Improved Ratio Estimators of Population Mean In Adaptive Cluster Sampling
|
|
- Brooke Hawkins
- 5 years ago
- Views:
Transcription
1 J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) 1 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural Improved Ratio Estimators of Populatio Mea I Adaptive Cluster Samplig Subhash Kumar Yadav 1, Sheela Misra, Sat Sara Mishra 1, ad ipapor Chutima 3 1 Departmet of Mathematics ad Statistics (A Cetre of Excellece o Advaced Computig)Dr. RML Avadh Uiversity, Faizabad- 4001, U.P., Idia Departmet of Statistics, Uiversity of Luckow, Luckow-6007, U.P., Idia 3 Departmet of Mathematics, Faculty of Sciece,Mahasarakham Uiversity, Maha Sarakham-44150, Thailad Received: 13 Aug. 015, Revised: 18 Sep. 015, Accepted: 3 Oct. 015 Published olie: 1 Ja. 016 Abstract: I this paper, we study the estimators of the populatio mea i adaptive cluster samplig by usig the iformatio of the auxiliary variable, that is, the populatio coefficiet of variatio, the coefficiet of skewess kurtosis of the auxiliary variable ad the correlatio coefficiet betwee mai variable ad the auxiliary variable. The large sample properties of the proposed estimators have bee studied up to the first order of approximatio. A umerical study is also carried out to judge the theoretical fidigs. The umerical example showed that if the populatio is rare ad hidde clustered populatio, all estimators i adaptive cluster samplig are more efficiet tha the estimators i simple radom samplig with the same coditio. Keywords: Adaptive Cluster Samplig, Auxiliary Variable, Mea squared error, Efficiecy 1 Itroductio Adaptive cluster samplig, proposed by Thompso(1990), is a efficiet method for samplig rare ad hidde clustered populatios. I adaptive cluster samplig, a iitial sample of uits is selected by simple radom samplig. If the value of the variable of iterest from a sampled uit satisfies a pre-specified coditio C,that is (i,y i > c) the the uit s eighborhood will also be added to the sample. If ay other uits that are adaptively added also satisfy the coditio C,the their eighborhoods are also added to the sample. This process is cotiued util o more uits that satisfy the coditio are foud. The set of all uits selected ad all eighborig uits that satisfy the coditio is called a etwork. The adaptive sample uits, which do ot satisfy the coditio, are called edge uits. A etwork ad its associated edge uits are called a cluster.if a uit is selected i the iitial sample ad does ot satisfy the coditio C, the there is oly oe uit i the etwork. It is well kow that the variable about which we have full iformatio is kow as auxiliary variable ad the iformatio is kow as auxiliary iformatio which is highly (positively or egatively) correlated with the variable uder study. Wheever auxiliary variable (iformatio) is kow, oe would like to use it at the desig or estimatio stage sice it is well kow ad established that the use of auxiliary iformatio i samplig theory ehaces the efficiecy of the estimators ad it is i use sice the use of samplig itself. I this paper, we will study the estimator of populatio mea i adaptive cluster samplig usig a auxiliary variable. Estimators Uder Simple Radom Samplig Let (x i,y i ),i = 1,,, be the pair of observatios for the auxiliary ad study variables, respectively for the populatio of size usig Simple Radom Samplig With Out Replacemet (SRSWOR). Let µ x ad µ y be the Correspodig author sat x003@yahoo.co.i c 016 SP atural Scieces Publishig Cor.
