Math 151. Rumbos Spring Solutions to Assignment #22

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Math 151. Rumbos Spring Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability that the sample mean will be less than 3. Solution: Let X k, for k = 1, 2, 3,..., denote the outcome of the k th roll of the die. We want to estimate PrX n 3, where X n is the sample mean with n = 81. In this case we have µ = EX k = 3.5 for all k and σ 2 = VarX k = 35/12 for all k. Since the outcomes of the rolls are independent and n = 81, we can use the Central Limit Theorem to approximate Xn µ PrX n 3 = Pr σ/ n / 12/ 81 PrZ 2.63, where Z Normal0, 1, by the Central Limit Theorem. Thus, using the symmetry of the pdf of Z Normal0, 1, PrX n 3 1 F Z 2.63 = , so that PrX n , or about 0.43%. 2. A random sample of size 49 is taken form a distribution with mean µ and variance σ 2. Estimate the probability that sample mean will be within 0.7 standard deviations from the mean of the distribution. Solution: We want to estimate Pr X n µ 0.7σ, where X n is the sample mean with n = 49. Since X n is the sample mean of a random sample and n is larger than 30, we can apply the Central Limit Theorem to approximate Xn µ Pr X n µ 0.7σ = Pr σ/ n 0.7σ σ/7 Pr Z 4.7, where Z Normal0, 1, by the Central Limit Theorem. We then have that Pr X n µ 0.7σ 1.

2 Math 151. Rumbos Spring A large freight elevator can transport a maximum of 9800 pounds. Suppose a load of cargo containing 49 boxes must be transported via the elevator. Experience has shown that the weight of boxes of this type of cargo follows a distribution with mean µ = 205 pounds and standard deviation σ = 15 pounds. Based on this information, what is the probability that all 49 boxes can be safely loaded onto the freight elevator and transported? Solution: Let X k, for k = 1, 2, 3,..., 49, denote the weight of each of the boxes. We want to estimate Pr X k 9800, where µ = EX k = 205 pounds, σ = VarX k = 15 pounds, and n = 49. We can therefore apply the Central Limit Theorem to estimate n Pr X k 9800 = Pr X k nµ , 045 nσ 715 PrZ 2.33, where Z Normal0, 1, by the Central Limit Theorem. Therefore, using the symmetry of the pdf of Normal0, 1, Pr X k F Z 2.33 = = Thus, the probability that all 49 boxes can be safely loaded onto the freight elevator and transported is about 0.99%, or less than 1%. 4. Forty nine measurements are recorded to several decimal places. Each of these 49 numbers is rounded off to the nearest integer. The sum of the original 49 numbers is approximated by the sum of those integers. Assume that the errors made in rounding off are independent, identically distributed random variables with a uniform distribution over the interval 0.5, 0.5. Compute approximately the probability that the sum of the integers is within two units of the true sum.

3 Math 151. Rumbos Spring Solution: Let X 1, X 2,..., X n, where n = 49, denote the 49 measurements, and Y 1, Y 2,..., Y n be the corresponding nearest integers after rounding off. Then X i = Y i + U i for i = 1, 2,..., n, where U 1, U 2,..., U n are independent identically distributed uniform random variables on the interval 0.5, 0.5. We then have that EU i = 0 for all i, and VarU i = 1 12, for all i. Thus, σ = 1/ 12. The sum, S, of the original measurements is S = Y i + U i, where Y i is the sum of the integer approximations. Put W = S Y i = U i. We would like to estimate Pr W 2. We will do this by applying the Central Limit Theorem to U 1, U 2, U 3,... Observe that W = nu n, where U n is the sample mean of the rounding off values U 1, U 2,..., U n. By the Central Limit Theorem U n Pr σ/ n z PrZ z, for all z R, where Z Normal0, 1. We therefore get that for z > 0. U n Pr σ/ n z Pr Z z,

4 Math 151. Rumbos Spring It then follows that Pr W 2 = Pr U n 2/n U n = Pr σ/ n 2 σ n Pr Z 2 σ n 2 = 2F Z σ 1 n = 2F Z F Z , or about 67.78%. Thus, the probability that the sum of the 49 integers is within 2 units of the true sum is about 67.78%. 5. Let X denote a random variable with pdf 1 if 1 < x <, f X x = x 2 0 otherwise. Consider a random sample of size 72 from this distribution. Compute approximately the probability that 50 or more observations of the random sample are less than 3. Solution: The probability that a random observation from the distribution is less than 3 is given by p = 3 f X x dx = x 2 dx = 2 3.

