Binomial Distribution


 Iris Ruth Woods
 1 years ago
 Views:
Transcription
1 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible values e.g. y 1,..., y or at most there is oe for every iteger e.g. y 1, y 2,.... Also recall that the probability distributio of a discrete r.v. is described by the probability of each possible value of the r.v. i.e. py = PY = y for all possible y. Some particular probability distributios occur ofte because they are useful descriptios of certai chace pheomeo uder study. For example, biomial distributio for discrete r.v. is very useful i practice. Toss a coi idepedetly three times ad record the umber of times that the coi laded o heads. Let Y = # of heads. The the sample space is: 1st toss 2d toss 3rd toss Y H H H 3 H H T 2 H T H 2 H T T 1 T H H 2 T H T 1 T T H 1 T T T 0 Suppose p = probability of heads ad q = probability of tails. The q = 1 p ad PY = 3 = p p p = p 3 PY = 2 = p p q + p q p + q p p = 3p 2 q PY = 1 = 3pq 2 PY = 0 = q 3 A biomial distributio must satisfy three 3 coditios: 1. There are trials, each of which ca result i oe of the two outcomes, either success or failure. i.e. the trials are dichotomous or Beroulli. 2. The probability of a success is costat p for all trials. 3. The trials are idepedet. Defie r.v. Y = # of successes i trials. The Y is said to have a biomial distributio with parameters ad p ad is writte as: Y B, p Let Y B, p. The formula for the biomial distributio is: where! PY = r = r! r! pr q r r = 0, 1, 2,..., are the possible umbers of successes. q = 1 p is the probability of failure. p r = p p p r times. q r = q q q r times. r =! r! r! is read as choose r. A quick review of factorial operatio k! = k k 1 1 4! = = 24 5! = = 120 0! = Examples Oe more example Properties Proportios Suppose Y B3, 1 2. PY = 1 = 3! = 3 1!2! Suppose Y B7, PY = 3 = 7! 3!4! = 3, p = 0.5 = = ! = 7, p = 0.6 Toss a fair die three times. What is the probability that there is at least oce a 6? Let Y B3, 1 6. The Pat least oe 6 = 1 Po 6 = 1 PY = 0. Sice PY = 0 = 3! = !3! 6 6 we have Pat least oe 6 = = = 3, p = Y B, p is a discrete r.v. The expectatio of Y is The variace of Y is µ Y = EY = p σ 2 Y = V ary = pq For example, suppose Y B10, 1 4, EY = = 10 4 = 2.5, V ary = = Aother example, suppose Y B100, 1 4, EY = = = 25, V ary = = I some situatios, oe is iterested i the proportio of successes amog trials. Defie where Y B, p. The W = Y EW = p, V arw = pq usig properties of biomial distributio i.e. expectatio ad variace of Y
2 z PZ>=1.5 PZ<=1.5 P0<=Z<=1.5 Overview Defiitio Defiitio ad properties Biomial distributio is a very importat probability model ad is ofte very useful. For example, cure rate of a ew drug, proportio of lakes that are suitable habitats for fish, survival rate of seedligs, etc. However, biomial distributio is ot suitable for all situatios. Are the trials dichotomous? Is the success probability costat? Are the trials idepedet? Cosider the fair die game, # of rats cured withi the same litter, etc. Always thik about the three assumptios before usig a biomial distributio. Recall that a r.v. is discrete if there are either a fiite umber of possible values e.g. y 1,..., y or at most there is oe for every iteger e.g. y 1, y 2,.... I cotrast a r.v. is cotiuous if the possible values are over a etire iterval of umbers. The probability distributio of a cotiuous r.v. is described by a probability desity curve such that area uder the curve correspods to probability. The most frequetly used type of cotiuous distributio is ormal distributio, which is also kow as Gaussia distributio. Normal distributio is a