STAT 111 Recitation 4
|
|
|
- Juliet Flowers
- 5 years ago
- Views:
Transcription
1 STAT 111 Recitation 4 Linjun Zhang September 29, 2017
2 Misc. Mid-term exam time: 6-8 pm, Wednesday, Oct. 11 The mid-term break is Oct. 5-8 The next recitation class sis Oct. 1 1
3 Recap of the concepts Long formula for the mean and variance of X. 2
4 Recap of the concepts Long formula for the mean and variance of X. Suppose that X satisfies binomials distribution Binomial(n, θ), then the mean of X is nθ, and the variance of X is nθ(1 θ). 2
5 Recap of the concepts Long formula for the mean and variance of X. Suppose that X satisfies binomials distribution Binomial(n, θ), then the mean of X is nθ, and the variance of X is nθ(1 θ). The mean and variance of T n and X (T n denotes the sum and X denotes the average) 2
6 Recap of the concepts Long formula for the mean and variance of X. Suppose that X satisfies binomials distribution Binomial(n, θ), then the mean of X is nθ, and the variance of X is nθ(1 θ). The mean and variance of T n and X (T n denotes the sum and X denotes the average) The mean and variance of D (D = X 1 X 2 denotes the difference between X 1 and X 2 ) 2
7 Mean and Variance of T n and X Given n i.i.d. random variables X 1, X 2,..., X n with mean µ and variance σ 2, we define two other random variables: The sum, denoted by T n : T n = X 1 + X X n ; The average, denoted by X, X = X1+X2+...+Xn n = Tn n. Must know formula for T n and X Mean of T n = nµ; Variance of T n = nσ 2 ; Mean of X = µ; Variance of X = σ 2 /n.
8 Difference of random variables Suppose there are two independent random variables X 1 with mean µ 1 and variance σ1 2, and X 2 with mean µ 2 and variance σ2 2. The difference D is defined by D = X 1 X 2. Must know formula for D Mean of D = µ 1 µ 2 ; Variance of D = σ σ2 2. 4
9 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X 5
10 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X 5
11 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? 5
12 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 5
13 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 Mean of T 10? 5
14 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 Mean of T 10? 0 5
15 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 Mean of T 10? 0 Variance of T 10? 5
16 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 Mean of T 10? 0 Variance of T 10? = 100 5
17 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 Mean of T 10? 0 Variance of T 10? = 100 Mean of X 1 X 2? 5
18 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 Mean of T 10? 0 Variance of T 10? = 100 Mean of X 1 X 2? 0 5
19 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 Mean of T 10? 0 Variance of T 10? = 100 Mean of X 1 X 2? 0 Variance of X 1 X 2? 5
20 Practice problems If X 1,..., X 10 are i.i.d. random variables with mean and variance 10 What s the mean of X Variance of X =? = 1 Mean of T 10? 0 Variance of T 10? = 100 Mean of X 1 X 2? 0 Variance of X 1 X 2? = 20 5
21 Roadmap Continuous random variables and normal distribution Z-chart (VERY IMPORTANT!!) 6
22 Continuous random variables A random variable X is continuous if it can take any value in some continuous range of values, denoted by (L, H). Every continuous random variable X has an associated density function f (x). 7
23 Continuous random variables A random variable X is continuous if it can take any value in some continuous range of values, denoted by (L, H). Every continuous random variable X has an associated density function f (x). The probability that the continuous random variable takes a value in the range (a, b) is 7
24 Continuous random variables From a calculus point of view this probability is obtained by integrating this density function over the range a to b. Prob(a < X < b) = b a f (x) dx. As a particular case, H L f (x) dx = 1. 8
25 The mean and variance of a continuous random variable The mean µ and variance σ 2 of a continuous random variable X having range (L, H) and density function f (x) are defined respectively by µ = H L xf (x) dx and σ 2 = H L (x µ) 2 f (x) dx = H L x 2 f (x) dx µ 2. 9
