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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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 of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution is a good approximation, and widely used. Contents 1 Specification 1.1 Probability mass function 1.2 Cumulative distribution function 2 Mean and variance 3 Mode and median 4 Covariance between two binomials 5 Relationship to other Probability mass function Cumulative distribution function Notation B(n, p) Parameters n N 0 number of trials p [0,1] success probability in each trial Support k { 0,, n } number of successes PMF CDF Mean np Median np or np Mode (n + 1)p or (n + 1)p 1 Variance np(1 p) Skewness 1/9

2 distributions 5.1 Sums of binomials 5.2 Conditional binomials 5.3 Bernoulli distribution 5.4 Poisson binomial distribution 5.5 Normal approximation 5.6 Poisson approximation 5.7 Limiting distributions 6 Generating binomial random variates 7 See also 8 References Ex. kurtosis Entropy MGF CF PGF Specification Probability mass function Binomial distribution for (blue), (green) and (red) In general, if the random variable K follows the binomial distribution with parameters n and p, we write K ~ B(n, p). The probability of getting exactly k successes in n trials is given by the probability mass function: Binomial distribution for with and as in Pascal's triangle The probability that a ball in a Galton box with 8 layers ( 2/9

3 . ) ends up in the central bin ( ) is for k = 0, 1, 2,..., n, where is the binomial coefficient (hence the name of the distribution) "n choose k", also denoted C(n, k), n C k, or n C k. The formula can be understood as follows: we want k successes (p k ) and n k failures (1 p) n k. However, the k successes can occur anywhere among the n trials, and there are C(n, k) different ways of distributing k successes in a sequence of n trials. In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as Looking at the expression ƒ(k, n, p) as a function of k, there is a k value that maximizes it. This k value can be found by calculating and comparing it to 1. There is always an integer M that satisfies ƒ(k, n, p) is monotone increasing for k < M and monotone decreasing for k > M, with the exception of the case where (n + 1)p is an integer. In this case, there are two values for which ƒ is maximal: (n + 1)p and (n + 1)p 1. M is the most probable (most likely) outcome of the Bernoulli trials and is called the mode. Note that the probability of it occurring can be fairly small. Cumulative distribution function The cumulative distribution function can be expressed as: where is the "floor" under x, i.e. the greatest integer less than or equal to x. 3/9

4 It can also be represented in terms of the regularized incomplete beta function, as follows: For k np, upper bounds for the lower tail of the distribution function can be derived. In particular, Hoeffding's inequality yields the bound and Chernoff's inequality can be used to derive the bound Moreover, these bounds are reasonably tight when p = 1/2, since the following expression holds for all k 3n/8 [1] Mean and variance If X ~ B(n, p) (that is, X is a binomially distributed random variable), then the expected value of X is and the variance is Mode and median Usually the mode of a binomial B(n, p) distribution is equal to, where is the floor function. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows: 4/9

5 In general, there is no single formula to find the median for a binomial distribution, and it may even be nonunique. However several special results have been established: If np is an integer, then the mean, median, and mode coincide and equal np. [2][3] Any median m must lie within the interval np m np. [4] A median m cannot lie too far away from the mean: m np min{ ln 2, max{p, 1 p} }. [5] The median is unique and equal to m = round(np) in cases when either p 1 ln 2 or p ln 2 or m np min{p, 1 p} (except for the case when p = ½ and n is odd). [4][5] When p = 1/2 and n is odd, any number m in the interval ½(n 1) m ½(n + 1) is a median of the binomial distribution. If p = 1/2 and n is even, then m = n/2 is the unique median. Covariance between two binomials If two binomially distributed random variables X and Y are observed together, estimating their covariance can be useful. Using the definition of covariance, in the case n = 1 (thus being Bernoulli trials) we have The first term is non zero only when both X and Y are one, and μ X and μ Y are equal to the two probabilities. Defining p B as the probability of both happening at the same time, this gives and for n such trials again due to independence If X and Y are the same variable, this reduces to the variance formula given above. Relationship to other distributions Sums of binomials If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable; its distribution is Conditional binomials 5/9