2 S. K. Yadav et al.: Improved ratio estimators of populatio mea... populatio meas of auxiliary ad study variables respectively ad x ad ȳ be the respective sample meas. Ratio estimators are used whe the lie of regressio of y o x passes through origi ad the variables X ad Y are positively correlated to each other, while product estimators are used whe X ad Y are egatively correlated to each other, otherwise regressio estimators are used to estimate the populatio parameters uder cosideratio. Cochra (1940) proposed the classical ratio estimator for estimatig the populatio mea,µ y = 1 variable as follows: where ȳ= 1 y i ad x= 1 x i, assumig that µ x = 1 ȳ R = ȳ µx x y i of the study x i, the populatio mea of the auxiliary variable is kow. The expressios for the Mea Squared Error (MSE) of the estimator give i (1) up to the first order of approximatio are respectively, as follows: MSE(ȳ R )= where C y = S y µ y, C x = S x µ x, S y = 1 S xy = 1 1 (x i µ x )(y i µ y ) 1 µ [ y +Cx ρc ] yc x (y i µ y ), f =, S x = 1 1 (1) () (x i µ x ), ρ xy = S xy S y S x, Sisodia ad Dwivedi (1981) proposed the followig estimator usig the coefficiet of variatio of auxiliary variable as: µx +C x ȳ R1 = ȳ (3) x+c x The MSE of the estimator ȳ R1, up to the first order of approximatio respectively are: MSE(ȳ R1 ) µ [ y + θ1c x ] θ 1 ρ x x,where θ1 = µ x (4) µ x +C x Upadhyay ad Sigh (1999) proposed two ratio type estimators utilizig the coefficiet of variatio ad the coefficiet of kurtosis of auxiliary variable β (x) β(x) µ x +C x ȳ R = ȳ (5) β (x) x+c x Cx µ x + β (x) ȳ R3 = ȳ (6) C x x+β (x) The MSE of the estimators ȳ R ad ȳ R3, up to the first order of approximatio respectively are: MSE(ȳ R ) MSE(ȳ R3 ) µ [ y + θ C x θ ] ρ x x µ [ y + θ3c x ] θ 3 ρ x x where θ = β (x) µ x β (x) µ x +C x ad θ 3 = C xµ x C x µ x +β (x) Sigh ad Tailor (003) proposed a ratio type estimator usig correlatio coefficiet betwee auxiliary variable ad the variable uder study as: µx + ρ xy ȳ R4 = ȳ (9) x+ ρ xy The MSE of the estimator ȳ R4, up to the first order of approximatio respectively are: MSE(ȳ R4 ) µ [ y + θ4 C x θ ] 4ρ x x,where θ4 = µ x (10) µ x + ρ xy (7) (8) c 016 SP atural Scieces Publishig Cor.
3 J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) / 3 Kadilar ad Cigi (003) suggested the followig estimator utilizig the auxiliary variable as: ( µ ) ȳ R5 = ȳ x x (11) The MSE of the estimator ȳ R5, up to the first order of approximatio respectively are: MSE(ȳ R5 ) µ [ y + 4Cx 4ρ ] xyc x (1) Ya ad Tia (010) proposed two ratio type estimators usig coefficiets of skewess β 1(x) ad kurtosis β(x) of auxiliary variable as: β(x) µ x + β 1(x) ȳ R6 = ȳ (13) β (x) x+β 1(x) β1(x) µ x + β (x) ȳ R7 = ȳ (14) β 1(x) x+β (x) The MSE of the estimators ȳ R6 ad ȳ R7, up to the first order of approximatio respectively are: MSE(ȳ R6 ) MSE(ȳ R7 ) µ [ y + θ6 C x θ ] 6ρ x x µ [ y + θ7c x ] θ 7 ρ x x (15) (16) where θ 6 = β (x) µ x β (x) µ x +β 1(x) ad θ 7 = β 1(x) µ x β 1(x) µ x +β (x) 3 Estimators Uder Adaptive Cluster Samplig Let the populatio cosists of distict idetifiable uits labeled from 1,,...,. Let y i ad x i (i = 1,,...,) deote the observatio o the characteristic x ad y respectively, uder study for the i th uit of the populatio. Let deote the iitial sample size ad ν deote the fial sample size. Let Ψ i deote the etwork that icludes uit i ad m i be the umber of uits i that etwork. The iitial sample of uits is selected by simple radom samplig without replacemet. The estimator of the populatio mea for the variable of iterest uder adaptive cluster samplig based o Hase- Hurwitz estimator as, ȳ ac = 1 (w y ) i (17) where, (w y ) i is the average of the variable y uder study i the etwork that icludes uit i of the iitial sample, that is:(w y ) i = m 1 i (y j ) j Ψ i The variace of ȳ ac is give by, V(ȳ ac )= ( 1) [(w y ) i µ y ] (18) Dryver ad Chao (007) proposed the followig ratio estimator i adaptive cluster samplig as, ȳ acr = ȳac x ac µ x (19) c 016 SP atural Scieces Publishig Cor.