26 Normal distribution The continuous random variable X has a normal distribution with mean µ and variance σ 2 (aka, N(µ, σ 2 )), then the density function f (x) is given by the formula f (x) = 1 2πσ e (x µ)2 2σ 2, < x <. 10
27 Normal distribution The continuous random variable X has a normal distribution with mean µ and variance σ 2 (aka, N(µ, σ 2 )), then the density function f (x) is given by the formula f (x) = 1 2πσ e (x µ)2 2σ 2, < x <. If a random variable has distribution N(0, 1), then it is called a standardized normal distribution, denoted by Z. 10
28 Normal distribution The continuous random variable X has a normal distribution with mean µ and variance σ 2 (aka, N(µ, σ 2 )), then the density function f (x) is given by the formula f (x) = 1 2πσ e (x µ)2 2σ 2, < x <. If a random variable has distribution N(0, 1), then it is called a standardized normal distribution, denoted by Z. If a random variable X has distribution N(µ, σ 2 ),then Z = X µ has standardized normal σ distribution. 10
29 Normal distribution The continuous random variable X has a normal distribution with mean µ and variance σ 2 (aka, N(µ, σ 2 )), then the density function f (x) is given by the formula f (x) = 1 2πσ e (x µ)2 2σ 2, < x <. If a random variable has distribution N(0, 1), then it is called a standardized normal distribution, denoted by Z. If a random variable X has distribution N(µ, σ 2 ),then Z = X µ has standardized normal σ distribution. Bell-shaped curve 10
30 Z-chart The Z-chart is designed for a standard normal random variable, i.e. N(0, 1). It gives less than probabilities for positive values of z, i.e. Prob(Z a) when a > 0. 11
31 Z-chart Z has normal distribution with mean 0 and variance 1 Prob(Z a) when a 0; Practice: Prob(Z 0.24), Prob(Z 0.51) 12
32 Z-chart Z has normal distribution with mean 0 and variance 1 Prob(Z a) when a 0; Practice: Find b, such that Prob(Z b) =
33 Z-chart Z has normal distribution with mean 0 and variance 1 Prob(Z a) when a < 0 Key fact: Prob(Z a) = 1 Prob(Z a). Practice: Prob(Z 0.24), Prob(Z 0.49) 12
34 Z-chart Z has normal distribution with mean 0 and variance 1 Prob(a < Z b) when a < b. Key fact: Prob(a < Z b) = Prob(Z b) Prob(Z a) Practice: Prob(0.24 < Z 0.51), Prob( 0.24 < Z 0.51) 12
35 Z-chart X has normal distribution with mean µ and variance σ 2. Prob(X a), Prob(a < X b). Key fact: X µ σ has distribution N(0, 1). Practice : Prob( < X 4) when X satisfies N(2, 16). 1
36 Z-chart X has normal distribution with mean µ and variance σ 2. Prob(X a), Prob(a < X b). Key fact: X µ σ has distribution N(0, 1). Practice : Find b, such that Prob(X b) = when X satisfies N(2, 16). 1
37 Z-chart One more: Prob(7 < X 10) when X has distribution N(8, 9). Z-chart 14
38 Z-chart One more: Prob(7 < X 10) when X has distribution N(8, 9). Solutions 1 Let Z = X 8 9 = X 8. 14
39 Z-chart One more: Prob(7 < X 10) when X has distribution N(8, 9). Solutions 1 Let Z = X 8 9 = X 8. 2 Prob(7 < X 10) = Prob( 7 8 < X ) = Prob( 0. < Z 0.67) 14
40 Z-chart One more: Prob(7 < X 10) when X has distribution N(8, 9). Solutions 1 Let Z = X 8 9 = X 8. 2 Prob(7 < X 10) = Prob( 7 8 < X ) = Prob( 0. < Z 0.67) Prob( 0. < Z 0.67) = Prob(Z 0.67) Prob(Z 0.) 14
41 Z-chart One more: Prob(7 < X 10) when X has distribution N(8, 9). Solutions 1 Let Z = X 8 9 = X 8. 2 Prob(7 < X 10) = Prob( 7 8 < X ) = Prob( 0. < Z 0.67) Prob( 0. < Z 0.67) = Prob(Z 0.67) Prob(Z 0.) 4 Prob(Z 0.67) (1 Prob(Z 0.)) = ( ) =
STAT 111 Recitation 3
STAT 111 Recitation 3 Linjun Zhang stat.wharton.upenn.edu/~linjunz/ September 23, 2017 Misc. The unpicked-up homeworks will be put in the STAT 111 box in the Stats Department lobby (It s on the 4th floor
STAT 111 Recitation 2
STAT 111 Recitation 2 Linjun Zhang October 10, 2017 Misc. Please collect homework 1 (graded). 1 Misc. Please collect homework 1 (graded). Office hours: 4:30-5:30pm every Monday, JMHH F86. 1 Misc. Please
Lecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
Normal Distribution. Notes. Normal Distribution. Standard Normal. Sums of Normal Random Variables. Normal. approximation of Binomial.