6 If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distribution Bernoulli distribution The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B(1, p) has the same meaning as X ~ Bern(p). Conversely, any binomial distribution, B(n, p), is the sum of n independent Bernoulli trials, Bern(p), each with the same probability p. Poisson binomial distribution The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent non identical Bernoulli trials Bern(p i ). If X has the Poisson binomial distribution with p 1 = = p n =p then X ~ B(n, p). Normal approximation If n is large enough, then the skew of the distribution is not too great. In this case a resonable approximation to B(n, p) is given by the normal distribution and this basic approximation can be improved in a simple way by using a suitable continuity correction. The basic approximation generally improves as n increases (at least 20) and is better when p is not near to 0 or 1. [6] Various rules of thumb may be used to decide whether n is large enough, and p is far enough from the extremes of zero or one: One rule is that both x=np and n(1 p) must be greater than 5. However, the specific number varies from source Binomial PDF and normal approximation to source, and depends on how good an approximation for n = 6 and p = 0.5 one wants; some sources give 10 which gives virtually the same results as the following rule for large n until n is very large (ex: x=11, n=7752). A second rule [6] is that for n > 5 the normal approximation is adequate if Another commonly used rule holds that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values, that is if 6/9

7 The following is an example of applying a continuity correction. Suppose one wishes to calculate Pr(X 8) for a binomial random variable X. If Y has a distribution given by the normal approximation, then Pr(X 8) is approximated by Pr(Y 8.5). The addition of 0.5 is the continuity correction; the uncorrected normal approximation gives considerably less accurate results. This approximation, known as de Moivre Laplace theorem, is a huge time saver when undertaking calculations by hand (exact calculations with large n are very onerous); historically, it was the first use of the normal distribution, introduced in Abraham de Moivre's book The Doctrine of Chances in Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a sum of n independent, identically distributed Bernoulli variables with parameter p. This fact is the basis of a hypothesis test, a "proportion z test," for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic. [7] For example, suppose one randomly samples n people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If groups of n people were sampled repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation σ = (p(1 p)/n) 1/2. Large sample sizes n are good because the standard deviation, as a proportion of the expected value, gets smaller, which allows a more precise estimate of the unknown parameter p. Poisson approximation The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed. Therefore the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to two rules of thumb, this approximation is good if n 20 and p 0.05, or if n 100 and np 10. [8] Limiting distributions Poisson limit theorem: As n approaches and p approaches 0 while np remains fixed at λ > 0 or at least np approaches λ > 0, then the Binomial(n, p) distribution approaches the Poisson distribution with expected value λ. de Moivre Laplace theorem: As n approaches while p remains fixed, the distribution of approaches the normal distribution with expected value 0 and variance 1. This result is sometimes loosely stated by saying that the distribution of X is asymtotically normal with expected 7/9

8 value np and variance np(1 p). This result is a specific case of the central limit theorem. Generating binomial random variates Methods for random number generation where the marginal distribution is a binomial distribution are wellestablished. [9][10] See also Binomial proportion confidence interval Logistic regression Multinomial distribution Negative binomial distribution References 1. ^ Matoušek, J, Vondrak, J: The Probabilistic Method (lecture notes) [1] ( 2. ^ Neumann, P. (1966). "Über den Median der Binomial and Poissonverteilung" (in German). Wissenschaftliche Zeitschrift der Technischen Universität Dresden 19: ^ Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", The Mathematical Gazette 94, ^ a b Kaas, R.; Buhrman, J.M. (1980). "Mean, Median and Mode in Binomial Distributions". Statistica Neerlandica 34 (1): doi: /j tb00681.x ( tb00681.x). 5. ^ a b Hamza, K. (1995). "The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions". Statistics & Probability Letters 23: doi: / (94)00090 U ( 7152%2894% U). 6. ^ a b Box, Hunter and Hunter (1978). Statistics for experimenters. Wiley. p ^ NIST/SEMATECH, " Does the proportion of defectives meet requirements?" ( e Handbook of Statistical Methods. 8. ^ NIST/SEMATECH, " Counts Control Charts" ( e Handbook of Statistical Methods. 9. ^ Devroye, Luc (1986) Non Uniform Random Variate Generation, New York: Springer Verlag. (See especially Chapter X, Discrete Univariate Distributions ( ) 10. ^ Kachitvichyanukul, V.; Schmeiser, B. W. (1988). "Binomial random variate generation". Communications of the ACM 31 (2): doi: / ( Retrieved from " Categories: Discrete distributions Factorial and binomial topics Distributions with conjugate priors Exponential family distributions This page was last modified on 6 May 2012 at 18:50. Text is available under the Creative Commons Attribution ShareAlike License; additional terms may 8/9

9 apply. See Terms of use for details. Wikipedia is a registered trademark of the Wikimedia Foundation, Inc., a non profit organization. 9/9

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

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

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

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 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

More information

Random Variables and Their Expected Values

Random Variables and Their Expected Values Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution

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

COURSE OUTLINE. Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4. Co- or Pre-requisite

COURSE OUTLINE. Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4. Co- or Pre-requisite COURSE OUTLINE Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4 Hours: Lecture/Lab/Other 4 Lecture Co- or Pre-requisite MAT151 or MAT149 with a minimum

More information

Lecture 5 : The Poisson Distribution. Jonathan Marchini

Lecture 5 : The Poisson Distribution. Jonathan Marchini Lecture 5 : The Poisson Distribution Jonathan Marchini Random events in time and space Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

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

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

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

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

Chapter 4 Expected Values

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

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

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

Review Exam Suppose that number of cars that passes through a certain rural intersection is a Poisson process with an average rate of 3 per day.