4 4 S. K. Yadav et al.: Improved ratio estimators of populatio mea... where x ac = 1 (w x ) i ad(w x ) i is the average of the auxiliary variable x i the etwork that icludes uit i of the iitial sample, that is, (w x ) i = m 1 i (x j ) j Ψ i The first order approximated MSE of ȳ acr is MSE(ȳ acr ) µ [ wy +Cwx ρ ] wx.wyc w wx Chutima (013) proposed the followig ratio type estimators of populatio mea usig parameters of the auxiliary iformatio based o Sisodia ad Dwivedi (1981) estimator ad Upadhyaya ad Sigh (1999) two estimators uder adaptive cluster samplig as, µy +C wx ȳ acr1 = ȳ ac (1) x ac +C wx µx β (wx )+C wx ȳ acr = ȳ ac () x ac β (wx )+C wx µx + β (wx ) ȳ acr3 = ȳ ac (3) x ac + β (wx ) where C wx is the populatio coefficiet of variatio of w x, β (wx ) is the populatio coefficiet of kurtosis of w x. Where S wy = 1 1 [(w y ) i µ y ], S wx = 1 1 [(w x ) i µ x ], S wx.wy = 1 1 [(w x ) i µ x ][(w y ) i µ y ]=ρ wx.wy S wx S wy The mea square errors of above estimators usig Taylor series method up to the first order of approximatios respectively are, MSE(ȳ acr1 ) MSE(ȳ acr ) MSE(ȳ acr3 ) µ [ wy + θw1 C wx θ ] w1ρ wx.w w wx µ [ wy + θw C wx θ ] wρ wx.w w wx µ [ wy + θw3 C wx θ ] w3ρ wx.w w wx where θ w1 = µ y µ y +C wx, θ w = µ xβ (wx) µ µ x β (wx) +C wx, θ w3 = x µ x +β ad R= µ x (wx) µ y (0) (4) (5) (6) 4 Proposed Estimators Motivated by Sigh ad Tailor (003), Kadilar ad Cigi (003) ad Ya ad Tia (010) estimators of populatio mea i simple radom samplig give above, we proposed the estimators based o these metioed estimators of populatio mea i adaptive cluster samplig as, µy + ρ wx.wy ȳ acr4 = ȳ ac (7) x ac + ρ wx.wy µ ȳ acr5 = ȳ x ac x (8) ac β(wx )µ x + β 1(wx ) ȳ acr6 = ȳ ac (9) β (wx ) x ac + β 1(wx ) ) ȳ acr7 = ȳ ac ( β1(wx )µ x + β (wx ) β 1(wx ) x ac + β (wx ) (30) c 016 SP atural Scieces Publishig Cor.
5 J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) / 5 Usig the method discussed bhutima (013), the mea square errors of above estimators up to the first order of approximatios respectively are, MSE(ȳ acr4 ) MSE(ȳ acr5 ) MSE(ȳ acr6 ) MSE(ȳ acr7 ) µ [ wy + θw4 C wx θ ] w4ρ wx.w w wx µ [ wy + 4Cwx ] 4ρ wx.w w wx µ [ wy + θw6 C wx θ ] w6ρ wx.w w wx µ [ wy + θw7c wx ] θ w7 ρ wx.w w wx (31) (3) (33) (34) where θ w4 = µ x µ x β µ x +ρ wx.wy, θ w6 = (wx) µ x β (wx) +β,θ w7 = 1(wx) µ x β 1(wx) µ x β 1(wx) +β (wx) 5 umerical Illustratio I this sectio, the simulatio x values ad y values from Chutima ad Kumpho (008) were studied. The data statistics of this populatios were show i Table 1.We take the sample size as =0 i Table, value of MSE which are computed usig equatios i Sectio 4, are give. Table 1: Data Statistics = 400 =0 µ y = 1.5 µ x = S y = θ 1 = S wy = 3.56 θ w1 = S x =.400 θ = S wx = θ w = S xy = θ 3 = 0.04 S wx.wy = 6.48 θ w3 = ρ xy = θ 4 = ρ wx.wy = 0.96 θ w4 = C y = θ 6 = C wy =.914 θ w6 = C x = 4.35 θ 7 = C wx = θ w7 = β 1(x) = 6.83 β (x) = β 1(wx ) = β (wx ) = Table : MSE Values of Estimators MSE(ȳ R1 )=0.966 MSE(ȳ R )=0.407 MSE(ȳ R3 )=1.117 MSE(ȳ R4 )=0.57 MSE(ȳ R5 )=1.904 MSE(ȳ R6 )=0.311 MSE(ȳ R7 )=1.068 MSE(ȳ acr1 )=0.435 MSE(ȳ acr )=0.309 MSE(ȳ acr3 )=0.595 Proposed Estimators MSE(ȳ acr4 )=0.1 MSE(ȳ acr5 )=1.010 MSE(ȳ acr6 )=0.093 MSE(ȳ acr7 )= Coclusio I the preset mauscript, we developed ew ratio type estimators for the estimatio of populatio mea by usig auxiliary iformatio i adaptive cluster samplig scheme. The bias ad the mea squared error of proposed estimators have bee obtaied up to the first order of approximatio. A empirical study is carried out ad from the estimated MSE of the estimators i Table-, it is clear that if the populatio is rare ad hidde clustered populatio, all estimators i adaptive cluster samplig are more efficiet tha the estimators of populatio mea i simple radom samplig, give the same coditio. Further amog the all metioed estimators of populatio mea alog with all proposed estimators i adaptive cluster samplig, the proposed estimator, ȳ acr6 has the smallest estimated mea square error. Therefore, it should preferably be adopted for the estimatio of populatio mea i adaptive cluster samplig scheme. c 016 SP atural Scieces Publishig Cor.
