Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Save this PDF as:

Size: px
Start display at page:

Download "Case Study. Normal and t Distributions. Density Plot. Normal Distributions"

Transcription

1 Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca be higher whe oe is ill with a fever, or durig or after physical exertio). However, if we measure the body temperature of a sigle healthy perso whe at rest, these measuremets vary little from day to day, ad we ca associate with each perso a idividual restig body temperture. There is, however, variatio amog idividuals of restig body temperture. A sample of = 130 idividuals had a average restig body temperature of degrees Fahreheit ad a stadard deviatio of 0.73 degrees Fahreheit. The ext slide shows a estimated desity plot from this sample. Normal 1 / 33 Normal Case Study Body Temperature 2 / 33 Desity Plot Normal Distributios Desity The estimated desity has these features: it is bell-shaped; it is early symmetric. May (but ot all) biological variables have similar shapes. Oe reaso is a geeralized the cetral limit theorem: radom variables that are formed by addig may radom effects will be approximately ormally distributed. Importat for iferece, eve whe uderlyig distributios are ot ormal, the samplig distributio of the sample mea is approximately ormal Restig Body Temperature (F) Normal Case Study Body Temperature 3 / 33 Normal Case Study Body Temperature 4 / 33

2 Example: Populatio Example: Samplig Distributio A populatio that is skewed. Populatio Samplig distributio of the sample mea whe = 130. Samplig Distributio, = Desity Desity x x Normal Case Study Body Temperature 5 / 33 Normal Case Study Body Temperature 6 / 33 Case Study: Questios The Big Picture Case Study How ca we use the sample data to estimate with cofidece the mea restig body temperture i a populatio? How would we test the ull hypothesis that the mea restig body temperture i the populatio is, i fact, equal to the well-kow 98.6 degrees Fahreheit? How robust are the methods of iferece to oormality i the uderlyig populatio? How large of a sample is eeded to esure that a cofidece iterval is o larger tha some specified amout? May iferece problems with a sigle quatitative, cotiuous variable may be modeled as a large populatio (bucket) of idividual umbers with a mea µ ad stadard deviatio σ. A radom sample of size has a sample mea x ad sample stadard deviatio s. Iferece about µ based o sample data assumes that the samplig distributio of x is approximately ormal with E( x) = µ ad SD( x) = σ/. To prepare to uderstad iferece methods for sigle samples of quatitative data, we eed to uderstad: the ormal ad related distributios; the samplig distributio of x. Normal Case Study Body Temperature 7 / 33 Normal Case Study Body Temperature 8 / 33

3 Cotiuous Distributios A cotiuous radom variable has possible values over a cotiuum. The total probability of oe is ot i discrete chuks at specific locatios, but rather is groud up like a very fie dust ad sprikled o the umber lie. We caot represet the distributio with a table of possible values ad the probability of each. Istead, we represet the distributio with a probability desity fuctio which measures the thickess of the probability dust. Probability is measured over itervals as the area uder the curve. A legal probability desity f : is ever egative (f (x) 0 for < x < ). has a total area uder the curve of oe ( f (x)dx = 1). The Stadard Normal Desity The stadard ormal desity is a symmetric, bell-shaped probability desity with equatio: φ(z) = 1 e z2 2, ( < z < ) 2π Desity Possible Values Normal Cotiuous Radom Variables Desity 9 / 33 Normal Stadard Normal Distributio Desity 10 / 33 Momets Bechmarks The mea of the stadard ormal distributio is µ = 0. This poit is the ceter of the desity ad the poit where the desity is highest. The stadard deviatio of the stadard ormal distributio is σ = 1. Notice that the poits 1 ad 1, which are respectively oe stadard deviatio below ad above the mea, are at poits of iflectio of the ormal curve. (This is useful for roughly estimatig the stadard deviatio from a plotted desity or histogram.) The area betwee 1 ad 1 uder a stadard ormal curve is approximately 68%. The area betwee 2 ad 2 uder a stadard ormal curve is approximately 95%. More precisely, the area betwee 1.96 ad = , which is why we have used 1.96 for 95% cofidece itervals for proportios. Normal Stadard Normal Distributio Desity 11 / 33 Normal Stadard Normal Distributio Probability Calculatios 12 / 33

