3. Continuous Random Variables


 Andra Sparks
 1 years ago
 Views:
Transcription
1 Statistics ad probability: Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay predetermied value x, P( X = x) = 0, sice if we measured X accurately eough, we are ever goig to hit the value x exactly. However the probability of some regio of values ear x ca be ozero. Probability desity fuctio (pdf): P(1.5 < X < 0.7 ) Probability of X i the rage a to b. Sice has to have some value Ad sice, For a pdf, for all. Cumulative distributio fuctio (cdf) : This is the probability of. Mea ad variace Expected value (mea) of : Variace of : Note that the mea ad variace may ot be well defied for distributios with broad tails. The mode is the value of where is maximum (which may ot be uique). The media is give by the value of x where.
2 Statistics ad probability: 3 Uiform distributio The cotiuous radom variable has the Uiform distributio betwee ad, with if f(x) { 1 x, for short. Roughly speakig,, if X ca oly take values betwee ad, ad ay value of withi these values is as likely as ay other value. Mea ad variace: for, ad Proof: Let y be the distace from the midpoit,. The sice meas add, ad the width be Usurprisigly the mea is the midpoit. * + ( ) Occurrece of the Uiform distributio 1) Waitig times from radom arrival time util a regular evet (see below) ) Egieerig toleraces: e.g. if a diameter is quoted "0.1mm", it sometimes assumed (probably icorrectly) that the error has a U(0.1, 0.1) distributio. 3) Simulatio: programmig laguages ofte have a stadard routie for simulatig the U(0, 1) distributio. This ca be used to simulate other probability distributios.
3 Statistics ad probability: 33 Example: Disk wait times I a hard disk drive, the disk rotates at 700rpm. The wait time is defied as the time betwee the read/write head movig ito positio ad the begiig of the required iformatio appearig uder the head. (a) Fid the distributio of the wait time. (b) Fid the mea ad stadard deviatio of the wait time. (c) Bootig a computer requires that 000 pieces of iformatio are read from radom positios. What is the total expected cotributio of the wait time to the boot time, ad rms deviatio? Solutio Rotatio time = 8.33ms. Wait time ca be aythig betwee 0 ad 8.33ms ad each time i this rage is as likely as ay other time. Therefore, distributio of the wait time is U(0, 8.33ms) (i.. 1 = 0 ad = 8.33ms). ; For 000 reads the mea time is ms = 8.3s. The variace is ms = 0.01s, so =0.11s. Expoetial distributio The cotiuous radom variable has the Expoetial distributio, parameter if: { Relatio to Poisso distributio: If a Poisso process has costat rate, the mea after a time is. The probability of ooccurreces i this time is
4 Statistics ad probability: 34 If is the pdf for the first occurrece, the the probability of o occurreces is also give by So equatig the two ways of calculatig the probability we have Now we ca differetiate with respect to givig hece :.the time util the first occurrece (ad betwee subsequet occurreces) has the Expoetial distributio, parameter. Occurrece 1) Time util the failure of a part. ) Times betwee radomly happeig evets Mea ad variace [ ] [ ] [ ] Example: Reliability The time till failure of a electroic compoet has a Expoetial distributio ad it is kow that 10% of compoets have failed by 1000 hours. (a) What is the probability that a compoet is still workig after 5000 hours? (b) Fid the mea ad stadard deviatio of the time till failure. Solutio (a) Let Y = time till failure i hours;
5 Statistics ad probability: 35 [ ] [ ] (b) Mea = = 9491 hours. Stadard deviatio = = = 9491 hours. Normal distributio The cotiuous radom variable has the Normal distributio if the pdf is: The parameter is the mea ad ad the variace is. The distributio is also sometimes called a Gaussia distributio. The pdf is symmetric about. X lies betwee ad with probability 0.95 i.e. X lies withi stadard deviatios of the mea approximately 95% of the time. Normalizatio [oexamiable] caot be itegrated aalytically for geeral rages, but the full rage ca be itegated as follows. Defie I ( x) dx e dx e x The switchig to polar coordiates we have
6 Statistics ad probability: 36 I dx e e x r 0 dy e y dxdy e x y 0 0 rdrd e r 0 rdre r Hece I = ad the ormal distributio itegrates to oe. Mea ad variace The mea is because the distributio is symmetric about (or you ca check explicitly by itegratig by parts). The variace ca be also be checked by itegratig by parts: [ ] Occurrece of the Normal distributio 1) Quite a few variables, e.g. huma height, measuremet errors, detector oise. (Bellshaped histogram). ) Sample meas ad totals  see below, Cetral Limit Theorem. 3) Approximatio to several other distributios  see below. Chage of variable The probability for X i a rage aroud is for a distributio is give by The probability should be the same if it is writte i terms of aother variable. Hece