6 6 S. K. Yadav et al.: Improved ratio estimators of populatio mea... Ackowledgemet The authors are very much thakful to the editor i chief ad the ukow leared referee for critically examiig the mauscript ad givig valuable suggestios to improve it. Refereces [1]. Chutima, Adaptive cluster samplig usig auxiliary variable,j.math.stat., 9, (013). []. Chutima ad B. Chutima, Ratio estimator usig auxiliary variable for adaptive cluster samplig, Joural of Thai Statistical Associatio, 6(),41-56 (008) [3] W.G.Cochra, The estimatio of the yields of the cereal experimets by samplig for the ratio of grai to total produce, Jour. Agri. Sci.,59,15-16 (1940). [4] A.L. Dryver ad C. Chao, Ratio estimators i adaptive cluster samplig, Evirometrics,18, (1967). [5] C. Kadilar, ad H. Cigi, A study o the chai ratio-type estimator,hacettepe Joural of Mathematics ad Statistics,3, (003). [6] H. P. Sigh ad R. Tailor, Use of kow correlatio coefficiet i estimatig the fiite populatio mea, Statistics i Trasitio,6, (003). [7] B.V.S. Sisodia ad V.K. Dwivedi, A modified ratio estimator usig coefficiet of variatio of auxiliary variable, J. Idia Soc. Agric. Statist., 33, (1981). [8] S.K. Thompso, Adaptive cluster samplig, J. Am. Statist. Assoc., 85, (1990). [9] L.. Upadhyaya ad H.P. Sigh, Use of trasformed auxiliary variable i estimatig the fiite populatio mea,biometri. J., 41, (1999). [10] Z. Ya ad B. Tia, Ratio Method to the Mea Estimatio Usig Co-efficiet of Skewess of Auxiliary Variable,, ICICA 010, Part II,CCIS 106, (010). Subhash Kumar Yadav is workig as assistat professor i the Departmet of Mathematics ad Statistics, Dr Ram Maohar Lohia Avadh Uiversity, Faizabad. He has got published may research papers i the field samplig of Statistics. Sheela Misra, the supervisor of Dr Subhash Kumar Yadav is workig as Professor i the Departmet of Statistics, Uiversity of Luckow, Luckow. She is a very good academicia as well as the admiistrator. She has successfully orgaized may iteratioal cofereces i Statistics. She has doe a lot of work i differet field of Statistics. Sat Shara Mishra is Associate Professor at Departmet of Mathematics ad Statistics, Dr Ram Maohar Lohia Avadh Uiversity, Faizabad. He is a seior faculty i the Departmet. He has supervised may Ph.D. i differet fields of Mathematics ad Statistics. He has got published a lot of research papers i differet jourals of repute. ipapor Chutima is workig as seior Faulty i the Departmet of Mathematics, Mahasarakham Uiversity, Maha Sarakham, Thailad. She is doig very good work i the field samplig of Statistics. She has got published may research papers i differet reputed jourals of Statistics. c 016 SP atural Scieces Publishig Cor.
Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable
Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya
More informationModified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis
America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry
More informationA General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)
Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig
More informationAlternative Ratio Estimator of Population Mean in Simple Random Sampling
Joural of Mathematics Research; Vol. 6, No. 3; 014 ISSN 1916-9795 E-ISSN 1916-9809 Published by Caadia Ceter of Sciece ad Educatio Alterative Ratio Estimator of Populatio Mea i Simple Radom Samplig Ekaette
More informationImproved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 4 Issue 2 Versio.0 Year 204 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA
More informationJambulingam Subramani 1, Gnanasegaran Kumarapandiyan 2 and Saminathan Balamurali 3
ISSN 1684-8403 Joural of Statistics Volume, 015. pp. 84-305 Abstract A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of
More informationA Family of Unbiased Estimators of Population Mean Using an Auxiliary Variable
Advaces i Computatioal Scieces ad Techolog ISSN 0973-6107 Volume 10, Number 1 (017 pp. 19-137 Research Idia Publicatios http://www.ripublicatio.com A Famil of Ubiased Estimators of Populatio Mea Usig a
More informationEstimation of the Population Mean in Presence of Non-Response
Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, 537 548 DOI: 0.535/CKSS.0.8.4.537 Estimatio of the Populatio Mea i Presece of No-Respose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,
More informationAbstract. Ranked set sampling, auxiliary variable, variance.
