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

Size: px
Start display at page:

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

Transcription

1 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 variable σ standard deviation [parameter] x value for random variable X (e.g., observed number of successes for a binomial random variable) X random variable X What is probability? The probability of an event is its relative frequency in the long run. If an event occurs x times out of n, then its probability will converge on X n as n becomes infinitely large. For example, if we flip a coin many times, we expect to see half the flips turn up heads. Note that when n is small, the observed relative frequency of an event will not be a reliable reflection of its probability. However, as the number of observations n increases, the observed frequency becomes a more reliable reflection of the probability. For example, if a coin is flipped 10 times, there is no guarantee that it will turn up as exactly 5 heads every time. However, if the coin is flipped 10,000 times, chances are pretty good that the proportions of heads will be pretty close to 0.5. Random variables (RVs) A random variable (RV) is a quantity take of various values depending on chance. There are two types of random variables: (1) discrete random variables and (2) continuous random variables. Discrete random variables form a countable set of outcomes. We will study binomial random variables as a way to familiarize ourselves with discrete random variable. Continuous random variables form an infinite continuum of possible outcomes. We will study normal (Gaussian) random variables as a way to familiarize ourselves with continuous random variables. Page 1 of probability.docx (5/18/2016)

2 Binomial random variables Definition Binomial random variables are discrete RVs of counts that describe the number of successes (X) in n independent Bernoulli trials, a where each Bernoulli trial has the probability of success p. Binomial random variables have two parameters: n the number of independent Bernoulli trials p the consistent probability of success for each trial EXAMPLE. As an example of a binomial RV, consider the number of successful treatments (X) in 3 patients (n) where the probability of success in each instance p is X can take on the discrete values of 0, 1, 2, or 3. Notation. Let b represent the binomial distribution and ~ represent distributed as. Thus, X~b(n, p) is said random variable X is distributed as a binomial random variable with parameters n and p. The binomial random variable in our example is X~b(3,.25). Let Pr(X = x) represent the probability random variable X takes on the value x. For example, Pr(X = 0) the probability of 0 successes for random variable X. Pr(X x) represent the probability random variable X takes on a value x or less. This is the cumulative probability of the event. DEFINITION. The probability mass function (pmf) of a discrete random variable assigns probabilities for all possible outcomes. EXAMPLE. The pmf for X~b(3,.25) is shown in Table 1. The exact probabilities of events are shown in the second column. The cumulative probability is shown in the third column. TABLE 1. The pmf for X~b(3,.25). X Number of successes Pr(X = x) Probability 0 (event A) (event B) (event C) (event D) Pr(X x) Cumulative Probability How we calculated these probabilities is not currently the issue. Instead, let us focus on meaning. The above pmf states that for X~b(3,.25) we expect to see 0 successes of the time, 1 success of the time, 2 successes of the time, and 3 successes of the time. To calculate binomial probabilities, use this app If you are curious or for some reason you need to calculate binomial probabilities by hand, use the formulas in Chapter 6 of Basic Biostatistics for Public Health Practice. a A Bernoulli trial is a random event that can take on one of two possible outcomes: success or failure. Page 2 of probability.docx (5/18/2016)

3 Rules for working with probabilities Let: A event A (e.g., let A represent 0 success out of 3 in our Example). B event B (e.g., let B represent 1 success out of 3). Pr(A) the probability of event A. (e.g., Pr(A) = ). Ā the complement of event A not A. the union of events (e.g., A B A or B). Rule 1: Probabilities can be no less than 0% and no more than 100%. an event with probability 0 never occurs and an event with probability 1 always occurs Note that an all the events in Table 1 obey this rule. 0 Pr(A) 1 Rule 2: All possible outcomes taken together have probability exactly equal to 1. Pr(all possible outcomes) = 1 Note that in Table 1, Pr(all possible outcomes) = = 1. Rule 3: When two events are disjoint (cannot occur together), the probability of their union is the sum of their individual probabilities. Pr(A B) = Pr(A) + Pr(B), if A and B are disjoint In Table 1 let A 0 successes and A 1 success. Pr(A B) = = Rule 4: The probability of a complement is equal to 1 minus the probability of the event. Pr(Ā) = 1 Pr(A) In Table 1, Ā (1, 2, or 3 successes) and Pr(Ā) = = Page 3 of probability.docx (5/18/2016)