7 Statistics ad probability: 37 Stadard Normal distributio There is o simple formula for, so umerical itegratio (or tables) must be used. The followig result meas that it is oly ecessary to have tables for oe value of ad. If, the This follows whe chagig variables sice hece Z is the stadardised value of X; N(0, 1) is the stadard Normal distributio. The Normal tables give values of Q=P(Z z), also called (z), for z betwee 0 ad Outside of exams this is probably best evaluated usig a computer package (e.g. Maple, Mathematica, Matlab, Excel); for historical reasos you still have to use tables. Example: Usig stadard Normal tables (o course web page ad i exams) If Z ~ N(0, 1): (a) (b) = = = (by symmetry) (c) = (d) = (1.5)  (0.5) = =
8 Statistics ad probability: 38 (e)  Usig iterpolatio: ( ) (f) Usig tables "i reverse",. (g) Fidig a rage of values withi which lies with probability 0.95: The aswer is ot uique; but suppose we wat a iterval which is symmetric about zero i.e. betwee d ad d. Tail area = 0.05 P(Z d) = (d) = Usig the tables "i reverse", d = rage is to P=0.05 P=0.05 Example: Maufacturig variability The outside diameter, X mm, of a copper pipe is N(15.00, 0.0 ) ad the fittigs for joiig the pipe have iside diameter Y mm, where Y ~ N(15.07, 0.0 ). (i) Fid the probability that X exceeds mm. (ii) Withi what rage will X lie with probability 0.95? (iii) Fid the probability that a radomly chose pipe fits ito a radomly chose fittig (i.e. X < Y). Solutio (i) ( ) (ii) From previous example lies i (1.96, 1.96) with probability i.e. ( )
9 Statistics ad probability: 39 i.e. the required rage is 14.96mm to 15.04mm. (iii) For we wat ). To aswer this we eed to kow the distributio of Distributio of the sum of Normal variates Remember tha meas ad variaces of idepedet radom variables just add. So if are idepedet ad each have a ormal distributio, we ca easily calculate the mea ad variace of the sum. A special property of the Normal distributio is that the distributio of the sum of Normal variates is also a Normal distributio. So if are costats the: c X c X c X ~ N( c c, c c c ) Proof that the distributio of the sum is Normal is beyod scope. Useful special cases for two variables are If all the X's have the same distributio i.e. 1 = =... = =, say ad 1 = =... = =, say, the: (iii) All c i = 1: X 1 + X X ~ N(, ) (iv) All c i = 1/: X = X X X 1 ~ N(, /) The last result tells you that if you average idetical idepedet oisy measuremets, the error decreases by. (variace goes dow as ). Example: Maufacturig variability (iii) Fid the probability that a radomly chose pipe fits ito a radomly chose fittig (i.e. X < Y). Usig the above results Hece ( )
10 Statistics ad probability: 310 Example: detector oise A detector o a satellite ca measure T+g, the temperature T of a source with a radom oise g, where g ~ N(0, 1K ). How may detectors with idepedet oise would you eed to measure T to a rms error of 0.1K? Aswer: We ca estimate the temperature from detectors by calculatig the mea from each. The variace of the mea will be 1K / where is the umber of detectors. A rms error of 0.1K correspods to a variace of 0.01 K, hece we eed =100 detectors. Normal approximatios Cetral Limit Theorem: If X 1, X,... are idepedet radom variables with the same distributio, which has mea ad variace (both fiite), the the sum X i i1 teds to the distributio as. Hece: The sample mea X = for large. 1 X i i 1 is distributed approximately as N(, /) For the approximatio to be good, has to be bigger tha 30 or more for skewed distributios, but ca be quite small for simple symmetric distributios. The approximatio teds to have much better fractioal accuracy ear the peak tha i the tails: do t rely o the approximatio to estimate the probability of very rare evets. Example: Average of samples from a uiform distributio:
11 Statistics ad probability: 311 Normal approximatio to the Biomial If X ~ B(, p) ad is large ad p is ot too ear 0 or 1, the X is approximately N(p, p(1p)). p p The probability of gettig from the Biomial distributio ca be approximated as the probability uder a Normal distributio for gettig i the rage from to. For example ca be approximated as where is the Normal distributio: Example: I toss a coi 1000 times, what is the probability that I get more tha 550 heads? Aswer: The umber of heads has a biomial distributio with mea p=500 ad variace So the umber of heads ca be approximated as. Hece ( )
12 Statistics ad probability: 31 Quality cotrol example: The maufacturig of computer chips produces 10% defective chips. 00 chips are radomly selected from a large productio batch. What is the probability that fewer tha 15 are defective? Aswer: the mea is, variace. So if is the umber of defective chips, approximately, hece ( ) [ ] This compares to the exact Biomial aswer. The Biomial aswer is easy to calculate o a computer, but the Normal approximatio is much easier if you have to do it by had. The Normal approximatio is about right, but ot accurate. Normal approximatio to the Poisso If Poisso parameter ad is large (> 7, say), the has approximately a distributio.