4 Stadard Normal Desity Geeral Areas Stadard Normal Desity Desity Area withi 1 = 0.68 Area withi 2 = 0.95 Area withi 3 = There is o formula to calculate geeral areas uder the stadard ormal curve. (The itegral of the desity has o closed form solutio.) We prefer to use R to fid probabilities. You also eed to lear to use ormal tables for exams Possible Values Normal Stadard Normal Distributio Probability Calculatios 13 / 33 Normal Stadard Normal Distributio Probability Calculatios 14 / 33 R The fuctio porm() calculates probabilities uder the stadard ormal curve by fidig the area to the left. For example, the area to the left of 1.57 is > porm(-1.57) [1] ad the area to the right of 2.12 is > 1 - porm(2.12) [1] Normal Stadard Normal Distributio Probability Calculatios 15 / 33 Tables The table o pages displays right tail probabilities for z = 0 to z = A poit o the axis rouded to two decimal places a.bc correspods to a row for a.b ad a colum for c. The umber i the table for this row ad colum is the area to the right. Symmetry of the ormal curve ad the fact that the total area is oe are eeded. The area to the left of 1.57 is the area to the right of 1.57 which is i the table. The area to the right of 2.12 is Whe usig the table, it is best to draw a rough sketch of the curve ad shade i the desired area. This practice allows oe to approximate the correct probability ad catch simple errors. Fid the area betwee z = 1.64 ad z = 2.55 o the board. Normal Stadard Normal Distributio Probability Calculatios 16 / 33

5 R Tables The fuctio qorm() is the iverse of porm() ad fids a quatile, or locatio where a give area is to the right. For example, the 0.9 quatile of the stadard ormal curve is > qorm(0.9) [1] ad the umber z so that the area betwee z ad z is 0.99 is > qorm(0.995) [1] sice the area to the left of z ad to the right of z must each be (1 0.99)/2 = ad = Draw a sketch! Fidig quatiles from the ormal table almost always requires some roud off error. To fid the umber z so that the area betwee z ad z is 0.99 requires fidig the probability i the middle of the table. We see z = 2.57 has a right tail area of ad z = 2.58 has a right ail area of , so the value of z we seek is betwee 2.57 ad For exam purposes, it is okay to pick the closest, here Use the table to fid the 0.03 quatile as accurately as possible. Draw a sketch! Normal Stadard Normal Distributio Quatile Calculatios 17 / 33 Normal Stadard Normal Distributio Quatile Calculatios 18 / 33 Geeral Normal Desity Geeral Normal Desity Normal Desity The geeral ormal desity with mea µ ad stadard deviatio σ is a symmetric, bell-shaped probability desity with equatio: ( ) 2 f (x) = 1 e 1 x µ 2 σ, ( < x < ) 2πσ Sketches of geeral ormal curves have the same shape as stadard ormal curves, but have rescaled axes. Desity Area withi 1 SD = 0.68 Area withi 2 SD = 0.95 Area withi 3 SD = µ 3σ µ 2σ µ σ µ µ + σ µ + 2σ µ + 3σ Possible Values Normal Geeral Norma Distributio Desity 19 / 33 Normal Geeral Norma Distributio Desity 20 / 33

6 All Normal Curves Have the Same Shape Normal Tail Probability All ormal curves have the same shape, ad are simply rescaled versios of the stadard ormal desity. Cosequetly, every area uder a geeral ormal curve correspods to a area uder the stadard ormal curve. The key stadardizatio formula is Solvig for x yields z = x µ σ x = µ + zσ which says algebraically that x is z stadard deviatios above the mea. Example If X N(100, 2), fid P(X > 97.5). Solutio: P(X > 97.5) = ( X 100 P > 2 = P(Z > 1.25) = 1 P(Z > 1.25) = ) Normal Geeral Norma Distributio Probability Calculatios 21 / 33 Normal Geeral Norma Distributio Probability Calculatios 22 / 33 Normal Quatiles Example If X N(100, 2), fid the cutoff values for the middle 70% of the distributio. Solutio: The cutoff poits will be the 0.15 ad 0.85 quatiles. From the table, 1.03 < z < 1.04 ad z = 1.04 is closest. Thus, the cutoff poits are the mea plus or mius 1.04 stadard deviatios (2) = 97.92, (2) = I R, a sigle call to qorm() fids these cutoffs. > qorm(c(0.15, 0.85), 100, 2) [1] Case Study Example I a populatio, suppose that: the mea restig body temperature is degrees Fahreheit; the stadard deviatio is 0.73 degrees Fahreheit; restig body temperatures are ormally distributed. Let X be the restig body temperature of a radomly chose idividual. Fid: 1 P(X < 98), the proportio of idividuals with temperature less tha P(98 < X < 100), the proportio of idividuals with temperature betwee 98 ad The 0.90 quatile of the distributio. 4 The cutoff values for the middle 50% of the distributio. Normal Geeral Norma Distributio Quatile Calculatios 23 / 33 Normal Geeral Norma Distributio Applicatio 24 / 33