Review Exam Suppose that number of cars that passes through a certain rural intersection is a Poisson process with an average rate of 3 per day. Review Exam 2 This is a sample of problems that would be good practice for the exam. This is by no means a guarantee that the problems on the exam will look identical to those on the exam but it should

More information

2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56

2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56 2. Describing Data We consider 1. Graphical methods 2. Numerical methods 1 / 56 General Use of Graphical and Numerical Methods Graphical methods can be used to visually and qualitatively present data and

More information

3.6: General Hypothesis Tests

3.6: General Hypothesis Tests 3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.

More information

Examination 110 Probability and Statistics Examination

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

More information

Quantitative Methods for Finance

Quantitative Methods for Finance Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain

More information

MATH 201. Final ANSWERS August 12, 2016

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.

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

Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

More information

1. Consider an untested batch of memory chips that have a known failure rate of 8% (yield = 92%).

1. Consider an untested batch of memory chips that have a known failure rate of 8% (yield = 92%). eview of Introduction to Probability and Statistics Chris Mack, http://www.lithoguru.com/scientist/statistics/review.html omework #2 Solutions 1. Consider an untested batch of memory chips that have a

More information

Common probability distributionsi Math 217/218 Probability and Statistics Prof. D. Joyce, 2016

Common probability distributionsi Math 217/218 Probability and Statistics Prof. D. Joyce, 2016 Introduction. ommon probability distributionsi Math 7/8 Probability and Statistics Prof. D. Joyce, 06 I summarize here some of the more common distributions used in probability and statistics. Some are

More information

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

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

More information

IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1

IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1 IEOR 4106: Introduction to Operations Research: Stochastic Models SOLUTIONS to Homework Assignment 1 Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability Models, by Sheldon

More information

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

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

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

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

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

3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.

3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved. 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

More information

7 Hypothesis testing - one sample tests

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

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

3. Continuous Random Variables

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

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

MATH4427 Notebook 3 Spring MATH4427 Notebook Hypotheses: Definitions and Examples... 3

MATH4427 Notebook 3 Spring MATH4427 Notebook Hypotheses: Definitions and Examples... 3 MATH4427 Notebook 3 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MATH4427 Notebook 3 3 3.1 Hypotheses: Definitions and Examples............................

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

Continuous Random Variables

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

More information

Math 431 An Introduction to Probability. Final Exam Solutions

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 <

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

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

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

CMPSCI611: Primality Testing Lecture 14

CMPSCI611: Primality Testing Lecture 14 CMPSCI611: Primality Testing Lecture 14 One of the most important domains for randomized algorithms is number theory, particularly with its applications to cryptography. Today we ll look at one of the

More information

Math 461 Fall 2006 Test 2 Solutions

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

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

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

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

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for

More information

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

More information

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. 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............................................

More information

The Chi-Square Distributions

The Chi-Square Distributions MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness

More information

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

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.

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

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

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

37.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes

37.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes The Poisson distribution 37.3 Introduction In this block we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and

More information

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

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,

More information

Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013 Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

More information

5.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes. Learning Style

5.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes. Learning Style The Poisson distribution 5.3 Introduction In this block we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and where

More information

WHERE DOES THE 10% CONDITION COME FROM?

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

More information

EE 302 Division 1. Homework 5 Solutions.

EE 302 Division 1. Homework 5 Solutions. EE 32 Division. Homework 5 Solutions. Problem. A fair four-sided die (with faces labeled,, 2, 3) is thrown once to determine how many times a fair coin is to be flipped: if N is the number that results

More information

( ) is proportional to ( 10 + x)!2. Calculate the

( ) is proportional to ( 10 + x)!2. Calculate the PRACTICE EXAMINATION NUMBER 6. An insurance company eamines its pool of auto insurance customers and gathers the following information: i) All customers insure at least one car. ii) 64 of the customers

More information

Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

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

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

More information

Practice Problems #4

Practice Problems #4 Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice

More information

Seminar paper Statistics

Seminar paper Statistics Seminar paper Statistics The seminar paper must contain: - the title page - the characterization of the data (origin, reason why you have chosen this analysis,...) - the list of the data (in the table)

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

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

BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

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

More information

Estimating and Finding Confidence Intervals

Estimating and Finding Confidence Intervals . Activity 7 Estimating and Finding Confidence Intervals Topic 33 (40) Estimating A Normal Population Mean μ (σ Known) A random sample of size 10 from a population of heights that has a normal distribution

