# A continuous random variable can take on any value in a specified interval or range

Save this PDF as:

Size: px
Start display at page:

Download "A continuous random variable can take on any value in a specified interval or range"

## Transcription

1 Continuous Probability Distributions A continuous random variable can take on any value in a specified interval or range Example: Let X be a random variable indicating the blood level of serum triglycerides, measured in mg/dl The probability distribution of X is represented by a smooth curve called a probability density function 1

2 The total area under the probability density function (pdf) is equal to 1 If the pdf is represented by f(x), then f(x) dx = 1 The area under the curve between any two points x 1 and x 2 is the probability that X takes a value between x 1 and x 2 2

3 Instead of assigning probabilities to specific outcomes of the random variable X, probabilities are assigned to ranges of values The probability associated with any one particular value is equal to 0 Therefore, P(X = x) =0 Also, P(X x) =P(X>x) The probability density function indicates which ranges of values are more likely to occur than others 3

4 The cumulative distribution function of X is F(x) = P(X x) = x f(x)dx Its value is the area under the probability density function to the left of x The population mean and variance of a continuous random variable X have the same meaning as they did for a discrete (but not categorical) random variable They are used to summarize the behavior of the random variable in terms of a measure of location and a measure of dispersion 4

5 The expected value E(X), or µ, is the average value taken on by the random variable X E(X) = xf(x) dx Var(X), or σ 2, is the average squared distance of each possible value of X from µ Var(X) = (x µ)2 f(x) dx = E(X µ) 2 The standard deviation of X is σ = Var(X) 5

6 The Normal Distribution The normal distribution is the most widely used probability distribution in statistics It is also called the Gaussian distribution or the bell-shaped curve Many random variables including blood pressure, weight, height, serum cholesterol level, and IQ score are approximately normally distributed However, the distribution s real importance will be seen in the areas of estimation and hypothesis testing The normal distribution can also be used as an approximation to many other distributions (such as the binomial) in various situations 6

7 The probability density function of a normal random variable X is given by f(x) = 1 2πσ e [ 1 2 ] (x µ) 2 σ 2 where <x< The symbol π (pi) represents a constant approximated by The letter e is also a constant, approximated by It is the base of the natural logarithms, such that ln(e x )=x 7

8 µ =E(X) σ 2 =Var(X) 8

9 µ and σ are the parameters of the normal distribution they completely define its shape 9

10 The normal distribution with mean µ and variance σ 2 is represented by N(µ, σ 2 ) To find P(X x), we would have to draw the probability density function of N(µ, σ 2 ) and determine the area to the left of x To do this, a table of areas calculated for the normal distribution can be used There is a problem here depending upon the application, the normal distribution could have any number of different values for µ and σ 2 We cannot tabulate every possible distribution In fact, only a single curve is tabulated 10

11 The standard normal distribution has mean 0 and variance 1, and is denoted by N(0, 1) Approximately 68% of the area under the standard normal curve lies between 1 and +1, about 95% between 2 and +2, and about 99% between 2.5 and

12 The cumulative distribution function for a standard normal curve is represented by where X N(0, 1) Φ(x) =F(x) =P(X x) 12

13 What is P( 1 X 1)? From column D, P( 1 X 1) = This is the probability that X takes a value within ±1 standard deviation of the mean 0 13

14 What is P(X 2)? From column B, P(X 2) = This is the probability that X takes a value at least 2 standard deviations above the mean Because the standard normal distribution is symmetric, P(X 2) = as well 14

15 For what value of x is it true that P( x X x) =0.95? From column D, x =1.96 The probability that X takes a value within ±1.96 standard deviations of the mean is 0.95 From column B, the probability that X takes a value more than 1.96 SD above the mean (or more than 1.96 SD below the mean) is

16 The 100 uth percentile of the standard normal distribution is represented by z u P(X <z u ) = u What is the 80th percentile of the standard normal distribution? We want the value z.80 for which P(X <z.80 ) = 0.80 From column A, P(X <0.84) = and P(X <0.85) =