very importat distributio for modelig cotiuous data, because data that are iflueced by may small ad urelated radom effects are approximately ormally distributed e.g. a studet s height. A ormal distributio is defied by a bellshaped curve with a probability desity fuctio fy = 1 e 1 2σ 2y µ2 2πσ fy y A r.v. Y is said to have a ormal distributio with parameters µ ad σ 2 ad is writte as: Y Nµ, σ 2 Total area uder the distributio curve is 1. The distributio curve is symmetric about the ceter µ. Values aroud µ are more probable. The area i the rage of µ ±σ is about 2/3 of the total area The distributio curve has a welldefied bell shape. The parameters µ ad σ 2 provide complete iformatio about the distributio curve. I fact, EY = µ, V ary = σ Stadard Stadard Stadard Stadard Defiitio ad properties More properties Fidig probabilities Examples A r.v. Z has a stadardized ormal distributio if Z Nµ, σ 2 with expectatio µ = 0 ad variace σ 2 = 1 ad is writte as: Z N0, 1. The distributio curve is symmetric about the ceter µ = 0. The area betwee 1 ad 1 i.e. µ ± σ is about 2/3. The area betwee 2 ad 2 i.e. µ ± 2σ is Suppose Z is a stadard ormal r.v. i.e. Z N0, 1. PZ 0 = 0.5. P < Z < = 1. PZ = 1 = 0 ad i geeral PZ = c = 0 for ay c. Hece PZ < 0 = PZ 0 = 0.5. Remark: Suppose Y B3, 0.5 the PY 0 PY > 0! Table A o page 408 of the course otes bluebook gives PZ z for ay give value z 0 also kow as the upper tail probability. The z values are o the outside of Table A ad the probabilities are i the iside of Table A. Read the z value off x.x from the row idex ad 0.0x from the colum idex. PZ 1.5 = PZ 1.5 = 1 PZ 1.5 = = P0 Z 1.5 = PZ 0 PZ 1.5 = = How to fid PZ z for ay give value z? Drawig pictures helps! fz
3 PZ>= 0.6 P 1.7<=Z<=1.7 mu = 20, sigma = 3 P20<=Y<= P 1.7<=Z<=1.7 P X < = 0.9 PZ<=?=0.90 z = 1.28 mu = 20, sigma = 3 PY<= Stadard Stadard Stadard Stadard Fidig quatiles PZ 0.6 = 1 PZ 0.6 = 1 PZ 0.6 = = P 1.7 Z 1.7 = 2 P0 Z 1.7 = = P 1.7 Z 1.7 = 1 PZ 1.7 or Z 1.7 = 1 2 PZ 1.7 = = = What value z would give PZ z = 0.90? This z value would also give PZ z = = Look up Table A for the z value that correspods to Whe z = 1.28, PZ z = Whe z = 1.29, PZ z = Thus z value sought after is a umber betwee 1.28 ad More directly the table o page 92 of the bluebook gives z = Iterpretatio: The z value sought after is the cutoff value where 90% of the populatio lies below i.e. the 0.9 th populatio quatile or the 90 th populatio percetile Geeral Geeral Geeral Geeral Defiitio ad stadardizatio Example A r.v. Y is said to have a ormal distributio with parameters µ ad σ 2 ad is writte as: Y Nµ, σ 2 For example, Y N20, 3 2 stads for ormal distributio with µ = 20, σ 2 = 9. FACT: Suppose Y Nµ, σ 2, the a very useful trasformatio is Z = Y µ N0, 1 σ For Y N20, 3 2 i.e. µ = 20, σ 2 = 9, we have P20 Y 23 = P20 µ Y µ 23 µ 20 µ = P Y µ 23 µ σ σ σ = P Z 3 3 = P0 Z 1 = PZ 0 PZ 1 = = Give a value y, the term z = y µ σ is called the zscore correspodig to y. It represets the umber of stadard deviatios y is from µ. For Y N20, 3 2, 23 is 1σ away from µ = is 0σ away from µ = 20 i.e. at µ = is 4 3σ away from µ = is 1σ away from µ = 20. Suppose Y N20, 3 2, what is PY 15.5? Y µ PY 15.5 = P 15.5 µ σ σ = P Z 3 = PZ 1.5 =