Lecture 21,22, 23 Text: A Course in Probability by Weiss 8.5 STAT 225 Introduction to Probability Models March 31, 2014 Standard Sums of Whitney Huang Purdue University 21,22, 23.1 Agenda 1 2 Standard
Central Limit Theorem, Joint Distributions Spring 2018
Central Limit Theorem, Joint Distributions 18.5 Spring 218.5.4.3.2.1-4 -3-2 -1 1 2 3 4 Exam next Wednesday Exam 1 on Wednesday March 7, regular room and time. Designed for 1 hour. You will have the full
Continuous Probability Distributions & Normal Distribution
Mathematical Methods Units 3/4 Student Learning Plan Continuous Probability Distributions & Normal Distribution 7 lessons Notes: Students need practice in recognising whether a problem involves a discrete
Confidence Intervals Introduction
Confidence Intervals Introduction A point estimate provides no information about the precision and reliability of estimation. For example, the sample mean X is a point estimate of the population mean μ
The Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
continuous rv Note for a legitimate pdf, we have f (x) 0 and f (x)dx = 1. For a continuous rv, P(X = c) = c f (x)dx = 0, hence
continuous rv Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a b, P(a X b) = b a f (x)dx.
Elementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
As you draw random samples of size n, as n increases, the sample means tend to be normally distributed.
The Central Limit Theorem The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. The clt says that if we collect samples of size n with a "large enough
Section 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
4.2 Probability Distributions
4.2 Probability Distributions Definition. A random variable is a variable whose value is a numerical outcome of a random phenomenon. The probability distribution of a random variable tells us what the
Version A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
Chapter 7: Estimation Sections
Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions Frequentist Methods: 7.5 Maximum Likelihood Estimators
What was in the last lecture?
What was in the last lecture? Normal distribution A continuous rv with bell-shaped density curve The pdf is given by f(x) = 1 2πσ e (x µ)2 2σ 2, < x < If X N(µ, σ 2 ), E(X) = µ and V (X) = σ 2 Standard
Central Limit Thm, Normal Approximations
Central Limit Thm, Normal Approximations Engineering Statistics Section 5.4 Josh Engwer TTU 23 March 2016 Josh Engwer (TTU) Central Limit Thm, Normal Approximations 23 March 2016 1 / 26 PART I PART I:
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10 Fall 2011 Lecture 8 Part 2 (Fall 2011) Probability Distributions Lecture 8 Part 2 1 / 23 Normal Density Function f
Chapter 3 Common Families of Distributions. Definition 3.4.1: A family of pmfs or pdfs is called exponential family if it can be expressed as
Lecture 0 on BST 63: Statistical Theory I Kui Zhang, 09/9/008 Review for the previous lecture Definition: Several continuous distributions, including uniform, gamma, normal, Beta, Cauchy, double exponential
Lecture Notes 6. Assume F belongs to a family of distributions, (e.g. F is Normal), indexed by some parameter θ.