13 Statistics ad probability: 313 Example: Stock Cotrol At a give hospital, patiets with a particular virus arrive at a average rate of oce every five days. Pills to treat the virus (oe per patiet) have to be ordered every 100 days. You are curretly out of pills; how may should you order if the probability of ruig out is to be less tha 0.005? Solutio Assume the patiets arrive idepedetly, so this is a Poisso process, with rate 0. / day. Therefore, Y, umber of pills eeded i 100 days, ~ Poisso, = 100 x 0. = 0. We wat, or ( ) uder the Normal approximatio, where a probability of correspods (from tables) to.575. Sice this correspods to., so we eed to order pills. Commet Let s say the virus is deadly, so we wat to make sure the probability is less tha 1 i a millio, A ormal approximatio would give 4.7 above the mea, so pills. But surely gettig just a bit above twice the average umber of cases is ot that ulikely?? Yes ideed, the assumptio of idepedece is extremely ulikely to be valid. Viruses ted to be ifectious, so occurreces are defiitely ot idepedet. There is likely to be a small but sigificat probability of a large umber of people beig ifected simultaeously a much larger umber of pills eeds to be stocked to be safe. Do t use approximatios that are too simple if their failure might be importat! Rare evets i particular are ofte a lot more likely tha predicted by (too) simple approximatios for the probability distributio.
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 informationOverview 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 informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationUniversity 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 Chisquare (χ ) distributio.
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo 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 information1. 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 : pvalue
More informationNormal 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 information8 The Poisson Distribution
8 The Poisso Distributio Let X biomial, p ). Recall that this meas that X has pmf ) p,p k) p k k p ) k for k 0,,...,. ) Agai, thik of X as the umber of successes i a series of idepedet experimets, each
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
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
More informationProperties 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 informationMeasures of Central Tendency
Measures of Cetral Tedecy A studet s grade will be determied by exam grades ( each exam couts twice ad there are three exams, HW average (couts oce, fial exam ( couts three times. Fid the average if the
More informationDerivation of the Poisson distribution
Gle Cowa RHUL Physics 1 December, 29 Derivatio of the Poisso distributio I this ote we derive the fuctioal form of the Poisso distributio ad ivestigate some of its properties. Cosider a time t i which
More informationChapter 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 informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationDescriptive statistics deals with the description or simple analysis of population or sample data.
Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small
More informationJoint Probability Distributions and Random Samples
STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margial
More informationChapter 5 Discrete Probability Distributions
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide Chapter 5 Discrete Probability Distributios Radom Variables Discrete Probability Distributios Epected Value ad Variace Poisso Distributio
More informationConfidence Intervals and Sample Size
8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGrawHill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 71 Cofidece Itervals for the
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationCHAPTER 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 informationUsing Excel to Construct Confidence Intervals
OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationThe Poisson Distribution
Lecture 5 The Poisso Distributio 5.1 Itroductio Example 5.1: Drowigs i Malta The book [Mou98] cites data from the St. Luke s Hospital Gazette, o the mothly umber of drowigs o Malta, over a period of early
More informationApproximating 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 informationSimulation and Monte Carlo integration
Chapter 3 Simulatio ad Mote Carlo itegratio I this chapter we itroduce the cocept of geeratig observatios from a specified distributio or sample, which is ofte called Mote Carlo geeratio. The ame of Mote
More informationBASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.
BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts
More informationLesson 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 informationPSYCHOLOGICAL 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 informationConfidence 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 informationConfidence Intervals for the Mean of Nonnormal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Cofidece Itervals for the Mea of Noormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio
More informationTHE 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 informationUC 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 informationIn 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 informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationSection 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 informationHypothesis 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 information5: 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 informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationChapter 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 informationSection 73 Estimating a Population. Requirements
Sectio 73 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio
More information0.7 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 KolmogorovSmirov 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 informationMaximum 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 information1 Hypothesis testing for a single mean
BST 140.65 Hypothesis Testig Review otes 1 Hypothesis testig for a sigle mea 1. The ull, or status quo, hypothesis is labeled H 0, the alterative H a or H 1 or H.... A type I error occurs whe we falsely
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationDefinition. Definition. 72 Estimating a Population Proportion. Definition. Definition
7 stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio
More informationMath 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 information1 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 information15.075 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 highdefiitio
More informationStat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals  the general concept
Statistics 104 Lecture 16 (IPS 6.1) Outlie for today Cofidece itervals Cofidece itervals for a mea, µ (kow σ) Cofidece itervals for a proportio, p Margi of error ad sample size Review of mai topics for
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationThe shaded region above represents the region in which z lies.
GCE A Level H Maths Solutio Paper SECTION A (PURE MATHEMATICS) (i) Im 3 Note: Uless required i the questio, it would be sufficiet to just idicate the cetre ad radius of the circle i such a locus drawig.
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More information7818 Interval estimation and hypothesis testing  Set
7 7818 Iterval estimatio ad hypothesis testig  Set revised Nov 9, 010 You might wat to read some of the chapter i MGB o Parametric Iterval Estimatio. There are subtle di ereces across questios. uderstad
More informationMEI 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 informationDistributions of Order Statistics
Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationThe 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 informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationStat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median.
Stat 04 Lecture Statistics 04 Lecture (IPS. &.) Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio Variables ad their distributios Variable:
More informationConfidence Intervals for One Mean with Tolerance Probability
Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with
More informationM06/5/MATME/SP2/ENG/TZ2/XX MATHEMATICS STANDARD LEVEL PAPER 2. Thursday 4 May 2006 (morning) 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES
IB MATHEMATICS STANDARD LEVEL PAPER 2 DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI 22067304 Thursday 4 May 2006 (morig) 1 hour 30 miutes INSTRUCTIONS TO CANDIDATES Do ot ope
More informationDetermining 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 informationIncremental 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 informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, 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 informationConfidence 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 informationPractice 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 informationx : X bar Mean (i.e. Average) of a sample
A quick referece for symbols ad formulas covered i COGS14: MEAN OF SAMPLE: x = x i x : X bar Mea (i.e. Average) of a sample x i : X sub i This stads for each idividual value you have i your sample. For
More informationConfidence Intervals for the Population Mean
Cofidece Itervals Math 283 Cofidece Itervals for the Populatio Mea Recall that from the empirical rule that the iterval of the mea plus/mius 2 times the stadard deviatio will cotai about 95% of the observatios.
More informationHypothesis Tests Applied to Means
The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationOutput 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 information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More informationDefinition. 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 information3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average
5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives
More informationMeasures 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 informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quatum Mechaics for Scietists ad Egieers David Miller Measuremet ad expectatio values Measuremet ad expectatio values Quatummechaical measuremet Probabilities ad expasio coefficiets Suppose we take some
More informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
More informationRadicals and Fractional Exponents
Radicals ad Roots Radicals ad Fractioal Expoets I math, may problems will ivolve what is called the radical symbol, X is proouced the th root of X, where is or greater, ad X is a positive umber. What it
More informationWe have seen that the physically observable properties of a quantum system are represented
Chapter 14 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationExample Consider the following set of data, showing the number of times a sample of 5 students check their per day:
Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationEconomics 140A Confidence Intervals and Hypothesis Testing
Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is
More information1 n. n > dt. t < n 1 + n=1
Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More information1 Review of Probability
Copyright c 27 by Karl Sigma 1 Review of Probability Radom variables are deoted by X, Y, Z, etc. The cumulative distributio fuctio (c.d.f.) of a radom variable X is deoted by F (x) = P (X x), < x
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationLecture Notes CMSC 251
We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1
More informationInference 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 informationMath 105: Review for Final Exam, Part II  SOLUTIONS
Math 5: Review for Fial Exam, Part II  SOLUTIONS. Cosider the fuctio fx) =x 3 l x o the iterval [/e, e ]. a) Fid the x ad ycoordiates of ay ad all local extrema ad classify each as a local maximum or
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationNotes on Hypothesis Testing
Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More information