More information

Standard Deviation Calculator

Standard Deviation Calculator CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or

More information

Chapter 2, part 2. Petter Mostad

Chapter 2, part 2. Petter Mostad Chapter 2, part 2 Petter Mostad mostad@chalmers.se Parametrical families of probability distributions How can we solve the problem of learning about the population distribution from the sample? Usual procedure:

More information

Some notes on the Poisson distribution

Some notes on the Poisson distribution Some notes on the Poisson distribution Ernie Croot October 2, 2008 1 Introduction The Poisson distribution is one of the most important that we will encounter in this course it is right up there with 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

1.1 Introduction, and Review of Probability Theory... 3. 1.1.1 Random Variable, Range, Types of Random Variables... 3. 1.1.2 CDF, PDF, Quantiles...

1.1 Introduction, and Review of Probability Theory... 3. 1.1.1 Random Variable, Range, Types of Random Variables... 3. 1.1.2 CDF, PDF, Quantiles... MATH4427 Notebook 1 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 1 MATH4427 Notebook 1 3 1.1 Introduction, and Review of Probability

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

Data. ECON 251 Research Methods. 1. Data and Descriptive Statistics (Review) Cross-Sectional and Time-Series Data. Population vs.

Data. ECON 251 Research Methods. 1. Data and Descriptive Statistics (Review) Cross-Sectional and Time-Series Data. Population vs. ECO 51 Research Methods 1. Data and Descriptive Statistics (Review) Data A variable - a characteristic of population or sample that is of interest for us. Data - the actual values of variables Quantitative

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

Principle of Data Reduction

Principle of Data Reduction Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then

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

Solution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute

Solution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the

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

Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

More information

Chapter 2: Systems of Linear Equations and Matrices:

Chapter 2: Systems of Linear Equations and Matrices: At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

More information

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

More information

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015. Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x

More information

PO906: Quantitative Data Analysis and Interpretation

PO906: Quantitative Data Analysis and Interpretation PO906: Quantitative Data Analysis and Interpretation Vera E. Troeger Office: 1.129 E-mail: v.e.troeger@warwick.ac.uk Office Hours: appointment by e-mail Quantitative Data Analysis Descriptive statistics:

More information

GCSE HIGHER Statistics Key Facts

GCSE HIGHER Statistics Key Facts GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information

More information

SOME APPLICATIONS OF THE POISSON DISTRIBUTION IN MORTALITY STUDIES. by W. F. SCOTT, M.A., Ph.D., F.F.A. SUMMARY

SOME APPLICATIONS OF THE POISSON DISTRIBUTION IN MORTALITY STUDIES. by W. F. SCOTT, M.A., Ph.D., F.F.A. SUMMARY 255 SOME APPLICATIONS OF THE POISSON DISTRIBUTION IN MORTALITY STUDIES by W. F. SCOTT, M.A., Ph.D., F.F.A. SUMMARY Let θ x, denote the number of deaths (from all causes or a certain cause) at age x last

More information

Sample Size Determination

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)

More information

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

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

More information

Fisher Information and Cramér-Rao Bound. 1 Fisher Information. Math 541: Statistical Theory II. Instructor: Songfeng Zheng

Fisher Information and Cramér-Rao Bound. 1 Fisher Information. Math 541: Statistical Theory II. Instructor: Songfeng Zheng Math 54: Statistical Theory II Fisher Information Cramér-Rao Bound Instructor: Songfeng Zheng In the parameter estimation problems, we obtain information about the parameter from a sample of data coming

More information

Random Variables of The Discrete Type

Random Variables of The Discrete Type Random Variables of The Discrete Type Definition: Given a random experiment with an outcome space S, a function X that assigns to each element s in S one and only one real number x X(s) is called a 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

INTRODUCTION TO PROBABILITY AND STATISTICS

INTRODUCTION TO PROBABILITY AND STATISTICS INTRODUCTION TO PROBABILITY AND STATISTICS Conditional probability and independent events.. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the first toss and a,,, or 4 on the

More information

Chapter 7. Estimates and Sample Size

Chapter 7. Estimates and Sample Size Chapter 7. Estimates and Sample Size Chapter Problem: How do we interpret a poll about global warming? Pew Research Center Poll: From what you ve read and heard, is there a solid evidence that the average

More information

MAS108 Probability I

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

More information

Summarizing Data: Measures of Variation

Summarizing Data: Measures of Variation Summarizing Data: Measures of Variation One aspect of most sets of data is that the values are not all alike; indeed, the extent to which they are unalike, or vary among themselves, is of basic importance

More information

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

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

More information