Basis for simulation techniques


 Karen Anderson
 1 years ago
 Views:
Transcription
1 Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios [3], page 47 Radom sample: The set of radom variables,,, is said to costitute a radom sample of size from the populatio with the distributio fuctio F( x) provided they are mutually idepedet ad idetically distributed with the distributio fuctio ( x) Fx ( ) for all i ad x. Statistic: Ay fuctio T (,,, ) of the observatios,,, is a statistic. Estimator: Ay statistic Θˆ Θ(, ˆ,, ) used to estimate the value of a parameter θ of the populatio is called a estimator of θ. A observed value θˆ θˆ ( x, x,, x ) is kow as a estimate of θ. F i. Estimatio methods. Method of momets  poit estimate Suppose oe or more parameters of the distributio of are to be estimated based o a radom sample of size. Defie the k th sample momet of to be: M k ' k k,,. () k The k th populatio momet is: µ k ' E [ k ] k,,, which is a fuctio of ukow parameters. ()
2 The method of momets cosists of equatig the first few populatio momets with the correspodig sample momets. Use as may equatios as ukows ad solve simultaeous equatios.. Cofidece itervals ([3], page 484)  iterval estimate We describe four ways of obtaiig cofidece itervals i the followig subsectios... Chebyshev s iequality Var[ Θˆ ] P( Θˆ ε < θ < Θˆ + ε) where Θˆ is the estimator of the parameter θ. This ca be used for ay estimator. ε (3) As a example, applyig this iequality to obtai the populatio mea θ µ, usig the sample mea as a estimator, Θˆ, ad the parameter beig estimated is the mea θ µ, the assumig the populatio variace is, Var[ ] σ, the iequality becomes: σ P ( ε < θ < + ε) Var[ ] ε σ P ( ε < θ < + ε) ε (4) (5) See Sectio 3. for why Var[ ] σ... Uderlyig radom variable has a ormal distributio + populatio variace is kow I geeral, cofidece itervals obtaied by Chebyshev s iequality ca be improved if the distributio of s kow. Here are the steps for obtaiig a cofidece iterval for parameter θ :. Fid a radom variable that is a fuctio of,,, : T T(,,, ; θ) (6) such that the distributio of T is kow.. Fid umbers a ad b such that: Pa ( < T< b) γ. (7)
3 3. After samplig the values x i of, determie the rage of values that θ ca take o while maitaiig the coditio This rage is a 00γ% cofidece iterval of θ. a < t( θ) < b, where (8) t( θ) Tx (, x,, x ; θ). (9) Thus, if s kow to have a ormal distributio, N ( µσ, ), the the sample mea is N ( µσ, ) ad Z (( µ ) ( σ ) ) has the stadard ormal distributio N( 0, ) (see Theorem 3.6 below). Therefore, Pa ( < Z< b) γ is the 00γ % CI (ote Z is the T fuctio of (6)). (0) x µ a < Z< b or a < < b or x bσ < µ < x a σ () σ By choosig a z ad b z, which is the umber of stadar deviatios from the mea oe must go i order to cotai 00γ % of the probability mass, we get the equatio i howtosimulate.doc: M zσ M µ M + zσ M () where σ M σ ( ) ad z is the umber of stadard deviatios from the mea oe must go i a ormal distributio N( 0, ) to cotai 00γ % of the probability mass. I other words PM ( zσ M µ M+ zσ M ) γ. So if γ 0.95, z.96. I other words, we are 95% sure that parameter µ, which is beig estimated, lies i that rage. Theorem 3.6 [3], page 70: Let,,, be mutually idepedet radom variables such that N ( µ i, σ i ), i,,,. The S is ormally distributed, that is S N ( µ, σ ), where