4 The area under the curve (AUC) concept Probability mass functions (pmf s) can be drawn. For our Example (pmf in Table 1): X~BINOMIAL(3,.25) X = 0 X = 1 X = 2 X = 3 Figure 1. X~b(3,.25). On the horizontal axis, the first bar stretches from 0 to 1. Therefore, this rectangle has base = 1. It has height = Thus, the area of this bar = h b = = = Pr(X = 0). The second bar also has a base of 1 (from 1 to 2), height of , and area = h b = = This corresponds to Pr(X = 1). The combined area of the first two bars = Pr(X = 0) or Pr(X = 1) = = In fact the area between any two values is equal to the probability of obtaining an outcome between these two values. This fact is referred to as the area under the curve (AUC) rule. We can also use the rule of complements to find AUCs (probabilities). Let A 0 successes. Therefore Ā one or more successes. By the rule of complements, Pr(Ā) = = If you add up the Pr(X = 1) + Pr(X = 2) + Pr(X = 3) you will see that this AUC is also equal to Page 4 of probability.docx (5/18/2016)

5 Normal (Gaussian) random variables Probability density functions (pdf) We will use normal random variables as a way to introduce continuous random variables. Probability density functions (pdf) assign probabilities for all possible outcomes for continuous random variables. A pdf cannot be easily shown in tabular form. I can however be drawn. While probability functions for discrete random variables (pmfs) are chunky pdfs are smooth curves. Normal pdf are recognized by their smooth, symmetrical bell shape. Normal random variables are a family of random variables. Each family member is are characterized by two parameters, μ ( mu ) and σ ( sigma ). μ the pdf s mean or expected value σ the pdf s standard deviation When μ changes, the location of the normal pdf changes. When σ changes, the spread of the normal pdf changes. Figure 2. σ1 > σ2 μ and σ are the analogues of xx and s for data distributions. However, you cannot calculate μ and σ. They describe probability functions, which are distinct from data. You can get an idea of the size of σ on a normal pdf by identifying the curve s points of inflection. This is where the curve begins to change slope. Trace the normal curve with your finger. You are skiing either the slope. The point at which the slope begins to flatten is a point of inflection. Page 5 of probability.docx (5/18/2016)

6 The left inflection point marks the location μ σ. This is one σ-unit below the mean. The right point of inflection marks the location of μ σ. This is one σ-unit below the mean. Page 6 of probability.docx (5/18/2016)

7 Normal probabilities and the area under the curve (AUC) Probabilities for continuous random variables are represented as the area under the curve (AUC) see prior section on AUCs! The rule helps get a grip on AUCs (probaiblities) from normal random variables. This rule applies to normal random variables only and says: 68% of the AUC for normal RVs lies in the region μ ± σ 95% of the AUC for normal RVs lies in the region μ ± 2σ 99.7% of the AUC for normal RVs lies in the region μ ± 3σ Visually, the 95 part of the rule looks like this: Think in terms of these landmarks: Although μ and σ vary from one normal random variable to the next, you can apply the rule to any normal random variable. Keep in mind that (a) AUC = probability (b) the total AUC = 1 (c) the values lie on the horizontal axis EXAMPLE. The Wechsler Intelligence Scale is used to measure intelligence. It is calibrated to produce a Normal distribution with μ = 100 and σ = 15 within each age group. NOTATION. Let X~N(µ, σ) represent a normal random variable with mean µ and standard deviation σ. Page 7 of probability.docx (5/18/2016)