Hacettepe Joural of Mathematics ad Statistics Volume (), 1 A class of Hartley-Ross type Ubiased estimators for Populatio Mea usig Raked Set Samplig Lakhkar Kha ad Javid Shabbir Abstract I this paper, we
More informationNew Ratio Estimators Using Correlation Coefficient
New atio Estimators Usig Correlatio Coefficiet Cem Kadilar ad Hula Cigi Hacettepe Uiversit, Departmet of tatistics, Betepe, 06800, Akara, Turke. e-mails : kadilar@hacettepe.edu.tr ; hcigi@hacettepe.edu.tr
More informationVaranasi , India. Corresponding author
A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet
More informationAClassofRegressionEstimatorwithCumDualProductEstimatorAsIntercept
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 15 Issue 3 Versio 1.0 Year 2015 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic.
More informationChain ratio-to-regression estimators in two-phase sampling in the presence of non-response
ProbStat Forum, Volume 08, July 015, Pages 95 10 ISS 0974-335 ProbStat Forum is a e-joural. For details please visit www.probstat.org.i Chai ratio-to-regressio estimators i two-phase samplig i the presece
More informationResearch Article An Alternative Estimator for Estimating the Finite Population Mean Using Auxiliary Information in Sample Surveys
Iteratioal Scholarly Research Network ISRN Probability ad Statistics Volume 01, Article ID 65768, 1 pages doi:10.50/01/65768 Research Article A Alterative Estimator for Estimatig the Fiite Populatio Mea
More informationSome Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation
; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 097-70)]; ISSN 0975-556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial Ratio-Product
More informationEnhancing ratio estimators for estimating population mean using maximum value of auxiliary variable
J.Nat.Sci.Foudatio Sri Laka 08 46 (: 45-46 DOI: http://d.doi.org/0.408/jsfsr.v46i.8498 RESEARCH ARTICLE Ehacig ratio estimators for estimatig populatio mea usig maimum value of auiliar variable Nasir Abbas,
More informationEnhancing the Mean Ratio Estimators for Estimating Population Mean Using Non-Conventional Location Parameters
evista Colombiaa de Estadística Jauary 016, Volume 39, Issue 1, pp. 63 to 79 DOI: http://dx.doi.org/10.15446/rce.v391.55139 Ehacig the Mea atio Estimators for Estimatig Populatio Mea Usig No-Covetioal
More informationJournal of Scientific Research Vol. 62, 2018 : Banaras Hindu University, Varanasi ISSN :
Joural of Scietific Research Vol. 6 8 : 3-34 Baaras Hidu Uiversity Varaasi ISS : 447-9483 Geeralized ad trasformed two phase samplig Ratio ad Product ype stimators for Populatio Mea Usig uiliary haracter
More informationImproved exponential estimator for population variance using two auxiliary variables
OCTOGON MATHEMATICAL MAGAZINE Vol. 7, No., October 009, pp 667-67 ISSN -5657, ISBN 97-973-55-5-0, www.hetfalu.ro/octogo 667 Improved expoetial estimator for populatio variace usig two auxiliar variables
More informationDual to Ratio Estimators for Mean Estimation in Successive Sampling using Auxiliary Information on Two Occasion
J. Stat. Appl. Pro. 7, o. 1, 49-58 (018) 49 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.18576/jsap/070105 Dual to Ratio Estimators for Mea Estimatio i Successive
More informationSYSTEMATIC SAMPLING FOR NON-LINEAR TREND IN MILK YIELD DATA
Joural of Reliability ad Statistical Studies; ISS (Prit): 0974-804, (Olie):9-5666 Vol. 7, Issue (04): 57-68 SYSTEMATIC SAMPLIG FOR O-LIEAR TRED I MILK YIELD DATA Tauj Kumar Padey ad Viod Kumar Departmet
More informationThe Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution
Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 1263-1277 HIKARI Ltd, www.m-hikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters
More informationApproximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece
More informationEstimation of Population Ratio in Post-Stratified Sampling Using Variable Transformation
Ope Joural o Statistics, 05, 5, -9 Published Olie Februar 05 i SciRes. http://www.scirp.org/joural/ojs http://dx.doi.org/0.436/ojs.05.500 Estimatio o Populatio Ratio i Post-Stratiied Samplig Usig Variable