4 µ µ σ σ i i (3) Example 3.30 [3]: Sample mea S. It ca be show that sice S has the distributio S Nµ (, σ ) ad f f S ( x), has the distributio N ( µσ, ) ad the radom variable (( µ ) ( σ ) ) has the distributio N( 0, ). To prove the statemet i the above example, use the results from Sectio 3., where is a radom variable that is a fuctio of aother radom variable S, whose pdf is kow (by theorem 3.6 above). Sice S, we ca write f f S ( x) usig the results of Sectio 3.. We kow that S Nµ (, σ ). Therefore, we have: f ( x) e π σ ( x µ ) σ (4) f ( x) e πσ ( x µ ) ( σ ). (5) Therefore, N ( µσ, ) I Sectio 3., we showed that the Var[ ] beig equal to σ does ot require the s to be ormally distributed. Here, whe the s are ormally distributed, we agai see that the variace of the r.v. s σ...3 Uderlyig radom variable has a ukow distributio but populatio variace is kow Eve without the s beig ormally distributed, because of the Cetral Limit Theorem, the sample mea, as a fuctio of radom variables, has a ormal distributio as. Cetral Limit Theorem ([3], page 7):
5 Let,,, be mutually idepedet radom variables with a fiite mea E [ i ] µ i ad a fiite variace Var[ ] σ i,,,,. We form the ormalized radom variable: Z so that EZ [ ] 0 ad Var[ Z ]. The, uder certai regularity coditios, the limitig distributio of Z is stadard ormal deoted Z N( 0, ), i.e. σ i µ i (6) lim () t PZ ( t) F Z t e y dy π (7) Special case: Let,,, be iid with a commo mea µ E [ i ] ad commo variace σ Var[ ], the (6) becomes Z ( µ ) σ (8) where is the sample mea. Therefore the sample mea from radom samples ted toward ormality as the sample size icreases. Give the Cetral Limit Theorem, the same cofidece iterval as i Sectio.. works here: PM ( zσ M µ M+ zσ M ) γ (9) where σ M σ ( )...4 Cofidece iterval whe populatio variace is ukow Example 3.35 of page 78 [3]: Assume that,,, be mutually idepedet idetically distributed radom variables such that N ( µσ, ). The it follows that V ( µ ) σ (0)
6 has the stadard ormal distributio (see example o first page). From example 3.33, ( )S W σ σ () has the chisquared distributio with degrees of freedom (prove this?). It follows that: T V ( µ )(( ) σ) has the distributio with W S ( µ ) t S ( ) ( ) σ degrees of freedom, where ( ) S () ( µ ) P t ; α < <, or (3) S ( ) t ; α α t ; α S µ t M + ; α σs M (4) where S M S ( ) ad t is the value of a tdistributed radom variable with ; α degrees of freedom such that 00( α) % of the probability mass of the radom variable is cotaied betwee ( t t ; α,. ; α ) Theorem 3.7 [3], page 7: If,,, is a sequece of mutually idepedet, stadard ormal variables, the Y (5) has the gamma distributio, GAM(, ), or the chisquare distributio with degrees of freedom Y χ.