4 P X < = 0.9 mu = 60, sigma = 10 PY<=?=0.90 z = Geeral Biomial ad s Fidig quatiles Key R commads Overview Defiitios Suppose Y N60, 10 2, what value y would give PY y = 0.90? From previous discussio, we kow that Thus PZ = 0.90 Y µ P σ = 0.90 PY µ 1.282σ = 0.90 PY µ σ = 0.90 Thus y = µ σ = = y is the stadard deviatio above the mea. > # Y~B3,0.5, compute PY=1 > dbiomx=1,size=3,prob=0.5 [1] > # Y~B7,0.6, compute PY=3 > dbiomx=3,size=7,prob=0.6 [1] > # Y~B3,1/6, compute PY>0 > pbiomq=0,size=3,prob=1/6,lower.tail=f [1] > # of by complemet > 1dbiomx=0,size=3,prob=1/6 [1] > # Z~N0,1, compute PZ>=1.5, PZ<=1.5, P0<=Z<=1.5, > porm1.5, lower.tail=f [1] > porm1.5, lower.tail=t [1] > porm0, lower.tail=fporm1.5, lower.tail=f [1] > # Z~N0,1, compute PZ>=0.6 > porm0.6, lower.tail=f [1] > # Z~N0,1, compute P1.7<=Z<=1.7 > 12*porm1.7, lower.tail=f [1] > # Z~N0,1, compute PZ<=?=0.90 > qorm0.9, lower.tail=t [1] > # Y~N20,9, compute P20<=Y<=23, PY<=15.5 > pormq=20,mea=20,sd=3,lower.tail=fporm23,mea=20,sd=3,lower.tail=f [1] > pormq=15.5,mea=20,sd=3,lower.tail=t [1] > # Y~N60,100, compute PY<=?=0.90 > qormp=0.9,mea=60,sd=10,lower.tail=t [1] Recall that a radom variable r.v. is a variable that depeds o the outcome of a chace situatio. The probability distributio of a r.v. is described by the probability of all possible outcomes of the r.v. Now we put data ad probability models together by viewig a observatio obs as a outcome of a r.v. We cosider obs as outcomes of r.v. s. The r.v. s Y 1,Y 2,...,Y formig the sample data are called a radom sample of size from a give distributio if 1 the Y i s are idepedet r.v. s; 2 the Y i s are idetically distributed i.e. every Y i has the same probability distributio. Coditios 1 ad 2 are sometimes abbreviated to the Y i s are iid. Further assume: 3 For each i, EY i = µ, V ary i = σ 2. A statistic is ay quatity whose value ca be calculated from sample data. It is a r.v. For example, sample mea is a statistic: Ȳ = Y i = Y 1 + Y Y The we have uder 1 3: EȲ = µ, σ2 V arȳ = As icreases, V arȳ decreases Sample mea FACT 1 FACT 2 FACT 3 By 2 ad 3, 1 EȲ = E Y 1 + Y Y = 1 EY 1 + Y Y = 1 EY 1 + EY EY = 1 µ + µ + + µ = µ By 1, 2, ad 3, 1 V arȳ = V ar Y 1 + Y Y = 1 2V ary 1 + Y Y = 1 V ary1 + V ary2 + + V ary 2 = 1 2σ2 + σ σ 2 = 1 2 σ2 = σ2 Suppose Y 1, Y 2,..., Y is a radom sample from Nµ,σ 2 distributio. The Ȳ Nµ, σ2 For example, if Y 1, Y 2,..., Y 40 are iid N3, 4, the Ȳ N3, 4 = N3, We may write Ȳ Nµ,σ2, where the variace of Ȳ is Ȳ σ 2 Ȳ = σ2 ad the correspodig stadard deviatio of Ȳ is σ Ȳ = σ also kow as the stadard error of the mea. Suppose Y 1,Y 2,...,Y is a radom sample from Nµ,σ 2 distributio. The Y i Nµ,σ 2 For example, if Y 1,Y 2,...,Y 40 are iid N3, 4, the 40 Thik of Y i = Ȳ : E Y i = µ, Y i N120, 160 V ar Y i = σ 2. Suppose Y 1, Y 2,...,Y is a radom sample from a distributio with mea µ ad variace σ 2. If is large, the the distributio of Ȳ is closely approximated by Ȳ Nµ, σ2 The larger is, the better the approximatio. This fact is also kow as the cetral limit theorem CLT for the sample mea
5 = 60, p = Example: fair die game FACT 4 A quick summary o CLT Recall the fair die game example: Y = wiig $ with probability distributio p0 = 1/2, p9 = 1/6,p15 = 1/3 ad mea µ = 6.5, σ 2 = If we repeat the game 1000 times idepedetly, the for Y 1, Y 2,..., Y 1000, we have EȲ = 6.5, V arȳ = 1000 By CLT, Ȳ is approximately N 6.5, For example, PȲ 7 P Z = PZ 2.32 = Suppose Y 1, Y 2,..., Y is a radom sample from a distributio with mea µ ad variace σ 2. If is large, the the distributio of Y i is closely approximated by Y i Nµ,σ 2 The larger is, the better the approximatio. This fact is also kow as the cetral limit theorem CLT for the sample sum. Example: fair die game Note that = 1000,E Y i = = 6500, V ar Y i = = By FACT 4, Y i is approximately N6500, The, P Y i 7000 P Z = PZ 2.32 = Radom sample, statistic, sample mea. If Y i iid Dµ, σ 2 σ2, EȲ = µ, V arȳ =. If Y i iid Nµ, σ 2 σ2, Ȳ Nµ,. If Y i iid Dµ, σ 2 σ2, approximately Ȳ Nµ, CLT. Similarly for Y i