Variables 1 Contents Descriptive.

Psychology 202a Advanced Psychological Statistics September 29, 2015.

Probability and Statistics Dr. Saeid Moloudzadeh Uniform Random Variable/ Normal Random Variable 1 Contents Descriptive Statistics Axioms.

1 Probability: Introduction Definitions,Definitions, Laws of ProbabilityLaws of Probability Random VariablesRandom Variables DistributionsDistributions.

Introduction to Probabilities Fall 2010 Dept. Electrical Engineering National Tsing Hua University 劉奕汶.

Chapter 6: Random Variables

Central Limit Theorem Let X 1, X 2, …, X n be n independent, identically distributed random variables with mean  and standard deviation . For large n:

Chapter 6 Large Random Samples Weiqi Luo ( 骆伟祺 ) School of Data & Computer Science Sun Yat-Sen University ：

7.2 Means & Variances of Random Variables AP Statistics.

Ver Chapter 5 Continuous Random Variables 1 Probability/Ch5.

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Business Statistics,

1 Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems.

Evaluating Hypotheses. Outline Empirically evaluating the accuracy of hypotheses is fundamental to machine learning – How well does this estimate its.

Conditional Expectation

Probability and Statistics Dr. Saeid Moloudzadeh Poisson Random Variable 1 Contents Descriptive Statistics Axioms of Probability Combinatorial.

Statistics and probability Dr. Khaled Ismael Almghari Phone No:

Chapter Five The Binomial Probability Distribution and Related Topics

Probabilistic Analysis of Computer Systems

Chapter 8: Fundamental Sampling Distributions and Data Descriptions:

AP Statistics: Chapter 7

TM 605: Probability for Telecom Managers

Random Sampling Population Random sample: Statistics Point estimate

Probability & Statistics Probability Theory Mathematical Probability Models Event Relationships Distributions of Random Variables Continuous Random.

ASV Chapters 1 - Sample Spaces and Probabilities

Probability and statistics I.

Ch. 6. Binomial Theory.

Chapter 8: Fundamental Sampling Distributions and Data Descriptions:

Fundamental Sampling Distributions and Data Descriptions

Presentation transcript:

Probability 2008 Rong-Jaye Chen

p2. Course Resources Webpage:

p3. Text S. Ghahramani, Fundamentals of Probability with stochastic processes 3rd Ed, Prentice-Hall, 2005

p4. Grading Scheme 1st Midterms 30% 2nd Midterms 30% Final Exam 40%

p5. TAs 1. TAs 徐順隆曾輔國 2. 霹靂博課程資訊 : 霹靂博 : 蔡佩娟上課時間 -4HY, 上課地點 -EC114, Office 網頁 :

p6. Syllabus Axioms of probability Combinatorial methods Conditional probability and independence Distributed functions and discrete random variables Special discrete distributions

p7. Continuous random variables Special continuous distributions Bivariate distribution Multivariate distribution More expectations and variances Sums of independent random variables and limit theorems

p8. Probability 1 Pr(at least two with the same birthday among 23 people)=?

p9. Sol: x =1-Pr(each two among 23 with different birthdays) =1-(365/365)(364/365) … (343/365) > 0.5 (Surprised?) It is called Birthday Paradox!

p10. Probability 2 On average, there are three misprints in every 10 pages of a particular book. If Chapter 1 contains 35 pages, what is the probability that Chapter 1 has 10 misprints?

p11. Sol: lamda=(3/10)35=10.5 It is a Poisson distribution so solution = e (10.5) 10 /10!=0.124

p12. Probability 3 Two random points are selected from (0,1) independently. Find the probability that one of them is at least three times the other.

p13. Sol: Let X 1 and X 2 be the points selected at random. Calculate f 12 (x, y), and use order statistic probilities in Sec 9.2 Sol = Pr(X (2) >=3X (1) ) = … = 1/3

p14. Probability 4 Toss a coin times, #(head)=5150 times. Is the coin unbiased?

p15. Sol: Suppose this coin is unbiased X=#(head) ~Binomial(n=10000,p=0.5) E(X)=np=5000 Var(X)=npq=2500 By Central Limit Theorem Pr(X>=5150) ~ Pr(Z>3)~0.001 (Z is the normal distributed random variable) So reject the hypothesis! The coin is biased!