Introduction to Probability and Statistics 131B Lecture 5

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Introduction to Probability and Statistics 131B Lecture 5
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Mathematics Lecture/Conference Right angle Student's t-test
Category of being Frequency Randomization Lecture/Conference Variety (linguistics) Multiplication sign Forcing (mathematics) Mereology Pressure Algebra Theory Resultant
Category of being Many-sorted logic Lecture/Conference Term (mathematics) Multiplication sign Archaeological field survey Energy level
Estimator Dependent and independent variables Lecture/Conference Maxima and minima Numerical analysis
Area Binomial heap Estimator Lecture/Conference Multiplication sign Square number Variance Right angle Lie group Mereology Grothendieck topology
Category of being Estimator Lecture/Conference
Area Logical constant Dependent and independent variables Functional (mathematics) Regulator gene Multiplication sign Mortality rate Numerical analysis Estimator Mathematics Lecture/Conference Different (Kate Ryan album) Linearization Right angle
Observational study Autocovariance Weight Projective plane Sampling (statistics) Stress (mechanics) Water vapor Numerical analysis Arithmetic mean Strategy game Lecture/Conference Different (Kate Ryan album) Term (mathematics) Average Right angle Object (grammar) Family Arc (geometry) Random variable
Group action Randomization State of matter Weight Multiplication sign Sampling (statistics) Independence (probability theory) Numerical analysis Estimator Strategy game Positional notation Lecture/Conference Different (Kate Ryan album) Average Analogy
Logical constant Trail Multiplication sign Sampling (statistics) Independence (probability theory) Basis <Mathematik> Mortality rate Rule of inference Explosion Lecture/Conference Different (Kate Ryan album) Right angle Series (mathematics) Pressure
Axiom of choice Building Group action Functional (mathematics) Divisor Equaliser (mathematics) 1 (number) Event horizon Frequency Estimator Lecture/Conference Different (Kate Ryan album) Thermal fluctuations Moving average Ranking Modulform Social class Area Constraint (mathematics) Sampling (statistics) Maxima and minima Mortality rate Cartesian coordinate system Variable (mathematics) Numerical analysis Order (biology)
Functional (mathematics) Lecture/Conference Direction (geometry) Square number Staff (military)
Estimator Square number Theory of everything Nichtlineares Gleichungssystem
Covering space Observational study Multiplication sign Connectivity (graph theory) Blind spot (vehicle) Forcing (mathematics) Model theory Archaeological field survey Sampling (statistics) Mortality rate Price index Numerical analysis Grothendieck topology Lecture/Conference Summierbarkeit Nichtlineares Gleichungssystem Musical ensemble
Axiom of choice Multiplication sign Weight Sheaf (mathematics) Sampling (statistics) Mereology Event horizon Numerical analysis Arithmetic mean Different (Kate Ryan album) Term (mathematics) Network topology Figurate number Table (information) Resultant
Ocean current Randomization Building Sigma-algebra Weight Decision theory Lemma (mathematics) Sampling (statistics) Independence (probability theory) Parameter (computer programming) Rule of inference Approximation Numerical analysis Category of being Estimator Many-sorted logic Lecture/Conference Network topology Reduction of order Summierbarkeit Mathematical optimization Random variable
Building Functional (mathematics) Group action Blind spot (vehicle) Multiplication sign 1 (number) Water vapor Parameter (computer programming) Rule of inference Graph coloring Event horizon Grothendieck topology Expected value Estimator Population density Causality Many-sorted logic Lecture/Conference Term (mathematics) Different (Kate Ryan album) Average Square number Energy level Set theory Alpha (investment) Random variable Potenz <Mathematik> Sigma-algebra Gamma function Moment (mathematics) Expression Curve Sampling (statistics) Independence (probability theory) Line (geometry) Numerical analysis Category of being Arithmetic mean Network topology Order (biology) Charge carrier Normal (geometry) Right angle Cycle (graph theory) Directed graph
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personnel for the race for the and here with Luelle and here we wouldn't factor this out now think OK so that's what it would look like ever Saturday as more typical so all our given these races that he get it but the English which that the 4 after all I think so but suppose that we know their existing ones elsewhere the thing you could take a sample besides you don't have unlimited resources some or all of the you choose and 1 through and how well it see what effect on does need choice of then went to a mail affect whether they be estimate at Rs it is unbiased as effect wears out by expected by expires new doesn't depend on the choice of the end the land think of it we had to worry about making estimated as far as unbiased by a peculiar choice of and wondering how does affect be variants yes it does doesn't like this doesn't affect backward does affect us and what would you like to do would you like to be big or small here estimates not so we wanna take the various and Isabel's so the race is a small spot the so that the race sex bias is minimized took a order we do that what is a function of capital L variables and we wanna find its minimum but its subject to constraints constraints ears were taking a sample besides rear 1 of minimize ever and 1 and 2 Iran's capital equal Song who will be wonder Caballo stigma little else where'd W. well over and Isabel subject to are you pull on but 4 of but but they so this is a minimization problem probably seen this kind of form after rate for