17 What is the 90th percentile of the standard normal distribution? This time, we want the value z.90 for which P(X <z.90 ) = 0.90 From column A, P(X <1.28) = What is the 99th percentile? From column A, P(X <2.33) =

18 Suppose that X N(10, 4) If 10 is subtracted from X, the distribution is shifted or translated X 10 N(0, 4) If X 10 is then divided by 2, the spread of the distribution is changed (the distribution is rescaled) X 10 2 N(0, 1) In general, if X N(µ, σ 2 ) and Z = X µ σ then Z N(0, 1) 18

19 By transforming X into Z, the table of areas for the standard normal distribution can be used to estimate probabilities associated with X Note: E(X + c) = E(X)+c E(cX) = c E(X) Var(X + c) = Var(X) Var(cX) = c 2 Var(X) These are the same properties that applied to the sample mean x and sample variance s 2 19

20 Therefore, ( X µ E(Z) = E σ ) = 1 σ E(X µ) = 1 σ [E(X) µ] = 1 σ [µ µ] = 0 and ( X µ Var(Z) = Var σ ) = 1 Var(X µ) σ2 = 1 σ 2 Var(X) = 1 σ 2 (σ2 ) = 1 20

21 Example: The diastolic blood pressures of males years of age are normally distributed with µ = 80 mm Hg and σ 2 = 144 mm Hg 2 σ = 12 mm Hg Therefore, a diastolic blood pressure of = 92 mm Hg lies 1 standard deviation above the mean Individuals with blood pressures above 95 mm Hg are considered to be hypertensive What is the probability that a randomly selected male has a blood pressure above 95 mm Hg? ( ) X P(X >95) = P > = P(Z>1.25) =

22 Approximately 10.6% of this population would be classified as hypertensive The value z =1.25 is called a z-score A z-score quantifies how far the value of interest lies from the mean µ, measured in units of the standard deviation σ Note that = 15 mm Hg One standard deviation is 12 mm Hg Therefore, 15 mm Hg is or 1.25 standard deviations above the mean of 80 mm Hg 22

23 What is the probability that a randomly selected male has a diastolic blood pressure above 110 mm Hg? ( ) X P(X >110) = P > = P(Z>2.50) = Approximately 0.6% of the population has a diastolic blood pressure above 110 mm Hg The z-score of 2.50 indicates that 110 lies 2.50 standard deviations above the mean of 80 mm Hg 23

24 What is the probability that a randomly selected male has a diastolic blood pressure below 60 mm Hg? ( X 80 P(X <60) = P < 12 = P(Z< 1.67) = P(Z>1.67) = ) Approximately 4.8% of the population has a diastolic blood pressure below 60 mm Hg The z-score of 1.67 indicates that 60 lies 1.67 standard deviations below the mean of 80 mm Hg 24

25 What value of diastolic blood pressure cuts off the upper 5% of this population? Using column B of the table for the standard normal distribution, the value Z = cuts off an area of 0.05 in the upper tail We want the value of X that corresponds to Z =1.645 Z = X µ σ = X X = 99.7 Approximately 5% of the men in this population have a diastolic blood pressure greater than 99.7 mm Hg 25

26 What value of diastolic blood pressure cuts off the lower 10% of the population? Using column B, the value Z =1.28 cuts off an area of 0.10 in the upper tail of the standard normal distribution Therefore, Z = 1.28 cuts off an area of 0.10 in the lower tail We want the value of X that corresponds to Z = 1.28 Z = X µ σ 1.28 = X X = 64.6 Approximately 10% of the men in this population have a diastolic blood pressure lower than 64.6 mm Hg 26

27 In some situations the normal distribution can be used as an approximation to the binomial distribution The approximation is most accurate when the probability distribution of the binomial random variable X is fairly symmetric This happens when the number of independent trials n is fairly large and p is not too close to either 0 or 1 27