7 Aswers (with R, table will be close) The χ 2 Distributio ad The χ 2 distributio is used to fid p-values for the test of idepedece ad the G-test we saw earlier for cotigecy tables. Now that the ormal distributio has bee itroduced, we ca better motivate the χ 2 distributio. Defiitio If Z 1,..., Z k are idepedet stadard ormal radom variables, the X 2 = Z Z 2 k has a χ 2 distributio with k degrees of freedom. Normal Geeral Norma Distributio Applicatio 25 / 33 Normal Other Distributios Chi-square Distributios 26 / 33 The χ 2 Distributio The fuctios pchisq() ad qchisq() fid probabilities ad quatiles, respectively, from the χ 2 distributios. The table o pages has the same iformatio for limited umbers of quatiles for each χ 2 distributio with 100 or fewer degrees of freedom. Ulike the ormal distributios where all ormal curves are just rescaligs of the stadard ormal curve, each χ 2 distributio is differet. t Distributio Defiitio If Z is a stadard ormal radom variable ad if X 2 is a χ 2 radom variable with k degrees of freedom, the T = Z X 2 /k has a t distributio with k degrees of freedom. t desities are symmetric, bell-shaped, ad cetered at 0 just like the stadard ormal desity, but are more spread out (higher variace). As the degrees of freedom icreases, the t distributios coverge to the stadard ormal. t distributios will be useful for statistical iferece for oe or more populatios of quatitative variables. Normal Other Distributios Chi-square Distributios 27 / 33 Normal Other Distributios t Distributios 28 / 33

8 The Cetral Limit Theorem Mea of the Samplig Distributio of X The Cetral Limit Theorem If X 1,..., X are a idepedet sample from a commo distributio F with mea E(X i ) = µ ad variace Var(X i ) = σ 2, (which eed ot be ormal), the X = i=1 X i is approximately ormal with E( X ) = µ ad Var( X ) = σ2 size is sufficietly large. if the sample The cetral limit theorem (ad its cousis) justifies almost all iferece methods the rest of the semester. The mea of the samplig distributio of X is foud usig the liearity properties of expectatio. ( i=1 E( X ) = E X ) i ) = E(X X ) ) (E(X1 = ) + + E(X ) ) ) = µ = µ Normal The Cetral Limit Theorem 29 / 33 Normal The Cetral Limit Theorem 30 / 33 Variace of the Samplig Distributio of X The variace of the samplig distributio of X is foud usig the properties of variaces of sums. Also, SE( X ) = σ. ( i=1 Var( X ) = Var X ) i ) 2Var(X1 = + + X ) ) 2(Var(X1 = ) + + Var(X ) ) ) 2σ 2 = = σ2 Case Study Example I a populatio, suppose that: the mea restig body temperature is degrees Fahreheit; the stadard deviatio is 0.73 degrees Fahreheit; restig body temperatures are ormally distributed. Let X 1,..., X 40 be the restig body temperatures of 40 radomly chose idividuals from the populatio. Fid: 1 P( X < 98), the probability that the sample mea is less tha P(98 < X < 100), the probability that the sample mea is betwee 98 ad the 0.90 quatile of the samplig distributio of X. 4 The cutoff values for the middle 50% of the samplig distributio of X. Normal The Cetral Limit Theorem 31 / 33 Normal The Cetral Limit Theorem 32 / 33

9 Aswers (with R, table will be close) ad Normal The Cetral Limit Theorem 33 / 33

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

Chapter 7: Confidence Interval and Sample Size

Chapter 7: Confidence Interval and Sample Size Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

More information

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

One-sample test of proportions

One-sample test of proportions Oe-sample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

More information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

More information

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number. GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

Normal Distribution.

Normal Distribution. Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

More information

Math C067 Sampling Distributions

Math C067 Sampling Distributions Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

More information

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011 15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes high-defiitio

More information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL STATISTICS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

STATISTICAL METHODS FOR BUSINESS

STATISTICAL METHODS FOR BUSINESS STATISTICAL METHODS FOR BUSINESS UNIT 7: INFERENTIAL TOOLS. DISTRIBUTIONS ASSOCIATED WITH SAMPLING 7.1.- Distributios associated with the samplig process. 7.2.- Iferetial processes ad relevat distributios.

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book) MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

More information

A Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as:

A Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as: A Test of Normality Textbook Referece: Chapter. (eighth editio, pages 59 ; seveth editio, pages 6 6). The calculatio of p values for hypothesis testig typically is based o the assumptio that the populatio

More information

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error

STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error STA 2023 Practice Questios Exam 2 Chapter 7- sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (-1) oe p ( 1 p) CI: prop.