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more provides sample or do sampling in such a way that pick this many from Telstra that will minimize get better so there's a room table the book think it's example a from Section 7 5 don't were the hospital example again hospitals are displayed the for nearly equally sized strata from small us largess in terms of number bets the number of hospitals in each Estrada is about the same because the number hospitals wasn't was a wife or what you'll hear so instead a 98 hospitals and the fewest number than the next guy would be composed of next 98 hospitals In terms of number of every hostile Mystranan has more beds in every hospital and to be the next 98 biggest very hostile here has more bets on any hostile there or there and these are the biggest 99 hospitals terms of number and not 90 acres numbering starter what many members of its members trees and each summer for the weights are also for telomerase 393 Sosa's . 2 4 9 but is a scissors and Isabel over capital and as 98 this is 98 over 393 Sabine 98 Over 393 To the means and each 1 is 1 . 9 26 . 9 9 56 . 3 1 5 9 2 1 . 2 as well choices 103 what for 204 . 8 243 . 5 4 1 9 . 2 and this the account results this is a need for he said in different races and different strata of of what we eventually force the respected the hospital has more events Sierra's as last year's 191 then there be more are so it is a number of discharges here the way we don't know the number of such as being a number of distrust among so the number because what they add In fact that this is the starter with the largest ever and barracks there was barges and so will you would use them this number and this number 2 From what was off sampling numbers for the 1st struck in the face of whites skirting these numbers quite corresponding wants to vote the Dragons just To death for Frank buff by U.S. Vermont sorry does not quite as stands at mouth bought by that figures ratio of his 2 she or not so want to decide what the sample size be sample size number is from Fatah 1 should be quite all part of what 106 times others may not be new user but maybe it proposes that this number taken from started to should be . 2 1 0 times Over Missile
minimize that they make a small by through readjusting samples that sort of all sizes from the various trust that's why you would moves stratified sampling and give you better and well Was it not followed up close you but this was it
did notice please edit truer things but this was you that you would not barings known sample but if you don't know the stone but somehow used 1 but not of the year don't always know things were them Our estimates release using samples but that means Avery samples and now do that peace is so what what do people do with the lack of knowledge of the sickness they do what they call proportional sap arm a disused E now but it or other races in the straps now it and we would want to be I need a sample from they'll strata the proportion of individuals L StrataCom hurdle population that these up when he gets all year these numbers up to set up these numbers what some of the and capitalist dollars Kaplan some of these becomes 1 the sellers become aware of don't most singers where users are you give up optimality but what else can you do reasonable OK so let's leave that behind and start the plant on the she but not OK so 0 end this stub chapter we consider our problems like this it's says she now that is because it is not is it is not it is a very important problem so why would the steepest honorable why but individually 1 because but also want 1 branches of the fact so we really have a whole lot of independence and rebels that is what we were in the red rules will allow leaves want honorary will briefly ran warned if Paulson In the next hour 0 doesn't so probably be the sum of all these random variables probably delete falls small so number of weeks has probably would be some reasonable size numbers so we use the EPA's approximation to Bernoulli and decided that should be Sunriver so there's some mathematical or scientific argument that this decision should be passed on to replaced please only from a tree by alpha particles by the book they use of the substance of their humor and France ERIC IUM itself articles well Oracle's admitted an hour would be be sauna and rulers based on mathematical reasoning but now our once they decided that we want an estimated 3 creditor in the sovereign E and there's another for example the book but where they talk about the current across a cell wall furnaces current by opening closing a lot of the channels and on things perhaps through the Charles with some probabilities polls of some other buildings there many many of them so do the would be decided by only these channels are open and so were adding up a bunch of Bernoulli random and so we might use a normal approximation of the net Red across cell wall would be by a R Warren to or what but lot aside what mu and sigma squared off many might in fact take some reduction of the properties of these channels that are opening was army of them are much faster would short order brought back to lot it will be much could sue of it a Avenue some