More informationEstimation of Population Mean in Presence of Non-Response in Double Sampling
J. Stat. Appl. Pro. 6, No. 2, 345-353 (2017) 345 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.18576/jsap/060209 Estimatio of Populatio Mea i Presece of No-Respose
More informationOn ratio and product methods with certain known population parameters of auxiliary variable in sample surveys
Statistics & Operatios Research Trasactios SORT 34 July-December 010, 157-180 ISSN: 1696-81 www.idescat.cat/sort/ Statistics & Operatios Research c Istitut d Estadística de Cataluya Trasactios sort@idescat.cat
More informationResearch Article A Two-Parameter Ratio-Product-Ratio Estimator Using Auxiliary Information
Iteratioal Scholarly Research Network ISRN Probability ad Statistics Volume, Article ID 386, 5 pages doi:.54//386 Research Article A Two-Parameter Ratio-Product-Ratio Estimator Usig Auxiliary Iformatio
More informationInvestigating the Significance of a Correlation Coefficient using Jackknife Estimates
Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN 2307-4531 (Prit & Olie) http://gssrr.org/idex.php?joural=jouralofbasicadapplied ---------------------------------------------------------------------------------------------------------------------------
More informationUse of Auxiliary Information for Estimating Population Mean in Systematic Sampling under Non- Response
Maoj K. haudhar, Sachi Malik, Rajesh Sigh Departmet of Statistics, Baaras Hidu Uiversit Varaasi-005, Idia Floreti Smaradache Uiversit of New Mexico, Gallup, USA Use of Auxiliar Iformatio for Estimatig
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationOn stratified randomized response sampling
Model Assisted Statistics ad Applicatios 1 (005,006) 31 36 31 IOS ress O stratified radomized respose samplig Jea-Bok Ryu a,, Jog-Mi Kim b, Tae-Youg Heo c ad Chu Gu ark d a Statistics, Divisio of Life
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 informationMethod of Estimation in the Presence of Nonresponse and Measurement Errors Simultaneously
Joural of Moder Applied Statistical Methods Volume 4 Issue Article 5--05 Method of Estimatio i the Presece of Norespose ad Measuremet Errors Simultaeousl Rajesh Sigh Sigh Baaras Hidu Uiversit, Varaasi,
More informationA Generalized Class of Estimators for Finite Population Variance in Presence of Measurement Errors
Joural of Moder Applied Statistical Methods Volume Issue Article 3 --03 A Geeralized Class of Estimators for Fiite Populatio Variace i Presece of Measuremet Errors Praas Sharma Baaras Hidu Uiversit, Varaasi,
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationDeveloping Efficient Ratio and Product Type Exponential Estimators of Population Mean under Two Phase Sampling for Stratification
America Joural of Operatioal Researc 05 5: -8 DOI: 0.593/j.ajor.05050.0 Developig Efficiet Ratio ad Product Type Epoetial Eimators of Populatio Mea uder Two Pase Samplig for Stratificatio Subas Kumar adav
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationCOM-Poisson Neyman Type A Distribution and its Properties
COM-oisso Neyma Type A Distributio ad its roperties S. Thilagarathiam Departmet of Mathematics Nehru Memorial College Trichy, Tamil Nadu, Idia. rathiam.thilaga93@gmail.com V. Saavithri Departmet of Mathematics
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality 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 informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationREVISTA INVESTIGACION OPERACIONAL VOL. 35, NO. 1, 49-57, 2014
EVISTA IVESTIGAIO OPEAIOAL VOL. 35, O., 9-57, 0 O A IMPOVED ATIO TYPE ESTIMATO OF FIITE POPULATIO MEA I SAMPLE SUVEYS A K P Swai Former Professor of Statistics, Utkal Uiversit, Bhubaeswar-7500, Idia ABSTAT
More informationModeling and Estimation of a Bivariate Pareto Distribution using the Principle of Maximum Entropy
Sri Laka Joural of Applied Statistics, Vol (5-3) Modelig ad Estimatio of a Bivariate Pareto Distributio usig the Priciple of Maximum Etropy Jagathath Krisha K.M. * Ecoomics Research Divisio, CSIR-Cetral
More informationGeneralized Exponential Type Estimator for Population Variance in Survey Sampling