28 Since the binomial random variable X has mean np and variance npq, its distribution can be approximated by the normal distribution N(np, npq) Example: In the United States, the probability that an individual exercises regularly is 0.42 For a group of 50 persons, what is the probability that exactly 20 exercise regularly? The total number of individuals who exercise, X, has a binomial distribution with n =50 and p =0.42 P(X = 20) = ( 50) (0.42) 20 (0.58) = This is the exact binomial probability 28

29 Now note that np = 50(0.42) = 21 and npq = 50(0.42)(0.58) = 12.2 Let Y be a normal random variable with mean 21 and variance 12.2 P(X = 20) can be approximated by the area under the N(21, 12.2) curve that lies between 19.5 and 20.5 The 0.5 that is both added to and subtracted from 20 is called a continuity correction factor It helps to compensate for the fact that a discrete binomial distribution is being approximated by a continuous normal distribution 29

30 ) P(19.5 Y 20.5) = P( Z = P( 0.43 Z 0.14) = P(0.14 Z 0.43) = P(Z 0.43) P(Z 0.14) = = Similarly, the probability that exactly 7 individuals exercise regularly is approximated by the area under the normal curve that lies between 6.5 and 7.5 P(X = 0) is approximated by P(Y 0.5) P(X = 50) is approximated by P(Y 49.5) 30

31 When does the normal approximation to the binomial distribution work? In general, N(np, npq) may be used to approximate Bin(n, p) when npq 5 If a computer is available, no approximation is necessary 31

### 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

### 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

### Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

### Statistics 100 Binomial and Normal Random Variables

Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random

### Lecture 8: Continuous random variables, expectation and variance

Lecture 8: Continuous random variables, expectation and variance Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde

### 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

### The basics of probability theory. Distribution of variables, some important distributions

The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a

### University of California, Los Angeles Department of Statistics. Normal distribution

University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Normal distribution The normal distribution is the most important distribution. It describes

### 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,

### Normal Distribution as an Approximation to the Binomial Distribution

Chapter 1 Student Lecture Notes 1-1 Normal Distribution as an Approximation to the Binomial Distribution : Goals ONE TWO THREE 2 Review Binomial Probability Distribution applies to a discrete random variable

### 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

### Continuous Random Variables

Continuous Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Continuous Random Variables 2 11 Introduction 2 12 Probability Density Functions 3 13 Transformations 5 2 Mean, Variance and Quantiles

### 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

### Probability Distributions

Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.

### 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

### 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

### given that among year old boys, carbohydrate intake is normally distributed, with a mean of 124 and a standard deviation of 20...

Probability - Chapter 5 given that among 12-14 year old boys, carbohydrate intake is normally distributed, with a mean of 124 and a standard deviation of 20... 5.6 What percentage of boys in this age range

### the number of organisms in the squares of a haemocytometer? the number of goals scored by a football team in a match?

Poisson Random Variables (Rees: 6.8 6.14) Examples: What is the distribution of: the number of organisms in the squares of a haemocytometer? the number of hits on a web site in one hour? the number of

### Bivariate Distributions

Chapter 4 Bivariate Distributions 4.1 Distributions of Two Random Variables In many practical cases it is desirable to take more than one measurement of a random observation: (brief examples) 1. What is

### Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 1.3 Homework Answers 1.80 If you ask a computer to generate "random numbers between 0 and 1, you uniform will get observations

### Random Variables. Chapter 2. Random Variables 1

Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets

### 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

### The Normal Curve. The Normal Curve and The Sampling Distribution

Discrete vs Continuous Data The Normal Curve and The Sampling Distribution We have seen examples of probability distributions for discrete variables X, such as the binomial distribution. We could use it

### CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

### CSE 312, 2011 Winter, W.L.Ruzzo. 7. continuous random variables

CSE 312, 2011 Winter, W.L.Ruzzo 7. continuous random variables continuous random variables Discrete random variable: takes values in a finite or countable set, e.g. X {1,2,..., 6} with equal probability

### DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS QM 120. Continuous Probability Distribution

DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Dr. Mohammad Zainal Continuous Probability Distribution 2 When a RV x is discrete,

### 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

### Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

### 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.

3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately

### Chapter 6 Continuous Probability Distributions

Continuous Probability Distributions Learning Objectives 1. Understand the difference between how probabilities are computed for discrete and continuous random variables. 2. Know how to compute probability

### 8. THE NORMAL DISTRIBUTION

8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,

### Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data.

Chapter 5 Section 5.1: Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Example 1: The test scores for a test were: 78, 81, 82, 76,

### Chapter 6 Random Variables

Chapter 6 Random Variables Day 1: 6.1 Discrete Random Variables Read 340-344 What is a random variable? Give some examples. A numerical variable that describes the outcomes of a chance process. Examples:

### 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

### 4: Probability. What is probability? Random variables (RVs)

4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

### Normal approximation to the Binomial

Chapter 5 Normal approximation to the Binomial 5.1 History In 1733, Abraham de Moivre presented an approximation to the Binomial distribution. He later (de Moivre, 1756, page 242 appended the derivation

### 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

### 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

### Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

### PHP 2510 Central limit theorem, confidence intervals. PHP 2510 October 20,

PHP 2510 Central limit theorem, confidence intervals PHP 2510 October 20, 2008 1 Distribution of the sample mean Case 1: Population distribution is normal For an individual in the population, X i N(µ,

### Normal Distribution. Definition A continuous random variable has a normal distribution if its probability density. f ( y ) = 1.

Normal Distribution Definition A continuous random variable has a normal distribution if its probability density e -(y -µ Y ) 2 2 / 2 σ function can be written as for < y < as Y f ( y ) = 1 σ Y 2 π Notation:

### 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

### Sampling Central Limit Theorem Proportions. Outline. 1 Sampling. 2 Central Limit Theorem. 3 Proportions

Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Populations and samples When we use statistics, we are trying to find out information about

### Chapter 4. iclicker Question 4.4 Pre-lecture. Part 2. Binomial Distribution. J.C. Wang. iclicker Question 4.4 Pre-lecture

Chapter 4 Part 2. Binomial Distribution J.C. Wang iclicker Question 4.4 Pre-lecture iclicker Question 4.4 Pre-lecture Outline Computing Binomial Probabilities Properties of a Binomial Distribution Computing

### UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

### Continuous random variables

Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the so-called continuous random variables. A random

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

### Review the following from Chapter 5

Bluman, Chapter 6 1 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that

### We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:

MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having

### Sampling Distribution of a Normal Variable

Ismor Fischer, 5/9/01 5.-1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,

### MEASURES OF VARIATION

NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

### Continuous Distributions, Mainly the Normal Distribution

Continuous Distributions, Mainly the Normal Distribution 1 Continuous Random Variables STA 281 Fall 2011 Discrete distributions place probability on specific numbers. A Bin(n,p) distribution, for example,

### E205 Final: Version B

Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

### Chapter 5. Random variables

Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

### CHAPTER 3. Statistics

CHAPTER 3 Statistics Statistics has many aspects. These include: data gathering data analysis statistical inference. For instance, politicians may wish to see whether a new law will be popular. They as

### Jointly Distributed Random Variables

Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................