Sufficient Statistics Lecture Notes 6 Sufficiency Data reduction in terms of a particular statistic can be thought of as a partition of the sample space X. Definition T is sufficient for θ if the conditional
A potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples
1.3 Regime switching models A potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples (or regimes). If the dates, the
Central Limit Theorem (cont d) 7/28/2006
Central Limit Theorem (cont d) 7/28/2006 Central Limit Theorem for Binomial Distributions Theorem. For the binomial distribution b(n, p, j) we have lim npq b(n, p, np + x npq ) = φ(x), n where φ(x) is
15.063: Communicating with Data Summer Recitation 4 Probability III
15.063: Communicating with Data Summer 2003 Recitation 4 Probability III Today s Content Normal RV Central Limit Theorem (CLT) Statistical Sampling 15.063, Summer '03 2 Normal Distribution Any normal RV
Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 207 Homework 5 Drew Armstrong Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Section 3., Exercises 3, 0. Section 3.3, Exercises 2, 3, 0,.
Homework: Due Wed, Feb 20 th. Chapter 8, # 60a + 62a (count together as 1), 74, 82
Announcements: Week 5 quiz begins at 4pm today and ends at 3pm on Wed If you take more than 20 minutes to complete your quiz, you will only receive partial credit. (It doesn t cut you off.) Today: Sections
Chapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 27 Continuous
Chapter 7: Estimation Sections
1 / 31 : Estimation Sections 7.1 Statistical Inference Bayesian Methods: 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators Frequentist Methods: 7.5 Maximum Likelihood
Chapter 7: Point Estimation and Sampling Distributions
Chapter 7: Point Estimation and Sampling Distributions Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 20 Motivation In chapter 3, we learned
Chapter 7 1. Random Variables
Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous
Probability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
Two hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
Section The Sampling Distribution of a Sample Mean
Section 5.2 - The Sampling Distribution of a Sample Mean Statistics 104 Autumn 2004 Copyright c 2004 by Mark E. Irwin The Sampling Distribution of a Sample Mean Example: Quality control check of light
Lecture Stat 302 Introduction to Probability - Slides 15
Lecture Stat 30 Introduction to Probability - Slides 15 AD March 010 AD () March 010 1 / 18 Continuous Random Variable Let X a (real-valued) continuous r.v.. It is characterized by its pdf f : R! [0, )
Chapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 28 One more
MLLunsford 1. Activity: Central Limit Theorem Theory and Computations
MLLunsford 1 Activity: Central Limit Theorem Theory and Computations Concepts: The Central Limit Theorem; computations using the Central Limit Theorem. Prerequisites: The student should be familiar with
Chapter 7 - Lecture 1 General concepts and criteria
Chapter 7 - Lecture 1 General concepts and criteria January 29th, 2010 Best estimator Mean Square error Unbiased estimators Example Unbiased estimators not unique Special case MVUE Bootstrap General Question
Point Estimation. Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
6 Point Estimation Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Point Estimation Statistical inference: directed toward conclusions about one or more parameters. We will use the generic
Chapter 4: Asymptotic Properties of MLE (Part 3)
Chapter 4: Asymptotic Properties of MLE (Part 3) Daniel O. Scharfstein 09/30/13 1 / 1 Breakdown of Assumptions Non-Existence of the MLE Multiple Solutions to Maximization Problem Multiple Solutions to
6 Central Limit Theorem. (Chs 6.4, 6.5)
6 Central Limit Theorem (Chs 6.4, 6.5) Motivating Example In the next few weeks, we will be focusing on making statistical inference about the true mean of a population by using sample datasets. Examples?