7 If,,, are ot ormal, the above theorem does ot hold as strogly as does the CLT for []. Theorem 3.0. If V ad W are idepedet radom variables such that V N( 0, ) ad W χ, the the radom variable: T V W (6) has the t distributio with degrees of freedom. Example 3.3: Let,,, be a sequece of mutually idepedet, ormal variables N µσ (, ). The radom variables Z i µ i are stadard ormal. Therefore σ Y Z i ( µ ) σ (7) has the chisquare distributio with degrees of freedom Y χ. We typically do ot kow µ, the populatio mea. Therefore, replace µ by the sample mea i. Defie a ra dom variable Referece [] gives a theorem: S σ σ (8) Theorem 6.: Let Let,,, be a sequece of mutually idepedet, ormal variables N ( µσ, ) for all i. Let  i be the sample mea, ad (9)
8 The: S be the sample variace. (30) ( i ). ad S are idepedet.. ( )S σ Chi square( ) σ Questio:. V is stadard ormal eve if s are ot ormally distributed provided is large because of CLT, but does Theorem 3.7 hold eve if Note: oormal distributios []: s are ot ormally distributed? The tbased cofidece iterval procedures are ofte applied whe,,, are ot draw from a ormal distributio. This is acceptable i large samples if the distributio of is reasoably symmetric. However the procedures are ot valid for highly skewed distributios. What about Pareto? File sizes follow the Pareto distributio!,,, Exercise: Work out Problem o page 49 of [3] i class...5 Depedet samples [3], page 503: I all the previous subsectios to fid cofidece itervals, we assumed that the samples were idepedet. But i most of our simulatios, this is ot likely to be true. I this case, the variace is o loger. If we assume that the sequece is widesese statioary, the autocovariace fuctio σ K j i E µ [( )( j µ )] Cov(, j ) (3) is fiite ad is a fuctio of oly i j. The variace of the sample mea is:
9 Var[ ]  Var[ ] + ( i, j ) ( i j) Cov(, j ) (3) because Var[ + Y] E[ (( + Y) E [ + Y] ) ] Var[ + Y] E[ (( + Y) E [ ] EY [ ]) ] Var[ + Y] E[ ( E[ ] ) + ( Y E[ Y] ) + ( E[ ] )( Y E[ Y] )] Var[ + Y] E[ ( E[ ] ) ] + E[ ( Y E[ Y] ) ] + E[ ( E[ ] )( Y E[ Y] )] (33) (34) (35) (36) Comig back to (3): Var[ + Y] Var[ ] + Var[ Y] + Cov(, Y) (37) Write out secod term i (3): Var[ ] σ j Kj (38) Cov (, ) j Cov( ) + Cov( 3 ) + + Cov( ) + ( ij, ) ( i j) (39) Cov( ) + Cov( 3 ) + + Cov( ) + + Cov( ) + Cov( ) + + Cov( ) Cosider how may terms have i j of. This is o the first lie, o the secod, for each subsequet lie util last but oe. The last lie agai has oly oe such term. Therefore, K occurs Cosider the multiplicative factor for + ( ). (40) K i (38). It is
10 . (4) ( ) Give the factor i the deomiator i (3), this checks out. Cosider the multiplicative factor of K. From (39), we see that the first two rows oly have such term as do the last two rows. The itermediate rows will have two such terms, e.g., 3 ad 3 5. Thus from (39), the multiplicative factor of K is + ( 4) + 4. From (38), we see this factor is: Checks out! So (38) is equivalet to (3). As,. (4) ( ) lim Var[ ] σ + K j aσ K, where a j (43) σ j It ca be show that uder rather geeral coditios, the statistic: j µ σ a  (44) of the correlated data approaches the stadard ormal distributio. Therefore, the 00( α) % CI for µ is give by: ± σz α a  (45) We ca avoid havig to estimate σ a by usig the method of idepedet replicatios. Replicate a experimet m times, with each experimet cotaiig observatios. If the seed i the m experimets are chose radomly, the the results of differet experimets will be idepedet though the observatios i a experimet will be depedet. Let the i th observatio i the j th
11 experimet be () j. Let the sample mea ad sample variace of the j th experimet be j () ad S () j, where: j ()  i () j ad (46) S () j [ i () j j ()] (47) Sice ( ), ( ), m ( ) are idepedet (ad idetically distributed) From the idividual sample meas, we obtai a estimator for the populatio mea µ as m  j () m j m m i () j j (48) m V [ j () ] m j ( j ()) m m ( ) m Sice a estimate of the variace is used, we use the tdistributio. Therefore the statistic m j (49) ( µ )(( m) V) is approximately tdistributed with ( m ) degrees of freedom. The 00( α) % CI is x ± (50) m This is actually oe of six approaches to hadle this depedece of samples problem [8] (see basic.pdf ad others.pdf file posted o web site, which are extracted pages from [8]). Also see [0]. They call this approach replicatio/deletio ad suggest deletig some iitial sample poits to get rid of trasiet behavior. t m ; α v Batch meas method: Igore the first few sample poits for trasiets. The take the whole (steady state) process of sample poits ad divide them ito k batches with size of k sam