8 Using this notation, Wechsler Intelligence scale scores is represented X~N(100, 15). The rule states that for X~N(100, 15): 68% of the AUC lies in the range μ ± σ = 100 ± 15 = 85 to % of the AUC lies in the range μ ± 2σ = 100 ± (2)(15) = 70 to % of the AUC lies in the range μ ± 3σ = 100 ± (3)(15) = 55 to 145 This next figure shows the AUC for X~N(100, 15). Notice the center of the curve is on µ. Also notice landmarks at ±1σ, ±2σ, ±3σ on the horizontal axis. Finding AUCs with an app for normal random variables The key concept here is the AUC between any two points corresponds to the probability of that observation. This applet will calculate AUCs between any two points on any X~N(μ, σ): Example. Plug in values for X~(100,15). The AUC between 130 and corresponds to the right tail of the pdf. Note that this AUC (probability) is (roughly 2.5%). Page 8 of probability.docx (5/18/2016)

Chapter 2. The Normal Distribution

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

More information

An Introduction to Basic Statistics and Probability

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

More information

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

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

More information

Chapter 3 Normal Distribution

Chapter 3 Normal Distribution Chapter 3 Normal Distribution Density curve A density curve is an idealized histogram, a mathematical model; the curve tells you what values the quantity can take and how likely they are. Example Height

More information

Comment on the Tree Diagrams Section

Comment on the Tree Diagrams Section Comment on the Tree Diagrams Section The reversal of conditional probabilities when using tree diagrams (calculating P (B A) from P (A B) and P (A B c )) is an example of Bayes formula, named after the

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

More information

Topic 2: Scalar random variables. Definition of random variables

Topic 2: Scalar random variables. Definition of random variables Topic 2: Scalar random variables Discrete and continuous random variables Probability distribution and densities (cdf, pmf, pdf) Important random variables Expectation, mean, variance, moments Markov and

More information

Stats on the TI 83 and TI 84 Calculator

Stats on the TI 83 and TI 84 Calculator Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and

More information

Chapter 10: Introducing Probability

Chapter 10: Introducing Probability Chapter 10: Introducing Probability Randomness and Probability So far, in the first half of the course, we have learned how to examine and obtain data. Now we turn to another very important aspect of Statistics

More information

Contents. TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics. Yuming Jiang. Basic Concepts Random Variables

Contents. TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics. Yuming Jiang. Basic Concepts Random Variables TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics Yuming Jiang 1 Some figures taken from the web. Contents Basic Concepts Random Variables Discrete Random Variables Continuous Random

More information

Joint Probability Distributions and Random Samples (Devore Chapter Five)

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

More information

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

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

More information

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

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.

More information

6.4 Normal Distribution

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

More information

Histograms and density curves

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

More information

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

More information

Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

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

More information

Normal distribution. ) 2 /2σ. 2π σ

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

More information

Unit 16 Normal Distributions

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

More information

Probability Distributions

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.

More information

Chapter 5. Random variables

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

More information

Stat 110. Unit 3: Random Variables Chapter 3 in the Text

Stat 110. Unit 3: Random Variables Chapter 3 in the Text Stat 110 Unit 3: Random Variables Chapter 3 in the Text 1 Unit 3 Outline Random Variables (RVs) and Distributions Discrete RVs and Probability Mass Functions (PMFs) Bernoulli and Binomial Distributions

More information

Bivariate Distributions

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

More information

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014 STAT511 Spring 2014 Lecture Notes 1 Chapter 3: Discrete Random Variable and Probability Distribution January 28, 2014 3 Discrete Random Variables Chapter Overview Random Variable (r.v. Definition Discrete

More information

8. THE NORMAL DISTRIBUTION

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,

More information

Continuous Distributions

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

More information

A review of the portions of probability useful for understanding experimental design and analysis.

A review of the portions of probability useful for understanding experimental design and analysis. Chapter 3 Review of Probability A review of the portions of probability useful for understanding experimental design and analysis. The material in this section is intended as a review of the topic of probability

More information

Lecture I. Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions.