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15. Lecture 10: Continuous RV Families. Prof. Vince Calhoun
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Lecture 10: Continuous RV Families Prof. Vince Calhoun 1 Reading This class: Section 4.4-4.5 Next class: Section 4.6-4.7 2 Homework 3.9, 3.49, 4.5,
Statistics 431 Spring 2007 P. Shaman. Preliminaries
Statistics 4 Spring 007 P. Shaman The Binomial Distribution Preliminaries A binomial experiment is defined by the following conditions: A sequence of n trials is conducted, with each trial having two possible
Tutorial 11: Limit Theorems. Baoxiang Wang & Yihan Zhang bxwang, April 10, 2017
Tutorial 11: Limit Theorems Baoxiang Wang & Yihan Zhang bxwang, yhzhang@cse.cuhk.edu.hk April 10, 2017 1 Outline The Central Limit Theorem (CLT) Normal Approximation Based on CLT De Moivre-Laplace Approximation
The Normal Distribution
The Normal Distribution The normal distribution plays a central role in probability theory and in statistics. It is often used as a model for the distribution of continuous random variables. Like all models,
Introduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous random variable)
5.3 Statistics and Their Distributions
Chapter 5 Joint Probability Distributions and Random Samples Instructor: Lingsong Zhang 1 Statistics and Their Distributions 5.3 Statistics and Their Distributions Statistics and Their Distributions Consider
Business Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
Chapter 8: Sampling distributions of estimators Sections
Chapter 8: Sampling distributions of estimators Sections 8.1 Sampling distribution of a statistic 8.2 The Chi-square distributions 8.3 Joint Distribution of the sample mean and sample variance Skip: p.
Statistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
Homework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a
Homework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a Announcements: There are some office hour changes for Nov 5, 8, 9 on website Week 5 quiz begins after class today and ends at
Chapter 8. Introduction to Statistical Inference
Chapter 8. Introduction to Statistical Inference Point Estimation Statistical inference is to draw some type of conclusion about one or more parameters(population characteristics). Now you know that a
Topic 6 - Continuous Distributions I. Discrete RVs. Probability Density. Continuous RVs. Background Reading. Recall the discrete distributions
Topic 6 - Continuous Distributions I Discrete RVs Recall the discrete distributions STAT 511 Professor Bruce Craig Binomial - X= number of successes (x =, 1,...,n) Geometric - X= number of trials (x =,...)
UQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions.
UQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions. Random Variables 2 A random variable X is a numerical (integer, real, complex, vector etc.) summary of the outcome of the random experiment.
4 Random Variables and Distributions
4 Random Variables and Distributions Random variables A random variable assigns each outcome in a sample space. e.g. called a realization of that variable to Note: We ll usually denote a random variable
Business Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
Chapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
Statistics 511 Supplemental Materials
Gaussian (or Normal) Random Variable In this section we introduce the Gaussian Random Variable, which is more commonly referred to as the Normal Random Variable. This is a random variable that has a bellshaped
15.063: Communicating with Data Summer Recitation 3 Probability II
15.063: Communicating with Data Summer 2003 Recitation 3 Probability II Today s Goal Binomial Random Variables (RV) Covariance and Correlation Sums of RV Normal RV 15.063, Summer '03 2 Random Variables
Chapter 7: Estimation Sections
1 / 40 Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: Chapter 7 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators Frequentist Methods:
The Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
Sampling Distribution
MAT 2379 (Spring 2012) Sampling Distribution Definition : Let X 1,..., X n be a collection of random variables. We say that they are identically distributed if they have a common distribution. Definition
Lecture 8. The Binomial Distribution. Binomial Distribution. Binomial Distribution. Probability Distributions: Normal and Binomial
Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed
Unit2: Probabilityanddistributions. 3. Normal and binomial distributions
Announcements Unit2: Probabilityanddistributions 3. Normal and binomial distributions Sta 101 - Fall 2017 Duke University, Department of Statistical Science Formatting of problem set submissions: Bad:
Drunken Birds, Brownian Motion, and Other Random Fun
Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales Outline Review of Basic Probability