with mea µ ad variace σ 2. The CLT is a remarkable result. It says that regardless of the shape of the origial distributio, the takig of averages or sums results i ormal distributio. Hece we oly eed to kow the populatio mea ad stadard deviatio to have a approximate distributio of Ȳ or Y i. Why is ormal distributio so useful? May atural pheomea give rise to ormal variables; may variables ca be trasformed to ormal distributio; ad CLT. Simulatios. How big must be? Usually whe 30, the CLT works well. But if Y i has a highly skewed distributio, the we eed larger, i which case we may cosider trasformatio to a symmetric distributio first. A applicatio of the CLT is to approximate a biomial distributio by a ormal distributio Normal approximatio of biomial Normal approximatio of biomial Normal approximatio of biomial Recall that if Y B, p, the µ = p, σ 2 = pq where q = 1 p. Let Y B60, 0.4. Thus = 60, p = 0.4, ad µ Y = 24, σ 2 Y = Compute PY 20 = PY = 0 + PY = PY = 20? Now, thik of Y as the sum of = 60 iid r.v. s, each of which is a 0/1 variable. Let Y i = 1 for a success with probability p ad Y i = 0 for a failure with probability q. The Y = Y i is the umber of successes i = 60 trials. By the CLT o sum FACT 4, we ca approximate the distributio of Y by ormal distributio with mea ad variace µ Y = p = 24 σ 2 Y = pq = 14.4 = Hece if Y B60, 0.4, the Y NA N24, Now, we approximate PY 20 by PY 20 PY NA 20 YNA = P = PZ = The exact probability is PY 20 = > pbiomq=20, size=60, prob=0.4, lower.tail=t > [1] The discrepacy is largely due to the fact that we are approximatig a discrete r.v. by a cotiuous r.v. We ca make the ormal approximatio better by usig cotiuity correctio cc. PY 20 PY NA 20.5 YNA = P = PZ = Normal approximatio is OK if p 5 ad q 5. Always check this before usig ormal distributio to approximate biomial distributio. I the example, Y B60, 0.4, p = 24,q = 36, thus it is OK to use ormal approximatio. Cosider proportio of success deoted as W before For Y B60, 0.4, ˆp = Y Eˆp = p = 0.4, V arˆp = pq = = Hece by CLT o average FACT 3, ˆp NA N0.4, Thus for the example Y B60, 0.4, Pˆp 0.48 Pˆp NA 0.48 ˆpNA = P = PZ = =
6 Sample variace Sample variace Example Suppose Y i are iid Nµ, σ 2. We kow that the sample mea Ȳ Nµ, σ2. What about the sample variace also a statistic FACT S 2 = 1 1 Y i Ȳ 2? If Y 1, Y 2,..., Y form a radom sample from Nµ, σ 2, the V 2 1S2 = χ 2 σ 2 1 where χ 2 1 is a chisquared distributio with 1 degrees of freedom d.f.. Shape of the χ 2 1 distributio curve depeds o the d.f. V 2 0 The distributio curve is skewed i.e. asymmetric. The mea ad variace of V 2 χ 2 1 are EV 2 = 1, V arv 2 = 2 1. Thik of V 2 as a scaled versio of S 2. Table B is i terms of V 2. Hece cautio: always scale to V 2 before usig Table B. Sice we have ES 2 = σ 2, σ2 S 2 = 1 V 2 V ars 2 = 2σ4 1. Oe reaso to use 1 i S 2 is to esure ES 2 = σ 2. As icreases, S 2 estimates σ 2 with more precisio. Let Y 1, Y 2,...,Y 7 be a radom sample from N6, 3 2 i.e. = 7. What is PS 2 > 21? 1S PS > 21 = P > σ 2 σ 2 = PV 2 > = PV 2 > 14 where V 2 χ 2 6. From Table B, we have So ad I R, PV 2 > = 0.05, PV 2 > = > pchisqq=14, df=6, lower.tail=f > [1] < PV 2 > 14 < < PS 2 > 21 < 0.05 I the example, if Y i N200, 3 2, the PS 2 > 21 is the same, because it does ot deped o µ. Agai cautio: rescale S 2 to V 2 before usig Table B. More o the d.f. for sample variace S 2 = 1 Y i 1 Ȳ 2 Cosider = 3 ad all the deviatios. Oce we kow that it follows that Y 1 Ȳ = 3, Y2 Ȳ = 1, Y 3 Ȳ = 2. Hece the complete iformatio about deviatios is cotaied i 1 of them. I geeral, d.f. represets the amout or pieces of iformatio available about the spread. We use Ȳ,S2 istead of media ad IQR, because they are easier to deal with