Lecture I. Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions. Lecture 1 1 Lecture I Definition 1. Statistics is the science of collecting, organizing, summarizing and analyzing the information in order to draw conclusions. It is a process consisting of 3 parts. Lecture

More information

The Normal Curve. The Normal Curve and The Sampling Distribution

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

More information

Random variables, probability distributions, binomial random variable

Random variables, probability distributions, binomial random variable Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

More information

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

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

More information

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

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

More information

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k. REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

More information

Statistics GCSE Higher Revision Sheet

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

More information

Section 5 Part 2. Probability Distributions for Discrete Random Variables

Section 5 Part 2. Probability Distributions for Discrete Random Variables Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability

More information

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

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,

More information

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 )

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

More information

Example. Two fair dice are tossed and the two outcomes recorded. As is familiar, we have

Example. Two fair dice are tossed and the two outcomes recorded. As is familiar, we have Lectures 9-10 jacques@ucsd.edu 5.1 Random Variables Let (Ω, F, P ) be a probability space. The Borel sets in R are the sets in the smallest σ- field on R that contains all countable unions and complements

More information

Introduction to Descriptive Statistics

Introduction to Descriptive Statistics Mathematics Learning Centre Introduction to Descriptive Statistics Jackie Nicholas c 1999 University of Sydney Acknowledgements Parts of this booklet were previously published in a booklet of the same

More information

Monte Carlo Method: Probability

Monte Carlo Method: Probability John (ARC/ICAM) Virginia Tech... Math/CS 4414: The Monte Carlo Method: PROBABILITY http://people.sc.fsu.edu/ jburkardt/presentations/ monte carlo probability.pdf... ARC: Advanced Research Computing ICAM:

More information

Chapter 4. Probability and Probability Distributions

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

More information

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

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

More information

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

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

More information

Probability distributions

Probability distributions Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.14-2.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,

More information

Discrete probability and the laws of chance

Discrete probability and the laws of chance Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

More information

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

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

More information

6-2 The Standard Normal Distribution. Uniform Distribution. Density Curve. Area and Probability. Using Area to Find Probability

6-2 The Standard Normal Distribution. Uniform Distribution. Density Curve. Area and Probability. Using Area to Find Probability 6-2 The Standard Normal Distribution This section presents the standard normal distribution which has three properties: 1. Its graph is bell-shaped. 2. Its mean is equal to 0 (μ = 0). 3. Its standard deviation

More information

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. 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,

More information

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

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,

More information

Review of Random Variables

Review of Random Variables Chapter 1 Review of Random Variables Updated: January 16, 2015 This chapter reviews basic probability concepts that are necessary for the modeling and statistical analysis of financial data. 1.1 Random

More information

6 3 The Standard Normal Distribution

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

More information

CHAPTER 6 NORMAL DISTIBUTIONS

CHAPTER 6 NORMAL DISTIBUTIONS CHAPTER 6 NORMAL DISTIBUTIONS GRAPHS OF NORMAL DISTRIBUTIONS (SECTION 6.1 OF UNDERSTANDABLE STATISTICS) The normal distribution is a continuous probability distribution determined by the value of µ and

More information

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 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

More information

People have thought about, and defined, probability in different ways. important to note the consequences of the definition:

People have thought about, and defined, probability in different ways. important to note the consequences of the definition: PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models

More information

Binomial distribution From Wikipedia, the free encyclopedia See also: Negative binomial distribution

Binomial distribution From Wikipedia, the free encyclopedia See also: Negative binomial distribution Binomial distribution From Wikipedia, the free encyclopedia See also: Negative binomial distribution In probability theory and statistics, the binomial distribution is the discrete probability distribution

More information

Discrete and Continuous Random Variables. Summer 2003

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

More information

Empirical Rule Confidence Intervals Finding a good sample size. Outline. 1 Empirical Rule. 2 Confidence Intervals. 3 Finding a good sample size