12 ples. The "sample meas" of those batches/segmets ca roughly be treated as idepedet, if k is sufficiet large. The key for usig batch meas is to select large eough batch sizes. See [9] for rule of thumb o how large to make the batch sizes. The PostNotes3.doc file [9] o CI calculatio explais the details. I that file, the umber of observatios per batch k is suggested to be at least 4t where choice of t is umber of observatios before correlatio dies out (decays to almost zero). This meas that correlatio should be computed ad the batch size determied. I simulatios that oe of my research studets ra, correlatio died out i 00 samples; this meas each batch size should be 800. Spectrum aalysis method [8]: ˆ Obtai a estimator for Var[ ] i (38) by replacig K j with a estimator K j obtaied from the samples: K ˆ j j [ ( )][ + j ( )] j (5) ad a estimator for σ i (38) as S M ( ). Plug K ˆ istead of ad istead of N j K j S σ ito (38) ad get a estimate v of the Var[ ]. The 00( α) % CI for the mea is x ± (5) m Matlab correlatio fuctios (maybe corr) ca be used to compute correlatio. Correlatio ρ Y Cov( x, Y) Var( )Var( Y) t m ; α v A example: For our research work o VBLS, here is what we did to determie how log to apply the replicatio/deletio approach. Each simulatio ru was executed for 6000sec because of the followig computatio. Our largest file size was GB file ad miimum badwidth was 00Mbps, which meas the maximum trasfer time is 80s. If we wat 00 such samples, we eed to at least simulate 00*80 sec 6000 sec. With a call arrival rate of 50 calls/sec, we eed at least 50*6000 or files. The first 0% of the data was dropped. This was arbitrary. My studet said it did t
13 impact results greatly. He could have dropped just 0%. The umber of replicatios was 5. I other words, my studet executed 5 rus for each lambda (call arrival rate). He computed the mea of file trasfer delays from all the sample poits i each ru. The he took these 5 meas ad computed aother mea ad the CI usig the tdistributio formula with 4 as the degrees of freedom. The five meas ca be assumed to be idepedet sice they are from differet rus. Builti matlab fuctio ormfit ca be used for CI calculatio. 3. Appedices 3. Appedix I: Fid the pdf ad CDF of a fuctio of a radom variable whose pdf ad CDF are kow Theorem 3. (page 40, [3]): Let be a cotiuous r.v. with desity that is ozero o a subset I of real umbers (that is f ( x) > 0, x I ad f ( x) 0 for x I. Let Φ be a differetiable mootoic fuctio whose domai is I ad whose rage is the set of reals. The Y Φ( ) is a f cotiuous radom variable with desity f Y give by: f Y ( y) f [ Φ ( y) ][ ( Φ )'( y) ] y Φ() I. (53) 0 otherwise Proof: Assume Φ( ) is a icreasig fuctio. F Y ( y) PY ( y) P( Φ( ) y) P ( Φ ( y) ) F ( Φ ( y) ) (54) dy dy du To get the desity fuctio (use chai rule: ): dx du dx f Y ( y) d FY ( y) d F ( Φ ( y) ) dy dy d d F ( Φ ( y) ) ( ) dy Φ y d Φ ( y) (55) f Y ( y) f [ Φ ( y) ][ ( Φ )'( y) ] (56) If Y a+ b, the
14 f Y ( y) y b f a a y ai + b 0 otherwise (57) The proof is similar for the case whe Φ( ) is decreasig. There whe P( Φ( ) y) P ( Φ ( y) ) F ( Φ ( y) ). But whe we take the derivatives we will get (53). The reaso for takig the absolute value is because pdf is +ve. 3. Appedix II: Derivatio for Var[Sample mea] Derivatio of Var[ ] σ : Var[ + Y] Var[ ] + Var[ Y] if ad Y are idepedet. Therefore Var Var[ ] σ ad Var[ ] Var i. ( σ ) σ This from page 93 of [3]. Here is how I verified it. Let Z Y, where Y Usig (57), sice Z Y, Var[ Z] EZ [ ] ( EZ [ ]) z f Z ( z) dz ( EZ [ ]) (58) f Z ( z) f Y ( z). (59) Therefore (58) becomes: Var[ Z]  fz ( z) dz ( EZ [ ]) f Y ( z) dz ( EZ [ ]) y y f Y ( z) d  y ( EZ [ ]) (60) y Var[ Z]  ( y f Y ( y) dy) ( EZ [ ]) (6) y EZ [ ] zf Z ( z) dz  f ( z ) d  y  yf Y Y ( y) dy EY [ ] (6)
15 Var[ Z] E[ Y EY [ ] ] σ Var ( Y) σ (63) 3.3 Distributios 3.3. Normal distributio If N ( µσ, ), its pdf is: f( x) exp  x µ < x < σ π σ (64) F Z ( z) z e y dy, where Z N( 0, ) (65) π x µ F ( x) F Z σ (66) 3.3. Gamma, chisquare ad t distributios Gamma pdf: GAM( λα, ), pg. 7/8 [3] f() t λ α t α e λt α > 0 t > 0. (67) Γα ( ) Γα ( ) x α e x dx, Γα ( ) ( α )Γ( α ) ad (68) 0 Γ( ) ( )! if is a positive iteger (69) The chisquare distributio is a special case of gamma distributio with α  ad λ , where is a positive iteger. Thus if GAM(, ), the it is said to have a chisquared distributio with degrees of freedom χ. Studett pdf (oe parameter ): f T () t Γ πγ + t ( + )  < t <  (70)