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationElements of Statistical Methods Lots of Data or Large Samples (Ch 8)
Elemets of Statistical Methods Lots of Data or Large Samples (Ch 8) Fritz Scholz Sprig Quarter 2010 February 26, 2010 x ad X We itroduced the sample mea x as the average of the observed sample values x
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationSampling Distributions, ZTests, Power
Samplig Distributios, ZTests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationIE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.
IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of
More informationSTAT 515 fa 2016 Lec Sampling distribution of the mean, part 2 (central limit theorem)
STAT 515 fa 2016 Lec 1516 Samplig distributio of the mea, part 2 cetral limit theorem Karl B. Gregory Moday, Sep 26th Cotets 1 The cetral limit theorem 1 1.1 The most importat theorem i statistics.............
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a realvalued radom variable
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationCHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics
CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Radom Samplig The basic idea of the statistical iferece is that we are allowed to draw ifereces or coclusios about a populatio based
More informationLecture 5. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture 5. Radom variable ad distributio of probability prof. dr hab.iż. Katarzya Zarzewsa Katedra Eletroii, AGH email: za@agh.edu.pl http://home.agh.edu.pl/~za
More informationLecture 18: Sampling distributions
Lecture 18: Samplig distributios I may applicatios, the populatio is oe or several ormal distributios (or approximately). We ow study properties of some importat statistics based o a radom sample from
More information71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationConfidence Intervals. Confidence Intervals
A overview Mot probability ditributio are idexed by oe me parameter. F example, N(µ,σ 2 ) B(, p). I igificace tet, we have ued poit etimat f parameter. F example, f iid Y 1,Y 2,...,Y N(µ,σ 2 ), Ȳ i a poit
More informationNOTES ON DISTRIBUTIONS
NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationProbability and Statistics
robability ad Statistics rof. Zheg Zheg Radom Variable A fiite sigle valued fuctio.) that maps the set of all eperimetal outcomes ito the set of real umbers R is a r.v., if the set ) is a evet F ) for
More informationEconomics Spring 2015
1 Ecoomics 400  Sprig 015 /17/015 pp. 3038; Ch. 7.1.47. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 17 of Groeber text ad all relevat lectures
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Tradeoff Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationLecture 4. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH email: za@agh.edu.pl http://home.agh.edu.pl/~za
More informationLecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables
CSCIB609: A Theorist s Toolkit, Fall 06 Aug 3 Lecture 0: the Cetral Limit Theorem Lecturer: Yua Zhou Scribe: Yua Xie & Yua Zhou Cetral Limit Theorem for iid radom variables Let us say that we wat to aalyze
More informationExponential Families and Bayesian Inference
Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a dparameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where
More informationCHAPTER 2. Mean This is the usual arithmetic mean or average and is equal to the sum of the measurements divided by number of measurements.