Empirical Rule Confidence Intervals Finding a good sample size. Outline. 1 Empirical Rule. 2 Confidence Intervals. 3 Finding a good sample size Outline 1 Empirical Rule 2 Confidence Intervals 3 Finding a good sample size Outline 1 Empirical Rule 2 Confidence Intervals 3 Finding a good sample size -3-2 -1 0 1 2 3 Question How much of the probability

More information

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

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

More information

Lecture 2: Discrete Distributions, Normal Distributions. Chapter 1

Lecture 2: Discrete Distributions, Normal Distributions. Chapter 1 Lecture 2: Discrete Distributions, Normal Distributions Chapter 1 Reminders Course website: www. stat.purdue.edu/~xuanyaoh/stat350 Office Hour: Mon 3:30-4:30, Wed 4-5 Bring a calculator, and copy Tables

More information

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

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

More information

4. Introduction to Statistics

4. Introduction to Statistics Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

More information

8.2 Confidence Intervals for One Population Mean When σ is Known

8.2 Confidence Intervals for One Population Mean When σ is Known 8.2 Confidence Intervals for One Population Mean When σ is Known Tom Lewis Fall Term 2009 8.2 Confidence Intervals for One Population Mean When σ isfall Known Term 2009 1 / 6 Outline 1 An example 2 Finding

More information

Chapter 7 What to do when you have the data

Chapter 7 What to do when you have the data Chapter 7 What to do when you have the data We saw in the previous chapters how to collect data. We will spend the rest of this course looking at how to analyse the data that we have collected. Stem and

More information

Math/Stat 370: Engineering Statistics, Washington State University

Math/Stat 370: Engineering Statistics, Washington State University Math/Stat 370: Engineering Statistics, Washington State University Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2 Haijun Li Math/Stat 370: Engineering Statistics,

More information

Lecture.7 Poisson Distributions - properties, Normal Distributions- properties. Theoretical Distributions. Discrete distribution

Lecture.7 Poisson Distributions - properties, Normal Distributions- properties. Theoretical Distributions. Discrete distribution Lecture.7 Poisson Distributions - properties, Normal Distributions- properties Theoretical distributions are Theoretical Distributions 1. Binomial distribution 2. Poisson distribution Discrete distribution

More information

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

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,

More information

The Normal Distribution

The Normal Distribution The Normal Distribution Continuous Distributions A continuous random variable is a variable whose possible values form some interval of numbers. Typically, a continuous variable involves a measurement

More information

EE 322: Probabilistic Methods for Electrical Engineers. Zhengdao Wang Department of Electrical and Computer Engineering Iowa State University

EE 322: Probabilistic Methods for Electrical Engineers. Zhengdao Wang Department of Electrical and Computer Engineering Iowa State University EE 322: Probabilistic Methods for Electrical Engineers Zhengdao Wang Department of Electrical and Computer Engineering Iowa State University Discrete Random Variables 1 Introduction to Random Variables

More information

Probability. Distribution. Outline

Probability. Distribution. Outline 7 The Normal Probability Distribution Outline 7.1 Properties of the Normal Distribution 7.2 The Standard Normal Distribution 7.3 Applications of the Normal Distribution 7.4 Assessing Normality 7.5 The

More information

Hypothesis Testing. Chapter Introduction

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

More information

Review. Lecture 3: Probability Distributions. Poisson Distribution. May 8, 2012 GENOME 560, Spring Su In Lee, CSE & GS

Review. Lecture 3: Probability Distributions. Poisson Distribution. May 8, 2012 GENOME 560, Spring Su In Lee, CSE & GS Lecture 3: Probability Distributions May 8, 202 GENOME 560, Spring 202 Su In Lee, CSE & GS suinlee@uw.edu Review Random variables Discrete: Probability mass function (pmf) Continuous: Probability density

More information

Important Probability Distributions OPRE 6301

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.