### This HW reviews the normal distribution, confidence intervals and the central limit theorem.

Homework 3 Solution This HW reviews the normal distribution, confidence intervals and the central limit theorem. (1) Suppose that X is a normally distributed random variable where X N(75, 3 2 ) (mean 75

### 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............................................

### Statistics GCSE Higher Revision Sheet

Statistics GCSE Higher Revision Sheet This document attempts to sum up the contents of the Higher Tier Statistics GCSE. There is one exam, two hours long. A calculator is allowed. It is worth 75% of the

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### Histograms and density curves

Histograms and density curves What s in our toolkit so far? Plot the data: histogram (or stemplot) Look for the overall pattern and identify deviations and outliers Numerical summary to briefly describe

### Tree Diagrams. on time. The man. by subway. From above tree diagram, we can get

1 Tree Diagrams Example: A man takes either a bus or the subway to work with probabilities 0.3 and 0.7, respectively. When he takes the bus, he is late 30% of the days. When he takes the subway, he is

### Continuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.

UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Neda Farzinnia, UCLA Statistics University of California,

### What is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference

0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures

### Math 2015 Lesson 21. We discuss the mean and the median, two important statistics about a distribution. p(x)dx = 0.5

ean and edian We discuss the mean and the median, two important statistics about a distribution. The edian The median is the halfway point of a distribution. It is the point where half the population has

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

### Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

### Sample Size Determination

Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

### Statistics 102 Problem Set 6 Due 28 March 2014

Statistics 102 Problem Set 6 Due 28 March 2014 Problem set policies: Please provide concise but clear answers for each question; just writing the result of a calculation (e.g., SD = 3.3 ) with no explanation

### Unit 16 Normal Distributions

Unit 16 Normal Distributions Objectives: To obtain relative frequencies (probabilities) and percentiles with a population having a normal distribution While there are many different types of distributions

### 10 BIVARIATE DISTRIBUTIONS

BIVARIATE DISTRIBUTIONS After some discussion of the Normal distribution, consideration is given to handling two continuous random variables. The Normal Distribution The probability density function f(x)

### Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.

Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Descriptive Statistics

Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

### Statistiek (WISB361)

Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17

### 2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

### The Normal Distribution

The Normal Distribution Cal State Northridge Ψ320 Andrew Ainsworth PhD The standard deviation Benefits: Uses measure of central tendency (i.e. mean) Uses all of the data points Has a special relationship

### Important Probability Distributions OPRE 6301

Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.

### Random Variables, Expectation, Distributions

Random Variables, Expectation, Distributions CS 5960/6960: Nonparametric Methods Tom Fletcher January 21, 2009 Review Random Variables Definition A random variable is a function defined on a probability

### Summary of Probability

Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

### Each exam covers lectures from since the previous exam and up to the exam date.

Sociology 301 Exam Review Liying Luo 03.22 Exam Review: Logistics Exams must be taken at the scheduled date and time unless 1. You provide verifiable documents of unforeseen illness or family emergency,

### 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

### STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I)

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable

### 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

### Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall

### Hypothesis Testing. Chapter Introduction

Contents 9 Hypothesis Testing 553 9.1 Introduction............................ 553 9.2 Hypothesis Test for a Mean................... 557 9.2.1 Steps in Hypothesis Testing............... 557 9.2.2 Diagrammatic

### BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential

### WEEK #22: PDFs and CDFs, Measures of Center and Spread

WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook

PROBLEM SET 1 For the first three answer true or false and explain your answer. A picture is often helpful. 1. Suppose the significance level of a hypothesis test is α=0.05. If the p-value of the test

### 5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

### 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

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### Random Variable: A function that assigns numerical values to all the outcomes in the sample space.

STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.

### Continuous Probability Distributions (120 Points)

Economics : Statistics for Economics Problem Set 5 - Suggested Solutions University of Notre Dame Instructor: Julio Garín Spring 22 Continuous Probability Distributions 2 Points. A management consulting

### 6 3 The Standard Normal Distribution

290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since

### You flip a fair coin four times, what is the probability that you obtain three heads.

Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

### Chapter 2. The Normal Distribution

Chapter 2 The Normal Distribution Lesson 2-1 Density Curve Review Graph the data Calculate a numerical summary of the data Describe the shape, center, spread and outliers of the data Histogram with Curve