χ 2 distributions and confidence intervals for population variance
χ 2 distributions and confidence intervals for population variance Let Z be a standard Normal random variable, i.e., Z N(0, 1). Define Y = Z 2. Y is a non-negative random variable. Its distribution is
Module 4: Probability
Module 4: Probability 1 / 22 Probability concepts in statistical inference Probability is a way of quantifying uncertainty associated with random events and is the basis for statistical inference. Inference
Exercise. Show the corrected sample variance is an unbiased estimator of population variance. S 2 = n i=1 (X i X ) 2 n 1. Exercise Estimation
Exercise Show the corrected sample variance is an unbiased estimator of population variance. S 2 = n i=1 (X i X ) 2 n 1 Exercise S 2 = = = = n i=1 (X i x) 2 n i=1 = (X i µ + µ X ) 2 = n 1 n 1 n i=1 ((X
Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS
Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS Part 1: Introduction Sampling Distributions & the Central Limit Theorem Point Estimation & Estimators Sections 7-1 to 7-2 Sample data
Tutorial 6. Sampling Distribution. ENGG2450A Tutors. 27 February The Chinese University of Hong Kong 1/6
Tutorial 6 Sampling Distribution ENGG2450A Tutors The Chinese University of Hong Kong 27 February 2017 1/6 Random Sample and Sampling Distribution 2/6 Random sample Consider a random variable X with distribution
MAKING SENSE OF DATA Essentials series
MAKING SENSE OF DATA Essentials series THE NORMAL DISTRIBUTION Copyright by City of Bradford MDC Prerequisites Descriptive statistics Charts and graphs The normal distribution Surveys and sampling Correlation
Unit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
Unit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
Week 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
Chapter 7. Sampling Distributions and the Central Limit Theorem
Chapter 7. Sampling Distributions and the Central Limit Theorem 1 Introduction 2 Sampling Distributions related to the normal distribution 3 The central limit theorem 4 The normal approximation to binomial
Basic Data Analysis. Stephen Turnbull Business Administration and Public Policy Lecture 4: May 2, Abstract
Basic Data Analysis Stephen Turnbull Business Administration and Public Policy Lecture 4: May 2, 2013 Abstract Introduct the normal distribution. Introduce basic notions of uncertainty, probability, events,
STAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
Statistical Tables Compiled by Alan J. Terry
Statistical Tables Compiled by Alan J. Terry School of Science and Sport University of the West of Scotland Paisley, Scotland Contents Table 1: Cumulative binomial probabilities Page 1 Table 2: Cumulative
Statistics, Their Distributions, and the Central Limit Theorem
Statistics, Their Distributions, and the Central Limit Theorem MATH 3342 Sections 5.3 and 5.4 Sample Means Suppose you sample from a popula0on 10 0mes. You record the following sample means: 10.1 9.5 9.6
Class 11. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 11 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 2017 by D.B. Rowe 1 Agenda: Recap Chapter 5.3 continued Lecture 6.1-6.2 Go over Eam 2. 2 5: Probability
MATH 3200 Exam 3 Dr. Syring
. Suppose n eligible voters are polled (randomly sampled) from a population of size N. The poll asks voters whether they support or do not support increasing local taxes to fund public parks. Let M be
The Multistep Binomial Model
Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test Friday 9th March - 12pm Examples Sheet 1 4 (not qu 3 or qu 5 on sheet 4) Lectures 1-9 10.1 A Discrete Model for Stock Price Reminder: The
Chapter 4 Random Variables & Probability. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random variable =
MATH 264 Problem Homework I
MATH Problem Homework I Due to December 9, 00@:0 PROBLEMS & SOLUTIONS. A student answers a multiple-choice examination question that offers four possible answers. Suppose that the probability that the
Chapter 5: Statistical Inference (in General)
Chapter 5: Statistical Inference (in General) Shiwen Shen University of South Carolina 2016 Fall Section 003 1 / 17 Motivation In chapter 3, we learn the discrete probability distributions, including Bernoulli,
Point Estimators. STATISTICS Lecture no. 10. Department of Econometrics FEM UO Brno office 69a, tel
STATISTICS Lecture no. 10 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 8. 12. 2009 Introduction Suppose that we manufacture lightbulbs and we want to state
2. The sum of all the probabilities in the sample space must add up to 1
Continuous Random Variables and Continuous Probability Distributions Continuous Random Variable: A variable X that can take values on an interval; key feature remember is that the values of the variable
Lecture 12. Some Useful Continuous Distributions. The most important continuous probability distribution in entire field of statistics.