More information

Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

More information

ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

More information

Random Variables. Consider a probability model (Ω, P ). Discrete Random Variables Chs. 2, 3, 4. Definition. A random variable is a function

Random Variables. Consider a probability model (Ω, P ). Discrete Random Variables Chs. 2, 3, 4. Definition. A random variable is a function Rom Variables Discrete Rom Variables Chs.,, 4 Rom Variables Probability Mass Functions Expectation: The Mean Variance Special Distributions Hypergeometric Binomial Poisson Joint Distributions Independence

More information

+ Section 6.2 and 6.3

+ Section 6.2 and 6.3 Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

More information

Binomial Distribution n = 20, p = 0.3

Binomial Distribution n = 20, p = 0.3 This document will describe how to use R to calculate probabilities associated with common distributions as well as to graph probability distributions. R has a number of built in functions for calculations

More information

TEACHER NOTES MATH NSPIRED

TEACHER NOTES MATH NSPIRED Math Objectives Students will understand that normal distributions can be used to approximate binomial distributions whenever both np and n(1 p) are sufficiently large. Students will understand that when

More information

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

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

More information

PROBABILITIES AND PROBABILITY DISTRIBUTIONS

PROBABILITIES AND PROBABILITY DISTRIBUTIONS Published in "Random Walks in Biology", 1983, Princeton University Press PROBABILITIES AND PROBABILITY DISTRIBUTIONS Howard C. Berg Table of Contents PROBABILITIES PROBABILITY DISTRIBUTIONS THE BINOMIAL

More information

Probability. Experiment is a process that results in an observation that cannot be determined

Probability. Experiment is a process that results in an observation that cannot be determined Probability Experiment is a process that results in an observation that cannot be determined with certainty in advance of the experiment. Each observation is called an outcome or a sample point which may

More information

Sample Questions for Mastery #5

Sample Questions for Mastery #5 Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could

More information

The Binomial Probability Distribution

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

More information

Statistics 100 Binomial and Normal Random Variables

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

More information

Probability and Statistics

Probability and Statistics CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

More information

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

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

More information

CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

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

More information

Week 4: Standard Error and Confidence Intervals

Week 4: Standard Error and Confidence Intervals Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

More information

7. Normal Distributions

7. Normal Distributions 7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped

More information

Chapter 1. Probability, Random Variables and Expectations. 1.1 Axiomatic Probability

Chapter 1. Probability, Random Variables and Expectations. 1.1 Axiomatic Probability Chapter 1 Probability, Random Variables and Expectations Note: The primary reference for these notes is Mittelhammer (1999. Other treatments of probability theory include Gallant (1997, Casella & Berger

More information

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution 8 October 2007 In this lecture we ll learn the following: 1. how continuous probability distributions differ

More information

MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables

MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

More information

III. Famous Discrete Distributions: The Binomial and Poisson Distributions

III. Famous Discrete Distributions: The Binomial and Poisson Distributions III. Famous Discrete Distributions: The Binomial and Poisson Distributions Up to this point, we have concerned ourselves with the general properties of categorical and continuous distributions, illustrated

More information

Confidence Intervals in Excel

Confidence Intervals in Excel Confidence Intervals in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

More information

Lesson 7 Z-Scores and Probability

Lesson 7 Z-Scores and Probability Lesson 7 Z-Scores and Probability Outline Introduction Areas Under the Normal Curve Using the Z-table Converting Z-score to area -area less than z/area greater than z/area between two z-values Converting

More information

Lesson 20. Probability and Cumulative Distribution Functions

Lesson 20. Probability and Cumulative Distribution Functions Lesson 20 Probability and Cumulative Distribution Functions Recall If p(x) is a density function for some characteristic of a population, then Recall If p(x) is a density function for some characteristic

More information

Chapter 9 Monté Carlo Simulation

Chapter 9 Monté Carlo Simulation MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?

More information

The Normal Distribution

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

More information

Chapter 5: Normal Probability Distributions - Solutions

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

More information

Biostatistics Lab Notes

Biostatistics Lab Notes Biostatistics Lab Notes Page 1 Lab 1: Measurement and Sampling Biostatistics Lab Notes Because we used a chance mechanism to select our sample, each sample will differ. My data set (GerstmanB.sav), looks

More information