Carmine the light turns green you wanna go right away but last summer will 1st car moves after some literal time later arose a 2nd car I'm very carriers you wanna revenue it's true because like white and red maybe even attempted the hot might say something anyway so there's some random on time we have to wait but if to show and this is a gamma rebel last that's 1 reason the going there reasons smile was the gamma and rural what is hardly gamma-ray rubles arise 1 way some of exponential independent expert and rubles think of that time from the movement of the 1st returning of the light green to the movement 1st being exponential honorary and then from the will of the 1st cartel assigning caramel as big another exponential honorable etc then you're order line with me some random will soon be 70 like a 2nd my 1st building are you wait so typically there was a sum of independent independent gets Vestron arose as gamma was might be a lot I ever but then the question water often soccer we make these things we take samples Anders Our chapter will take independent samples of the number of world estimated to be predators of the underlying this is still alive times they'll be arguments and tell you what shape or type of underlying this mission now and then you have to try find parameters for that type of on their normal there there's another argument for capped a E yet the case here are debuted watch 1 oak tree for a while and we are ready to take a lot of different oak tree for our Member differ oak tree may be tried victim them all same this is the number of these from Oak Tree wanted an hour hour and now or you might observed and different cells and measure the current on the Cross won't sell 1 to install 2 etc here you might serve and different drivers on the way so the problem of estimating parameters can sometimes be done it is to world also by expressing the parameters as functions at the moment she did say it was close curve for off so effects as on Trevorrow ladder the 1st moment of Exeter's level cycle that the 1st moment of maybe we use new the subscript or we use newest script new to the site of the 1st moment of acts as news of want equal emphasis on this case express expressed the parameters as a function of being 1st but and what is it what is a function f of me 1 half of acts that expose axes normal with mean new and racing was where everyone is expected by and you too is expected by squared Harper expressed new and was burdened terms of these moments is me 1 right this case as of new 1 is new ones have what we need to functions as time Singh was weirdo expressed as a function of new wanted me to well they were squared is new to minus new square the 2nd bias 1st onwards there we express the proud annually in terms of the Turns out just personal savior every wraps the 2nd parameter sigma squared in terms of the moment and the 1st example of its the gamma but the expected value acts it is remembering because I was passed in 1 of the final set and what's expected of square so a new 1 and mutual all right now what's a density fire but squared and what about the Dead Sea by X squared we all over again will go and what's is scam love alpha plus 2 2 over lender to the which could be
damn Elf was through it is L 4 plus 1 times at times camel them well for us to overdo would be of terms offer their Cleverdale feathers when the 1st 2 letters OK and so on how who expressed some real express Alvarez a function at 1 of new ones and too and they want express members of different culture of you ones mutual yes but we are express leather Valez functionally what would do that at you can't of course is a trick raisins into unknowns Irish Casarez Q. but so what I take a mere 1 divided by this Miners weird I get less than what you check now Pierce's Elfers land at times me 1 in year ago they expressed land in shares of me wanted me to fight is not quite this time 2 that's ever 1 everyone Sony to these cases even express fairly prayers Media parameters in terms of the moments now when you take samples you can being smaller and replied those estimates into these functions the will Logan estimates for these parameters as for the method of moments but Dubai it Quebec we want to the talks the text key for us Frank In this sense that asked if he she was slower large numbers so they are but new 1 is an average of these random variables finally expectations that average worker burst of expectation Texas too was an average of independent around rules if there's an expectations it would be mutual and of our sources to have defensive end court we some you what happened to have abused want to an end if we express parameters as functions of these moments we can plug in business for the moment we get as much for the rapporteur so for example must sort of Versailles the 1st is sort a parameter to reverse moments as the 1st moments toward Our landowner that both knew what that is later for a while In the case of 4 a normal and they're all face color OAS make new causes the new 1 alone packets BY hats but an auto estimates say well report new 1 had your meal to that year I just need to have my have squared she 4 To 0 for court for and Chris for normal and finally for gamma hard estimator Alpha 12 will be joined squared to have my near 1 expert at he and remember recorded and you want me to add years gives new 1 over me to have my eyes U.S. work it sigh up to 4 next time will talk about being statistical properties of these estimators using a lot large numbers of Zora saddled the thermal or odd lot rare events things so we treat these as really this these estimators David method of