More information

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

Practice Problems for Test 3

Practice Problems for Test 3 Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all

More information

Now here is the important step

Now here is the important step LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

More information

Quadrat Sampling in Population Ecology

Quadrat Sampling in Population Ecology Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu Multi-server Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio -coectio

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information

Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS

Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS Uit 8: Iferece for Proortios Chaters 8 & 9 i IPS Lecture Outlie Iferece for a Proortio (oe samle) Iferece for Two Proortios (two samles) Cotigecy Tables ad the χ test Iferece for Proortios IPS, Chater

More information

Mann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)

Mann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test) No-Parametric ivariate Statistics: Wilcoxo-Ma-Whitey 2 Sample Test 1 Ma-Whitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo-) Ma-Whitey (WMW) test is the o-parametric equivalet of a pooled

More information

OMG! Excessive Texting Tied to Risky Teen Behaviors

OMG! Excessive Texting Tied to Risky Teen Behaviors BUSIESS WEEK: EXECUTIVE EALT ovember 09, 2010 OMG! Excessive Textig Tied to Risky Tee Behaviors Kids who sed more tha 120 a day more likely to try drugs, alcohol ad sex, researchers fid TUESDAY, ov. 9

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

Parametric (theoretical) probability distributions. (Wilks, Ch. 4) Discrete distributions: (e.g., yes/no; above normal, normal, below normal)

Parametric (theoretical) probability distributions. (Wilks, Ch. 4) Discrete distributions: (e.g., yes/no; above normal, normal, below normal) 6 Parametric (theoretical) probability distributios. (Wilks, Ch. 4) Note: parametric: assume a theoretical distributio (e.g., Gauss) No-parametric: o assumptio made about the distributio Advatages of assumig

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC. SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol

More information

Research Method (I) --Knowledge on Sampling (Simple Random Sampling)

Research Method (I) --Knowledge on Sampling (Simple Random Sampling) Research Method (I) --Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

THE TWO-VARIABLE LINEAR REGRESSION MODEL

THE TWO-VARIABLE LINEAR REGRESSION MODEL THE TWO-VARIABLE LINEAR REGRESSION MODEL Herma J. Bieres Pesylvaia State Uiversity April 30, 202. Itroductio Suppose you are a ecoomics or busiess maor i a college close to the beach i the souther part

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

Central Limit Theorem and Its Applications to Baseball

Central Limit Theorem and Its Applications to Baseball Cetral Limit Theorem ad Its Applicatios to Baseball by Nicole Aderso A project submitted to the Departmet of Mathematical Scieces i coformity with the requiremets for Math 4301 (Hoours Semiar) Lakehead

More information

Exploratory Data Analysis

Exploratory Data Analysis 1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Research Article Sign Data Derivative Recovery

Research Article Sign Data Derivative Recovery Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

More information

A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING

A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING MATTHEW ACTIPES Abstract. This paper begis by defiig a probability space ad establishig probability fuctios i this space over discrete radom variables.

More information

BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)

BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1) BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet

More information

TI-83, TI-83 Plus or TI-84 for Non-Business Statistics

TI-83, TI-83 Plus or TI-84 for Non-Business Statistics TI-83, TI-83 Plu or TI-84 for No-Buie Statitic Chapter 3 Eterig Data Pre [STAT] the firt optio i already highlighted (:Edit) o you ca either pre [ENTER] or. Make ure the curor i i the lit, ot o the lit

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

Topic 5: Confidence Intervals (Chapter 9)

Topic 5: Confidence Intervals (Chapter 9) Topic 5: Cofidece Iterval (Chapter 9) 1. Itroductio The two geeral area of tatitical iferece are: 1) etimatio of parameter(), ch. 9 ) hypothei tetig of parameter(), ch. 10 Let X be ome radom variable with

More information

COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS

COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

Parameter estimation for nonlinear models: Numerical approaches to solving the inverse problem. Lecture 11 04/01/2008. Sven Zenker

Parameter estimation for nonlinear models: Numerical approaches to solving the inverse problem. Lecture 11 04/01/2008. Sven Zenker Parameter estimatio for oliear models: Numerical approaches to solvig the iverse problem Lecture 11 04/01/2008 Sve Zeker Review: Trasformatio of radom variables Cosider probability distributio of a radom

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9%

0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% Sectio 10 Aswer Key: 0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% 1) A simple radom sample of New Yorkers fids that 87 are

More information

AP Calculus BC 2003 Scoring Guidelines Form B

AP Calculus BC 2003 Scoring Guidelines Form B AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet

More information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

TI-89, TI-92 Plus or Voyage 200 for Non-Business Statistics

TI-89, TI-92 Plus or Voyage 200 for Non-Business Statistics Chapter 3 TI-89, TI-9 Plu or Voyage 00 for No-Buie Statitic Eterig Data Pre [APPS], elect FlahApp the pre [ENTER]. Highlight Stat/Lit Editor the pre [ENTER]. Pre [ENTER] agai to elect the mai folder. (Note:

More information

Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond

Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information