16 Refereces [] K. S. Trivedi, Probability, Statistics with Reliability, Queueig ad Computer Sciece Applicatios, Secod Editio, Wiley, 00, ISBN [] R. Yates ad D. Goodma, Probability ad Stochastic Processes, Wiley, ISBN [3] K. S. Trivedi, Probability, Statistics with Reliability, Queueig ad Computer Sciece Applicatios, First Editio, Pretice Hall, 98, ISBN r. [4] D. Bertsekas ad R. Gallager, Data Networks, Pretice Hall, 986, ISBN [5] Prof. Boorsty s otes, Polytechic Uiversity, NY. [6] A. Leo Garcia ad I. Widjaja, Commuicatio Networks, McGraw Hill, 000, First Editio. [7] Mischa Schwartz, Telecommuicatios Networks, Protocols, Modelig ad Aalysis, Addiso Wesley, 987. [8] Averill M. Law, W. David Kelto, Simulatio modelig ad aalysis, McGraw Hill, 000. [9] Output Aalysis of a Sigle System, IE 305 Simulatio, distributed /9/03 (posted o web site as PostNotes3.doc). [0] Prof. William Saders, UIUC otes (posted o web site). [] R. Fewster, Stats 0, chapter 6.
Topic 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 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 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 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 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 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 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 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 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 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 informationThis section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol DiscreteEvent System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol DiscreteEvet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
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 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 informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
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 informationLecture 3. Properties of Summary Statistics: Sampling Distribution
Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary
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 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 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 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 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 informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
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 information1.010 Uncertainty in Engineering Fall 2008
MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00  Brief Notes # 9 Poit ad Iterval
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 informationFACULTY 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 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 informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 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 be doe
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 informationMathematical Statistics  MS
Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
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 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 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 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 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 informationThere is no straightforward approach for choosing the warmup period l.
B. Maddah INDE 504 DiscreteEvet Simulatio Output Aalysis () Statistical Aalysis for SteadyState Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.
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 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 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 informationhttp://www.xelca.l/articles/ufo_ladigsbaa_houte.aspx imulatio Output aalysis 3/4/06 This lecture Output: A simulatio determies the value of some performace measures, e.g. productio per hour, average queue
More informationDepartment of Mathematics
Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets
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 informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
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 informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hardcopy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
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 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 informationECONOMETRIC THEORY. MODULE XIII Lecture  34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture  34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationGoodnessofFit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
GoodessofFit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH25201: Probability ad Statistics II Sprig 2019 Cotets 1 ChiSquared Tests with Kow Probabilities 1 1.1 ChiSquared Testig................
More informationEE 4TM4: Digital Communications II Probability Theory
1 EE 4TM4: Digital Commuicatios II Probability Theory I. RANDOM VARIABLES A radom variable is a realvalued fuctio defied o the sample space. Example: Suppose that our experimet cosists of tossig two fair
More informationChapter 6. Sampling and Estimation
Samplig ad Estimatio  34 Chapter 6. Samplig ad Estimatio 6.. Itroductio Frequetly the egieer is uable to completely characterize the etire populatio. She/he must be satisfied with examiig some subset
More informationA Question. Output Analysis. Example. What Are We Doing Wrong? Result from throwing a die. Let X be the random variable
A Questio Output Aalysis Let X be the radom variable Result from throwig a die 5.. Questio: What is E (X? Would you throw just oce ad take the result as your aswer? Itroductio to Simulatio WS/  L 7 /
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationENGI 4421 Confidence Intervals (Two Samples) Page 1201
ENGI 44 Cofidece Itervals (Two Samples) Page 0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.45.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
More informationAdvanced Stochastic Processes.
Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.
More informationInterval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),
Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We
More informationExam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 57pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234
STA 291 Lecture 19 Exam II Next Tuesday 57pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Locatio CB 234 STA 291  Lecture 19 1 Exam II Covers Chapter 9 10.1; 10.2; 10.3; 10.4; 10.6
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
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 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 informationThe Sample Variance Formula: A Detailed Study of an Old Controversy
The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study realvalued fuctios defied
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 informationGeneralized Semi Markov Processes (GSMP)
Geeralized Semi Markov Processes (GSMP) Summary Some Defiitios Markov ad SemiMarkov Processes The Poisso Process Properties of the Poisso Process Iterarrival times Memoryless property ad the residual
More informationLimit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1p).
Limit Theorems Covergece i Probability Let X be the umber of heads observed i tosses. The, E[X] = p ad Var[X] = p(p). L O This P x p NM QP P x p should be close to uity for large if our ituitio is correct.
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 informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
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 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 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 informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
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 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 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 information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
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 informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: YeChi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
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 informationOutput Analysis and RunLength Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad RuLegth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More informationMath 61CM  Solutions to homework 3
Math 6CM  Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More information6.041/6.431 Spring 2009 Final Exam Thursday, May 21, 1:304:30 PM.
6.041/6.431 Sprig 2009 Fial Exam Thursday, May 21, 1:304:30 PM. Name: Recitatio Istructor: Questio Part Score Out of 0 2 1 all 18 2 all 24 3 a 4 b 4 c 4 4 a 6 b 6 c 6 5 a 6 b 6 6 a 4 b 4 c 4 d 5 e 5 7
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
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 informationLecture 7: Density Estimation: knearest Neighbor and Basis Approach
STAT 425: Itroductio to Noparametric Statistics Witer 28 Lecture 7: Desity Estimatio: knearest Neighbor ad Basis Approach Istructor: YeChi Che Referece: Sectio 8.4 of All of Noparametric Statistics.
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Itroductio to Probability ad Statistics Lecture 23: Cotiuous radom variables Iequalities, CLT Puramrita Sarkar Departmet of Statistics ad Data Sciece The Uiversity of Texas at Austi www.cs.cmu.edu/
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 3 9//203 Large deviatios Theory. Cramér s Theorem Cotet.. Cramér s Theorem. 2. Rate fuctio ad properties. 3. Chage of measure techique.
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 informationSection 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis
Sectio 9.2 Tests About a Populatio Proportio P H A N T O M S Parameters Hypothesis Assess Coditios Name the Test Test Statistic (Calculate) Obtai P value Make a decisio State coclusio Sectio 9.2 Tests
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 informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More information