CHAPTER 2 umerical Measures Graphical method may ot always be sufficiet for describig data. You ca use the data to calculate a set of umbers that will covey a good metal picture of the frequecy distributio.
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationRule of probability. Let A and B be two events (sets of elementary events). 11. If P (AB) = P (A)P (B), then A and B are independent.
Percetile: the αth percetile of a populatio is the value x 0, such that P (X x 0 ) α% For example the 5th is the x 0, such that P (X x 0 ) 5% 05 Rule of probability Let A ad B be two evets (sets of elemetary
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationStat 421SP2012 Interval Estimation Section
Stat 41SP01 Iterval Estimatio Sectio 11.111. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationIntroduction to Probability and Statistics Twelfth Edition
Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver Itroductio to Probability ad Statistics Twelfth
More informationDistribution of Random Samples & Limit theorems
STAT/MATH 395 A  PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to
More informationAMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 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 ukow mea µ = E(X) of a distributio by
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationStat 225 Lecture Notes Week 7, Chapter 8 and 11
Normal Distributio Stat 5 Lecture Notes Week 7, Chapter 8 ad Please also prit out the ormal radom variable table from the Stat 5 homepage. The ormal distributio is by far the most importat distributio
More informationTests of Hypotheses Based on a Single Sample (Devore Chapter Eight)
Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH25201: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with ztests 1 1.1 Overview of Hypothesis Testig..........
More informationCentral Limit Theorem the Meaning and the Usage
Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationApril 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE
April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationBig Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.
5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece
More informationSTAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)
STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated
More informationHomework 5 Solutions
Homework 5 Solutios p329 # 12 No. To estimate the chace you eed the expected value ad stadard error. To do get the expected value you eed the average of the box ad to get the stadard error you eed the
More informationBayesian Methods: Introduction to Multiparameter Models
Bayesia Methods: Itroductio to Multiparameter Models Parameter: θ = ( θ, θ) Give Likelihood p(y θ) ad prior p(θ ), the posterior p proportioal to p(y θ) x p(θ ) Margial posterior ( θ, θ y) is Iterested
More informationLecture 12: November 13, 2018
Mathematical Toolkit Autum 2018 Lecturer: Madhur Tulsiai Lecture 12: November 13, 2018 1 Radomized polyomial idetity testig We will use our kowledge of coditioal probability to prove the followig lemma,
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationLECTURE 8: ASYMPTOTICS I
LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece
More information(6) Fundamental Sampling Distribution and Data Discription
34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye 8.1 Radom Samplig: Populatio:
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter & Teachig Material.
More information32 estimating the cumulative distribution function
32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationz is the upper tail critical value from the normal distribution
Statistical Iferece drawig coclusios about a populatio parameter, based o a sample estimate. Populatio: GRE results for a ew eam format o the quatitative sectio Sample: =30 test scores Populatio Samplig
More informationMath 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency
Math 152. Rumbos Fall 2009 1 Solutios to Review Problems for Exam #2 1. I the book Experimetatio ad Measuremet, by W. J. Youde ad published by the by the Natioal Sciece Teachers Associatio i 1962, the
More informationConfidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M.
MATH1005 Statistics Lecture 24 M. Stewart School of Mathematics ad Statistics Uiversity of Sydey Outlie Cofidece itervals summary Coservative ad approximate cofidece itervals for a biomial p The aïve iterval
More informationEco411 Lab: Central Limit Theorem, Normal Distribution, and Journey to Girl State
Eco411 Lab: Cetral Limit Theorem, Normal Distributio, ad Jourey to Girl State 1. Some studets may woder why the magic umber 1.96 or 2 (called critical values) is so importat i statistics. Where do they
More informationTMA4245 Statistics. Corrected 30 May and 4 June Norwegian University of Science and Technology Department of Mathematical Sciences.