ENM 207 Lecture 12 Some Useful Continuous Distributions Normal Distribution The most important continuous probability distribution in entire field of statistics. Its graph, called the normal curve, is
The Normal Distribution
Will Monroe CS 09 The Normal Distribution Lecture Notes # July 9, 207 Based on a chapter by Chris Piech The single most important random variable type is the normal a.k.a. Gaussian) random variable, parametrized
Figure 1: 2πσ is said to have a normal distribution with mean µ and standard deviation σ. This is also denoted
Figure 1: Math 223 Lecture Notes 4/1/04 Section 4.10 The normal distribution Recall that a continuous random variable X with probability distribution function f(x) = 1 µ)2 (x e 2σ 2πσ is said to have a
A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
1. Covariance between two variables X and Y is denoted by Cov(X, Y) and defined by. Cov(X, Y ) = E(X E(X))(Y E(Y ))
Correlation & Estimation - Class 7 January 28, 2014 Debdeep Pati Association between two variables 1. Covariance between two variables X and Y is denoted by Cov(X, Y) and defined by Cov(X, Y ) = E(X E(X))(Y
NORMAL APPROXIMATION. In the last chapter we discovered that, when sampling from almost any distribution, e r2 2 rdrdϕ = 2π e u du =2π.
NOMAL APPOXIMATION Standardized Normal Distribution Standardized implies that its mean is eual to and the standard deviation is eual to. We will always use Z as a name of this V, N (, ) will be our symbolic
Conjugate priors: Beta and normal Class 15, Jeremy Orloff and Jonathan Bloom
1 Learning Goals Conjugate s: Beta and normal Class 15, 18.05 Jeremy Orloff and Jonathan Bloom 1. Understand the benefits of conjugate s.. Be able to update a beta given a Bernoulli, binomial, or geometric
Introduction to Statistical Data Analysis II
Introduction to Statistical Data Analysis II JULY 2011 Afsaneh Yazdani Preface Major branches of Statistics: - Descriptive Statistics - Inferential Statistics Preface What is Inferential Statistics? Preface
ECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
Statistics, Measures of Central Tendency I
Statistics, Measures of Central Tendency I We are considering a random variable X with a probability distribution which has some parameters. We want to get an idea what these parameters are. We perfom
Chapter 5. Statistical inference for Parametric Models
Chapter 5. Statistical inference for Parametric Models Outline Overview Parameter estimation Method of moments How good are method of moments estimates? Interval estimation Statistical Inference for Parametric
THE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management
THE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management BA 386T Tom Shively PROBABILITY CONCEPTS AND NORMAL DISTRIBUTIONS The fundamental idea underlying any statistical
Midterm Exam III Review
Midterm Exam III Review Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Midterm Exam III Review 1 / 25 Permutations and Combinations ORDER In order to count the number of possible ways
Counting Basics. Venn diagrams
Counting Basics Sets Ways of specifying sets Union and intersection Universal set and complements Empty set and disjoint sets Venn diagrams Counting Inclusion-exclusion Multiplication principle Addition
Geometric Brownian Motions
Chapter 6 Geometric Brownian Motions 1 Normal Distributions We begin by recalling the normal distribution briefly. Let Z be a random variable distributed as standard normal, i.e., Z N(0, 1). The probability