5 Math 151. Rumbos Spring Let Y denote the number of observation out of the 72 which are less than 3. Then, Y Binomialp, n, where n = 72. It then follows that the mean of Y is µ = np = 48 and the standard deviation of Y is σ = np1 p = 4. By the Central Limit Theorem we then get that Y np Pr z PrZ z, np1 p for all z R, where Z Normal0, 1, or Y 48 Pr z PrZ z. 4 We want to estimate PrY > 49 = 1 PrY 49, where Y PrY 49 = Pr 4 4 Pr Z Thus, PrY > = or about 40.13%.

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

Definition 6.1.1. A r.v. X has a normal distribution with mean µ and variance σ 2, where µ R, and σ > 0, if its density is f(x) = 1. 2σ 2.

Chapter 6 Brownian Motion 6. Normal Distribution Definition 6... A r.v. X has a normal distribution with mean µ and variance σ, where µ R, and σ > 0, if its density is fx = πσ e x µ σ. The previous definition

Math 431 An Introduction to Probability. Final Exam Solutions

Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

Math 461 Fall 2006 Test 2 Solutions

Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two

An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

( ) = P Z > = P( Z > 1) = 1 Φ(1) = 1 0.8413 = 0.1587 P X > 17

4.6 I company that manufactures and bottles of apple juice uses a machine that automatically fills 6 ounce bottles. There is some variation, however, in the amounts of liquid dispensed into the bottles

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

Practice Problems for Homework #6. Normal distribution and Central Limit Theorem.

Practice Problems for Homework #6. Normal distribution and Central Limit Theorem. 1. Read Section 3.4.6 about the Normal distribution and Section 4.7 about the Central Limit Theorem. 2. Solve the practice

Using pivots to construct confidence intervals. In Example 41 we used the fact that

Using pivots to construct confidence intervals In Example 41 we used the fact that Q( X, µ) = X µ σ/ n N(0, 1) for all µ. We then said Q( X, µ) z α/2 with probability 1 α, and converted this into a statement

University of California, Berkeley, Statistics 134: Concepts of Probability

University of California, Berkeley, Statistics 134: Concepts of Probability Michael Lugo, Spring 211 Exam 2 solutions 1. A fair twenty-sided die has its faces labeled 1, 2, 3,..., 2. The die is rolled

5. Continuous Random Variables

5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem

MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

2.6. Probability. In general the probability density of a random variable satisfies two conditions:

2.6. PROBABILITY 66 2.6. Probability 2.6.. Continuous Random Variables. A random variable a real-valued function defined on some set of possible outcomes of a random experiment; e.g. the number of points

Math 425 (Fall 08) Solutions Midterm 2 November 6, 2008

Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the

Math 58. Rumbos Fall Solutions to Assignment #5

Math 58. Rumbos Fall 2008 1 Solutions to Assignment #5 1. (Typographical Errors 1 ) Typographical and spelling errors can be either nonword errors or word errors. A nonword error is not a real word, as

ISyE 6761 Fall 2012 Homework #2 Solutions

1 1. The joint p.m.f. of X and Y is (a) Find E[X Y ] for 1, 2, 3. (b) Find E[E[X Y ]]. (c) Are X and Y independent? ISE 6761 Fall 212 Homework #2 Solutions f(x, ) x 1 x 2 x 3 1 1/9 1/3 1/9 2 1/9 1/18 3

Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

Math 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304. jones/courses/141

Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Variance: expected squared deviation from the mean: Standard

Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

University of California, Los Angeles Department of Statistics. Random variables

University of California, Los Angeles Department of Statistics Statistics Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

The Normal Distribution. Alan T. Arnholt Department of Mathematical Sciences Appalachian State University

The Normal Distribution Alan T. Arnholt Department of Mathematical Sciences Appalachian State University arnholt@math.appstate.edu Spring 2006 R Notes 1 Copyright c 2006 Alan T. Arnholt 2 Continuous Random

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

The Normal distribution

The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ

Practice problems for Homework 11 - Point Estimation

Practice problems for Homework 11 - Point Estimation 1. (10 marks) Suppose we want to select a random sample of size 5 from the current CS 3341 students. Which of the following strategies is the best:

ECE302 Spring 2006 HW4 Solutions February 6, 2006 1

ECE302 Spring 2006 HW4 Solutions February 6, 2006 1 Solutions to HW4 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

6. Jointly Distributed Random Variables

6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.

Chapter 4. Probability Distributions

Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive

Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 5 Solutions

Math 370/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 5 Solutions About this problem set: These are problems from Course 1/P actuarial exams that I have collected over the

MAS108 Probability I

1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density

HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 18. A Brief Introduction to Continuous Probability

CS 7 Discrete Mathematics and Probability Theory Fall 29 Satish Rao, David Tse Note 8 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

For a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )

Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll

WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

Hypothesis Testing for Beginners

Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes

3.4 The Normal Distribution

3.4 The Normal Distribution All of the probability distributions we have found so far have been for finite random variables. (We could use rectangles in a histogram.) A probability distribution for a continuous

Aggregate Loss Models

Aggregate Loss Models Chapter 9 Stat 477 - Loss Models Chapter 9 (Stat 477) Aggregate Loss Models Brian Hartman - BYU 1 / 22 Objectives Objectives Individual risk model Collective risk model Computing

Chapter 5: Normal Probability Distributions - Solutions

Chapter 5: Normal Probability Distributions - Solutions Note: All areas and z-scores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

Binomial random variables

Binomial and Poisson Random Variables Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Binomial random variables 1. A certain coin has a 5% of landing heads, and a 75% chance

Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions 1. The following table contains a probability distribution for a random variable X. a. Find the expected value (mean) of X. x 1 2

The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

Generating Random Data. Alan T. Arnholt Department of Mathematical Sciences Appalachian State University arnholt@math.appstate.edu

Generating Random Data Alan T. Arnholt Department of Mathematical Sciences Appalachian State University arnholt@math.appstate.edu Spring 2006 R Notes 1 Copyright c 2006 Alan T. Arnholt 2 Generating Random

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:

Chapter 7 Notes - Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a

Feb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172-179)

Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172-179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities

1 Prior Probability and Posterior Probability

Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which

4.1 4.2 Probability Distribution for Discrete Random Variables

4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

Math/Stats 342: Solutions to Homework

Math/Stats 342: Solutions to Homework Steven Miller (sjm1@williams.edu) November 17, 2011 Abstract Below are solutions / sketches of solutions to the homework problems from Math/Stats 342: Probability

Markov Chains for the RISK Board Game Revisited. Introduction. The Markov Chain. Jason A. Osborne North Carolina State University Raleigh, NC 27695

Markov Chains for the RISK Board Game Revisited Jason A. Osborne North Carolina State University Raleigh, NC 27695 Introduction Probabilistic reasoning goes a long way in many popular board games. Abbott

Round to Decimal Places

Day 1 1 Round 5.0126 to 2 decimal 2 Round 10.3217 to 3 decimal 3 Round 0.1371 to 3 decimal 4 Round 23.4004 to 2 decimal 5 Round 8.1889 to 2 decimal 6 Round 9.4275 to 2 decimal 7 Round 22.8173 to 1 decimal

P (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )

Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =

16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION

6. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION It is sometimes difficult to directly compute probabilities for a binomial (n, p) random variable, X. We need a different table for each value of

Hypothesis Testing Introduction

Hypothesis Testing Introduction Hypothesis: A conjecture about the distribution of some random variables. For example, a claim about the value of a parameter of the statistical model. A hypothesis can

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Open book and note Calculator OK Multiple Choice 1 point each MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data.