Norwegia Uiversity of Sciece ad Techology Departmet of Mathematical Scieces Corrected 3 May ad 4 Jue Solutios TMA445 Statistics Saturday 6 May 9: 3: Problem Sow desity a The probability is.9.5 6x x dx
More informationSTAC51: Categorical data Analysis
STAC51: Categorical data Aalysis Mahida Samarakoo Jauary 28, 2016 Mahida Samarakoo STAC51: Categorical data Aalysis 1 / 35 Table of cotets Iferece for Proportios 1 Iferece for Proportios Mahida Samarakoo
More information0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =
PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several wellow discrete probability distributios ad study some of their properties. Some of these distributios, lie
More informationChapter 18 Summary Sampling Distribution Models
Uit 5 Itroductio to Iferece Chapter 18 Summary Samplig Distributio Models What have we leared? Sample proportios ad meas will vary from sample to sample that s samplig error (samplig variability). Samplig
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter 2 & Teachig
More information5. INEQUALITIES, LIMIT THEOREMS AND GEOMETRIC PROBABILITY
IA Probability Let Term 5 INEQUALITIES, LIMIT THEOREMS AND GEOMETRIC PROBABILITY 51 Iequalities Suppose that X 0 is a radom variable takig oegative values ad that c > 0 is a costat The P X c E X, c is
More informationConfidence Level We want to estimate the true mean of a random variable X economically and with confidence.
Cofidece Iterval 700 Samples Sample Mea 03 Cofidece Level 095 Margi of Error 0037 We wat to estimate the true mea of a radom variable X ecoomically ad with cofidece True Mea μ from the Etire Populatio
More informationDiscrete probability distributions
Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop
More informationComputing Confidence Intervals for Sample Data
Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios
More informationProbability 2  Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
Probability 2  Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
More informationExample: Find the SD of the set {x j } = {2, 4, 5, 8, 5, 11, 7}.
1 (*) If a lot of the data is far from the mea, the may of the (x j x) 2 terms will be quite large, so the mea of these terms will be large ad the SD of the data will be large. (*) I particular, outliers
More informationData Description. Measure of Central Tendency. Data Description. Chapter x i
Data Descriptio Describe Distributio with Numbers Example: Birth weights (i lb) of 5 babies bor from two groups of wome uder differet care programs. Group : 7, 6, 8, 7, 7 Group : 3, 4, 8, 9, Chapter 3
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationf(x)dx = 1 and f(x) 0 for all x.
OCR Statistics 2 Module Revisio Sheet The S2 exam is 1 hour 30 miutes log. You are allowed a graphics calculator. Before you go ito the exam make sureyou are fully aware of the cotets of theformula booklet
More informationDirection: This test is worth 150 points. You are required to complete this test within 55 minutes.
Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem
More informationLecture Chapter 6: Convergence of Random Sequences
ECE5: Aalysis of Radom Sigals Fall 6 Lecture Chapter 6: Covergece of Radom Sequeces Dr Salim El Rouayheb Scribe: Abhay Ashutosh Doel, Qibo Zhag, Peiwe Tia, Pegzhe Wag, Lu Liu Radom sequece Defiitio A ifiite
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationSTAT431 Review. X = n. n )
STAT43 Review I. Results related to ormal distributio Expected value ad variace. (a) E(aXbY) = aex bey, Var(aXbY) = a VarX b VarY provided X ad Y are idepedet. Normal distributios: (a) Z N(, ) (b) X N(µ,
More informationModeling and Performance Analysis with DiscreteEvent Simulation
Simulatio Modelig ad Performace Aalysis with DiscreteEvet Simulatio Chapter 5 Statistical Models i Simulatio Cotets Basic Probability Theory Cocepts Useful Statistical Models Discrete Distributios Cotiuous
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More informationLecture 2: Poisson Sta*s*cs Probability Density Func*ons Expecta*on and Variance Es*mators
Lecture 2: Poisso Sta*s*cs Probability Desity Fuc*os Expecta*o ad Variace Es*mators Biomial Distribu*o: P (k successes i attempts) =! k!( k)! p k s( p s ) k prob of each success Poisso Distributio Note
More information