Revista Colombiaa de Estadística Juio 2014, volume 37, o. 1, pp. 211 a 222 Geeralized Expoetial Type Estimator for Populatio Variace i Survey Samplig Estimadores tipo expoecial geeralizado para la variaza
More informationJ. Stat. Appl. Pro. Lett. 2, No. 1, (2015) 15
J. Stat. Appl. Pro. Lett. 2, No. 1, 15-22 2015 15 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.12785/jsapl/020102 Martigale Method for Rui Probabilityi
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationEstimation of Gumbel Parameters under Ranked Set Sampling
Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 11-2014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com
More informationStability Analysis of the Euler Discretization for SIR Epidemic Model
Stability Aalysis of the Euler Discretizatio for SIR Epidemic Model Agus Suryato Departmet of Mathematics, Faculty of Scieces, Brawijaya Uiversity, Jl Vetera Malag 6545 Idoesia Abstract I this paper we
More informationCentral limit theorem and almost sure central limit theorem for the product of some partial sums
Proc. Idia Acad. Sci. Math. Sci. Vol. 8, No. 2, May 2008, pp. 289 294. Prited i Idia Cetral it theorem ad almost sure cetral it theorem for the product of some partial sums YU MIAO College of Mathematics
More informationAkaike Information Criterion and Fourth-Order Kernel Method for Line Transect Sampling (LTS)
Appl. Math. If. Sci. 10, No. 1, 267-271 (2016 267 Applied Mathematics & Iformatio Scieces A Iteratioal Joural http://dx.doi.org/10.18576/amis/100127 Akaike Iformatio Criterio ad Fourth-Order Kerel Method
More informationAccess to the published version may require journal subscription. Published with permission from: Elsevier.
This is a author produced versio of a paper published i Statistics ad Probability Letters. This paper has bee peer-reviewed, it does ot iclude the joural pagiatio. Citatio for the published paper: Forkma,
More informationConfidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation
Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. 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 informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationLinear Regression Models
Liear Regressio Models Dr. Joh Mellor-Crummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
More informationA Generalized Class of Unbiased Estimators for Population Mean Using Auxiliary Information on an Attribute and an Auxiliary Variable
Iteratioal Joural of Computatioal ad Applied Mathematics. ISSN 89-4966 Volume, Number 07, pp. -8 Research Idia ublicatios http://www.ripublicatio.com A Geeralized Class of Ubiased Estimators for opulatio
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 informationBayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function
Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume 12, Number 4 (217), pp. 791-796 Research Idia Publicatios http://www.ripublicatio.com Bayesia ad E- Bayesia Method of Estimatio of Parameter
More informationGeneralization of Samuelson s inequality and location of eigenvalues
Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,
More informationRAINFALL PREDICTION BY WAVELET DECOMPOSITION
RAIFALL PREDICTIO BY WAVELET DECOMPOSITIO A. W. JAYAWARDEA Departmet of Civil Egieerig, The Uiversit of Hog Kog, Hog Kog, Chia P. C. XU Academ of Mathematics ad Sstem Scieces, Chiese Academ of Scieces,
More informationConstrained Inverse Adaptive Cluster Sampling
Joural of Of cial Statistics, Vol. 19, No. 1, 2003, pp. 45±57 Costraied Iverse Adaptive Cluster Samplig Emilia Rocco 1 Adaptive cluster samplig ca be a useful desig for samplig rare ad clustered populatios.
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 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 informationControl Charts for Mean for Non-Normally Correlated Data
Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article 5 5-1-017 Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies
More informationMechanical Efficiency of Planetary Gear Trains: An Estimate
Mechaical Efficiecy of Plaetary Gear Trais: A Estimate Dr. A. Sriath Professor, Dept. of Mechaical Egieerig K L Uiversity, A.P, Idia E-mail: sriath_me@klce.ac.i G. Yedukodalu Assistat Professor, Dept.
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 informationComparison of Minimum Initial Capital with Investment and Non-investment Discrete Time Surplus Processes
The 22 d Aual Meetig i Mathematics (AMM 207) Departmet of Mathematics, Faculty of Sciece Chiag Mai Uiversity, Chiag Mai, Thailad Compariso of Miimum Iitial Capital with Ivestmet ad -ivestmet Discrete Time
More informationFastest mixing Markov chain on a path
Fastest mixig Markov chai o a path Stephe Boyd Persi Diacois Ju Su Li Xiao Revised July 2004 Abstract We ider the problem of assigig trasitio probabilities to the edges of a path, so the resultig Markov
More informationChapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p
Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE Part 3: Summary of CI for µ Cofidece Iterval for a Populatio Proportio p Sectio 8-4 Summary for creatig a 100(1-α)% CI for µ: Whe σ 2 is kow ad paret
More informationSome Results on Certain Symmetric Circulant Matrices
Joural of Iformatics ad Mathematical Scieces Vol 7, No, pp 81 86, 015 ISSN 0975-5748 olie; 0974-875X prit Pulished y RGN Pulicatios http://wwwrgpulicatioscom Some Results o Certai Symmetric Circulat Matrices
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 informationPOWER AKASH DISTRIBUTION AND ITS APPLICATION
POWER AKASH DISTRIBUTION AND ITS APPLICATION Rama SHANKER PhD, Uiversity Professor, Departmet of Statistics, College of Sciece, Eritrea Istitute of Techology, Asmara, Eritrea E-mail: shakerrama009@gmail.com
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 information10-701/ Machine Learning Mid-term Exam Solution
0-70/5-78 Machie Learig Mid-term Exam Solutio Your Name: Your Adrew ID: True or False (Give oe setece explaatio) (20%). (F) For a cotiuous radom variable x ad its probability distributio fuctio p(x), it
More informationIntroducing a Novel Bivariate Generalized Skew-Symmetric Normal Distribution
Joural of mathematics ad computer Sciece 7 (03) 66-7 Article history: Received April 03 Accepted May 03 Available olie Jue 03 Itroducig a Novel Bivariate Geeralized Skew-Symmetric Normal Distributio Behrouz
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 informationWorksheet 23 ( ) Introduction to Simple Linear Regression (continued)
Worksheet 3 ( 11.5-11.8) Itroductio to Simple Liear Regressio (cotiued) This worksheet is a cotiuatio of Discussio Sheet 3; please complete that discussio sheet first if you have ot already doe so. This
More informationSolution of Differential Equation from the Transform Technique
Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from
More informationEvapotranspiration Estimation Using Support Vector Machines and Hargreaves-Samani Equation for St. Johns, FL, USA
Evirometal Egieerig 0th Iteratioal Coferece eissn 2029-7092 / eisbn 978-609-476-044-0 Vilius Gedimias Techical Uiversity Lithuaia, 27 28 April 207 Article ID: eviro.207.094 http://eviro.vgtu.lt DOI: https://doi.org/0.3846/eviro.207.094
More informationA New Mixed Randomized Response Model
Iteratioal Joural of Busiess ad Social Sciece ol No ; October 00 A New Mixed adomized espose Model Aesha Nazuk NUST Busiess School Islamabad, Paksta E-mail: Aeshaazuk@bsedupk Phoe: 009-5-9085-367 Abstract
More informationBenaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco
EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha
More informationPreponderantly increasing/decreasing data in regression analysis
Croatia Operatioal Research Review 269 CRORR 7(2016), 269 276 Prepoderatly icreasig/decreasig data i regressio aalysis Darija Marković 1, 1 Departmet of Mathematics, J. J. Strossmayer Uiversity of Osijek,
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 informationA goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality
A goodess-of-fit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
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 informationHigher-order iterative methods by using Householder's method for solving certain nonlinear equations
Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem
More informationEstimation of Backward Perturbation Bounds For Linear Least Squares Problem
dvaced Sciece ad Techology Letters Vol.53 (ITS 4), pp.47-476 http://dx.doi.org/.457/astl.4.53.96 Estimatio of Bacward Perturbatio Bouds For Liear Least Squares Problem Xixiu Li School of Natural Scieces,
More informationInformation-based Feature Selection
Iformatio-based Feature Selectio Farza Faria, Abbas Kazeroui, Afshi Babveyh Email: {faria,abbask,afshib}@staford.edu 1 Itroductio Feature selectio is a topic of great iterest i applicatios dealig with
More informationResearch Article A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property
Discrete Dyamics i Nature ad Society Volume 2011, Article ID 360583, 6 pages doi:10.1155/2011/360583 Research Article A Note o Ergodicity of Systems with the Asymptotic Average Shadowig Property Risog
More informationON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS
Joural of Algebra, Number Theory: Advaces ad Applicatios Volume, Number, 00, Pages 7-89 ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS OLCAY KARAATLI ad REFİK KESKİN Departmet
More informationLecture 11 Simple Linear Regression
Lecture 11 Simple Liear Regressio Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech Midterm 2 mea: 91.2 media: 93.75 std: 6.5 2 Meddicorp
More informationWarped, Chirp Z-Transform: Radar Signal Processing
arped, Chirp Z-Trasform: Radar Sigal Processig by Garimella Ramamurthy Report o: IIIT/TR// Cetre for Commuicatios Iteratioal Istitute of Iformatio Techology Hyderabad - 5 3, IDIA Jauary ARPED, CHIRP Z
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationSince X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain
Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the
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 informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
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 information