3.2 Measures of Spread In some data sets the observations are close together, while in others they are more spread out. In addition to measures of the center, it's often important to measure the spread

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / October

The normal approximation to the binomial

The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very

Chapter 3 Joint Distributions

Chapter 3 Joint Distributions 3.6 Functions of Jointly Distributed Random Variables Discrete Random Variables: Let f(x, y) denote the joint pdf of random variables X and Y with A denoting the two-dimensional

MATH 201. Final ANSWERS August 12, 2016

MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6-sided dice, five 8-sided dice, and six 0-sided dice. A die is drawn from the bag and then rolled.

3. Continuous Random Variables

3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such

Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

ECE302 Spring 2006 HW7 Solutions March 11, 2006 1

ECE32 Spring 26 HW7 Solutions March, 26 Solutions to HW7 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where

Lecture 10: Depicting Sampling Distributions of a Sample Proportion

Lecture 10: Depicting Sampling Distributions of a Sample Proportion Chapter 5: Probability and Sampling Distributions 2/10/12 Lecture 10 1 Sample Proportion 1 is assigned to population members having a

MAT 155. Key Concept. September 27, 2010. 155S5.5_3 Poisson Probability Distributions. Chapter 5 Probability Distributions

MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance and Standard

1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

Chapter 4 Expected Values

Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges

The Standard Normal distribution

The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

1) What is the probability that the random variable has a value greater than 2? A) 0.750 B) 0.625 C) 0.875 D) 0.700

Practice for Chapter 6 & 7 Math 227 This is merely an aid to help you study. The actual exam is not multiple choice nor is it limited to these types of questions. Using the following uniform density curve,

Binomial random variables (Review)

Poisson / Empirical Rule Approximations / Hypergeometric Solutions STAT-UB.3 Statistics for Business Control and Regression Models Binomial random variables (Review. Suppose that you are rolling a die

13.2 Measures of Central Tendency

13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers

Solutions to Worksheet on Hypothesis Tests

s to Worksheet on Hypothesis Tests. A production line produces rulers that are supposed to be inches long. A sample of 49 of the rulers had a mean of. and a standard deviation of.5 inches. The quality

Discrete and Continuous Random Variables. Summer 2003

Discrete and Continuous Random Variables Summer 003 Random Variables A random variable is a rule that assigns a numerical value to each possible outcome of a probabilistic experiment. We denote a random

JANUARY 2016 EXAMINATIONS. Life Insurance I

PAPER CODE NO. MATH 273 EXAMINER: Dr. C. Boado-Penas TEL.NO. 44026 DEPARTMENT: Mathematical Sciences JANUARY 2016 EXAMINATIONS Life Insurance I Time allowed: Two and a half hours INSTRUCTIONS TO CANDIDATES:

Variances and covariances

Chapter 4 Variances and covariances 4.1 Overview The expected value of a random variable gives a crude measure for the center of location of the distribution of that random variable. For instance, if the

Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

Fifth Problem Assignment

EECS 40 PROBLEM (24 points) Discrete random variable X is described by the PMF { K x p X (x) = 2, if x = 0,, 2 0, for all other values of x Due on Feb 9, 2007 Let D, D 2,..., D N represent N successive

Key Concept. Density Curve

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal

Week 3&4: Z tables and the Sampling Distribution of X

Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal

Section 5.1 Continuous Random Variables: Introduction

Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,

Continuous Distributions

MAT 2379 3X (Summer 2012) Continuous Distributions Up to now we have been working with discrete random variables whose R X is finite or countable. However we will have to allow for variables that can take

ECE302 Spring 2006 HW5 Solutions February 21, 2006 1

ECE3 Spring 6 HW5 Solutions February 1, 6 1 Solutions to HW5 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics

Lecture 13: Martingales

Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

12.5: CHI-SQUARE GOODNESS OF FIT TESTS

125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability