Introduction to Probability and Statistics 131B Lecture 5
Formal Metadata
Title 
Introduction to Probability and Statistics 131B Lecture 5

Title of Series  
Part Number 
5

Number of Parts 
15

Author 

License 
CC Attribution  ShareAlike 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. 
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Publisher 

Release Date 
2013

Language 
English

Content Metadata
Subject Area 
00:00
Mathematics
Right angle
Student's ttest
01:22
Category of being
Frequency
Randomization
Variety (linguistics)
Multiplication sign
Forcing (mathematics)
Mereology
Pressure
Algebra
Theory
Resultant
07:38
Category of being
Manysorted logic
Term (mathematics)
Multiplication sign
Archaeological field survey
Energy level
10:09
Estimator
Dependent and independent variables
Maxima and minima
Number
13:43
Area
Binomial heap
Estimator
Multiplication sign
Square number
Variance
Right angle
Lie group
Mereology
Grothendieck topology
16:08
Category of being
Estimator
17:27
Logical constant
Area
Functional (mathematics)
Dependent and independent variables
Regulator gene
Multiplication sign
Mortality rate
Number
Estimator
Mathematics
Different (Kate Ryan album)
Linearization
Right angle
25:13
Observational study
Autocovariance
Weight
Projective plane
Sampling (statistics)
Stress (mechanics)
Water vapor
Number
Arithmetic mean
Strategy game
Different (Kate Ryan album)
Term (mathematics)
Average
Right angle
Object (grammar)
Family
Arc (geometry)
Random variable
33:36
Area
Group action
Randomization
State of matter
Multiplication sign
Weight
Sampling (statistics)
Independence (probability theory)
Number
Estimator
Strategy game
Positional notation
Different (Kate Ryan album)
Average
Analogy
38:20
Logical constant
Trail
Multiplication sign
Sampling (statistics)
Independence (probability theory)
Basis <Mathematik>
Mortality rate
Rule of inference
Explosion
Different (Kate Ryan album)
Right angle
Series (mathematics)
Pressure
42:34
Axiom of choice
Group action
Building
Functional (mathematics)
Divisor
Equaliser (mathematics)
1 (number)
Event horizon
Number
Frequency
Estimator
Different (Kate Ryan album)
Thermal fluctuations
Moving average
Ranking
Modulform
Social class
Area
Constraint (mathematics)
Sampling (statistics)
Maxima and minima
Mortality rate
Variable (mathematics)
Cartesian coordinate system
Order (biology)
53:36
Functional (mathematics)
Direction (geometry)
Square number
Staff (military)
55:37
Estimator
Square number
Theory of everything
Nichtlineares Gleichungssystem
58:01
Covering space
Observational study
Connectivity (graph theory)
Multiplication sign
Forcing (mathematics)
Blind spot (vehicle)
Model theory
Archaeological field survey
Sampling (statistics)
Price index
Mortality rate
Grothendieck topology
Number
Summierbarkeit
Nichtlineares Gleichungssystem
Musical ensemble
1:05:12
Axiom of choice
Multiplication sign
Weight
Sheaf (mathematics)
Sampling (statistics)
Mereology
Event horizon
Number
Arithmetic mean
Different (Kate Ryan album)
Term (mathematics)
Network topology
Figurate number
Table (information)
Resultant
1:11:59
Ocean current
Building
Randomization
Sigmaalgebra
Decision theory
Weight
Lemma (mathematics)
Sampling (statistics)
Independence (probability theory)
Parameter (computer programming)
Rule of inference
Approximation
Number
Category of being
Estimator
Manysorted logic
Network topology
Reduction of order
Summierbarkeit
Mathematical optimization
Random variable
1:22:05
Functional (mathematics)
Group action
Building
Multiplication sign
Blind spot (vehicle)
1 (number)
Set (mathematics)
Water vapor
Parameter (computer programming)
Event horizon
Rule of inference
Graph coloring
Grothendieck topology
Number
Expected value
Estimator
Population density
Manysorted logic
Causality
Term (mathematics)
Different (Kate Ryan album)
Average
Square number
Energy level
Alpha (investment)
Random variable
Potenz <Mathematik>
Sigmaalgebra
Gamma function
Moment (mathematics)
Expression
Curve
Sampling (statistics)
Independence (probability theory)
Line (geometry)
Category of being
Arithmetic mean
Network topology
Order (biology)
Charge carrier
Normal (geometry)
Right angle
Cycle (graph theory)
Directed graph
00:07
so I had a request From 1 of these students in the past too often 28 29 problem book because problem 30 users 28 cracked back or I find that wrote on these 2 problems right talk it so or maybe writers perhaps this is the
01:15
technique in and no of people answer questions truthfully when they might be embarrassed by answering questions they questions that he
01:23
used performanceenhancing drugs may NEW as fast as their red these are embarrassed part of the 1st thing you have to answer truthfully studied give you give them questioned randomly question 1 is that is a true characters a gay did a question to visit drove you didn't do it already laser show you have property a her as a true you don't have property a something like that so they have personally interviewed is given 1 of these 2 questions randomly and answers question 1 or questions to my theory of yours was recording answer doesn't know what questions the person answering and it personally realizes that Eurydice noticed there's result because there's a lot of so but this leaves a year time period or the problem of deciding what we need to find out for example calmly people today or covering people health care 1st aid from the outcome of so tour something like it cloying is fast he is probably the question was is asked to ever course 1 might with probability question to its for we do this month time and time again Apple or would be portion of it a a pause of answers 2 it where trial of the interview so we this out capital are from reviews to end on side chief but for self today or have characters that they have worked varieties are somehow to make some refers about cure so if we know the spa a 0 he parent Q what is the probability be responded answers yes but or are cooked go back well With the probability of a yes or his wife to demand that their very best guess what's probably of yes given the
06:43
question 1 is like every were nearly repeaters eh answer so the question is did you do aII or to have property I'll what forces populist has property a so probably answering yes ask question 1 would be that because person random but assume they assert truthfully and the spoke church people answer is a city that somebody at random what's probably property a worthy of a pressure that and will be probably yes there answering questions too yet 1
07:45
might assume because that's force that work not have property what's a probability that question 1 is the 1 asked me and the probably question to a sort asked 1 might hear sold the probably a yes answer would be the probability a yes answer given question 1 of asked probably question what asked plus a probability of a yes answer given question to asked the public question was asked others of being killed times 3 plus 1 minus Q times 1 might think and if back lot like that a 2nd hard out there vaccinating which plucked that's terms there are due I see a stone book surveys ahead fartlek beat terms fewer Davis listeners a member of a few times what well he and in the end minus 1 plus in what's left 1 West and that's probably a guest spots at the local level so
10:04
we can find to reduce all 4 Q to get Bess saw virtue
10:38
because that's the thing oranges Ali Larter finding estimates for both it she took or he at can salvage their daily but they can be summarized by FOR that's already is the proportion of respondents who answered yes well capital R is an estimator for little or no for this always Capitola are Parker with described as follows with and people we all ever and we record the information this way excise 1 if I we said yes 0 said no overseas said no and an are would be what 1 or the number of views interviewees Kansas this will be account of harmonious as 1 or number of interviewees times the number of guesses available portion of it yes what's a probability that stuff excise want sir probably
13:35
it's little and of course that's 1 less a public
13:43
excise 0 so what's expected value of Capitola on a day Specter value of some Agassi I wanna rent and expected by the son a lawyer over and time some of the expected values what's expected each turned its all are right wrote this expect delays are so they get are only time and time when they expected by AT & by an off and where a lot of areas of sold is too long what's in his vision of astronomer herbal binomial of primary demand are rights of expected life that would be an hour with the variants of this and ah times on my are for deliveries of this thing quite as if squares br 1 lies are over at and because this is a final milk from an arm injury be and bar oneliners are only multiply by this squares where comes saying variances ordering times and times are promise I thing site that was part of the or was a C C artsy was all find you or parts that just
15:46
part cleaners artsy parties finally estimator for ah therefore D what Steve all the various OK that's Physidae
16:09
follows a is the 2 of and now he is fine and estimator for so this is cabbala is an unbiased estimator for little R and has a nice various small when you hear a lot of developers build means there as much as some accuracy Woodward estimated that Joe as In the
16:48
final to force the population that for example property us while we look there we see how the news related to the real value Will are only witness may little are by what venerable capital R so if we define capital to that To me
17:30
caviar plus became swung over to obtain minus 1 that should be addressed Mero he only because this is are and when you Lara look you there and this is a pretty reasonable function of this variable right you add some abide by linear function fact of being able but what about the expected value of q as is an unbiased estimator Of those few yet well it also do the during that and what's expected I travel the lock but little are here and then I rewrite it way before it becomes a la he minus 1 over to buyers want and what set at little so it's an unbiased estimator Q a lot of areas of Chapel Hill deutch let me ask a question if C is a constant was rates posse Razorbacks wise at exactly the 5th saying doesn't change Charlotte spread out so it here we have something like that we're doing so are all over to appease minus 1 plus minus 1 over to a minus 1 area around plus a constant adding additional a constant doesn't change America would drop it and over a multiplier render a lot constant take the reins pull a concert but when it comes out it becomes square ever areas are Over there every day at the La oneliners are over and so how depict would you pass toss a coin doing this firmer you're doing these things would be point inside with me we're duties answer questions wonder question to at Delaware the race if he is path because of that and regulator said so it shouldn't take a coin toss and yearold died from 1 1 the what 2 are now on 29 with similar going difference after a few years my into wasn't did you visit to did not do a more like doomed mobilized or were you born in June so some others question which your knows answer maybe former time mother born following a alive that some issues were you born in July so they probably yes given questioned its activity oneliners for more me a some other number that's known so for example of question was were you born an out should be went off and I you go through all this weird place 1 oneliners health care policy so Elf of goes here changes the so that off the are meant to us all this a regular that this replaced by Elf is probably guess question to it at solving 4 killed are still males these 2 things true or are probably yes responses so end Fox make little or capital R which comes from the data that's the thing and then Capel he would be cab minuses over a similar thing or the other unbiased
24:29
estimator questions on 28 per nite quite so let's returned to India released God Estrada fired sapwood auto called the a it but force talk
26:33
populations as some of them the numbers of the various and the weight of strata it's the same proportion of the population that's an also strapped photo of 34 . 4 4 4 the population means his list of Italy he what the song Omega opera reduced some is the population all right retake the told strata 1 and the total instructs who tones rather L Sara but it but some or that yeah I would probably call the Meehan stray Elvis saying right study population mean and stressed out are get from the abide by told by a number of members for war she or since TEL as Mueller to replace Halaby but not remove the water inside terms awaits the that way as we did the summaries of the population mean is a weighted average of these Stratton means and the waiters portion of members of that struck an old of such a fresh way scepter so the object is to air approximated but but but but but but but he but and let me inserted here by a simple random sampling study in each stratum we perform simple random sampling we take a bill in Seville individuals on else struck me random variables for health strategy but are denoted X 1 a X and so the sector 1 indicates what predator from 1st 1 indicates which different individuals and stress other ends of LV they're not they're not independent why not quiet is not independent from simple random Sampras there's a slight and so we know for example that the covariance acts I L X jails would be minor Cygnus word over the Was it might want well by Cabela's thanks air Pfizer from project these are not independent of the simple ran the family but I must stop classical sample but 2 what do but but if I compare for different I also
33:24
will be in the same it will could be gone if
34:01
however from useful because then the estimator for may also be accessed by ah poetry just gave young so there is a greater new Eagle some Ali who wanted none of is again I switched from capital Allakaket led his ever a sector a solicitors New is this so we want estimated USA prettier merely take a sample after we knew this was strapped however we estimated new L potato sample average of else strata we of way don't add up over a different strategy analogous as mayor for Sosa B W L X are where ex parte down that's notation from the book yes it is where B samples from the by the number of samples announced nest mayor for new with certain here there was spared from you and in fact weighty expected value excess bar white what what what's expected value XL violent Isabel selected expected by this and party expected and this is a constant state expected artists this is Nero so I knew it expected I've excess so excess bar still gives us a an unbiased estimator and using independence from 5 different strapped E is very excess far smaller E. what's a areas of XL this is a simple case of official random sampling of population size capital a mail sampling the lives of individuals and shorter hours they were or a city elsewhere Bell well all these generals of the same thing so Singh was where rents belt times population growth the ball was a because the effect bets from just plain
38:13
old Noah simple random acts
38:21
ballots computer rates that are asked us for stratified well what what B X L R the rebels overran right means from the different from the retina rules about art independent so sample means the difference that it will be independent and reports say would oversee excess bar is a cell of independent rhetoric and Sullivan ran rebels would into some of these France's take a constant times Arender will take the public cars not wear think series independence of the different Arender will set samples from different is different from their independence within the Stratton there's not independently than the start of what were not using that reducing independence of different strata of what's this largess up to that end a fact players all all guilty of advancer person it was Iowa was right Juniper Trail that pressure also the sickness where it is a evasive any individual ran 1 soldier member of race basis of X was quick and simple random that's the singers were at the out of the race will be the same Atlanta rebels from different individual Renner rules with different types of it and if you ignore these population pressure German has reasonable look for it so we don't was last time there's some full . corrections for holes the in the key book or case now so there's our various found that they would ignore the key the various arrest air is approximately dead that's
42:04
0 0 OK so long as it is a bit various business OK so this is all true if embraces the same plant
43:00
then I really necessary dactyloides I at the various comes in desert area or periods of estimator I or here doubled in rates as later health struck but now it's not always the case in applications that the ranks in the same indifference that show but so what which have some examples being not the hostile example if there were a few hospital was not very many beds and compared the discharges in a month for hospitals smaller events with the discharges in a month hospitals with huge number beds which 1 would have a larger various warrior building was ordered be about the same hospital with 100 beds and be discharging the variability in how it hit the discharge for months would be about the same as for hospital with thousand beds different want more beds should have larger periods or if you're asking me in a population of army languages you speak fluently and U.S. this question in North Dakota what easing the variability would be there you may not have the not support but you might have some image in the mind of what it's like in North Dakota the people there would be a lot of different languages that probably not going to be won and lost mostly would 1 now come to California and ask people homily was very fluid in to be more will be there quite so that I think it'd be of other examples where you might the population into different groups and in these different groups would be different it but when you're making but as E the court and the race is different in different strata another question might be Gosselin was household incomes and nautical order and also low incomes and Southern California Nordic order there would be a lot of people at the very high and I don't think mostly be built prospering class incomes were not recur much but in Southern California fluctuation incomes have you seen RollsRoyces and Lamborghini sir may be seen cars that look like they very moved a little slower as another example of the base might be different in different strap so if we come back here that would mean that year we would
48:05
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
53:42
organization he may recall the check 0 per worthy ingredients are criteria same direction so up to compete with the of these 2 functions staff greater bath but the he is the tho not no squares on the started each China gathers there should be there
55:43
E squares missing here somewhere minimizes estimates that get from Noyer population there is a W where can and will always have it was
56:11
detained Sunday and Isabel was 1 over and l wasn't records there were no 1 around that might wanna renown a toe toe toe at I Reggie is this Middle Elskamp autoimmune drove juice Bertrand's Abell produced by cancer both want and what are these are lined up that day of this equation but why at what is this
58:02
equation saying it says that lawyers here you must be able to land at times a components here for the quarter all lenders trolleys all have the same value and or is it don't depend on so far fire
58:51
but that but and face Kurutz to pay off thank you he my head that well Was I said but wait I would say no are the call do too course Miss these information and show what Mayor calls land weightier tell what what if we solve receipt of refined Illinois and so but well it's a model so we don't ordered there some number Lambda set that we want this to be true that the better miser study is because some number land for a while so that means that AOL Islam signal WL brought every summer is now at the of this is less In a land at times a some Kelly 1 say well at 11 it this band Over sound over 1 to cover artist Eagle want Kappa Sigma case that was hurt because we know that these actor minimized the number and l or doesn't depend on surveys must be equal the sum fixed number and I guess all their friends well and is a fact that the sum of their nails must be an they were I use the fact that this numbers that number express enters the I CHANGES AT index when I go from here to here because I'm gonna use a different acts committed by using less land selects ends about its plans at times as some multiply this by single LWL OK so that's a that's overseas about starts work force notably sample that should be taken from this portion of the sample was taken from the health threat calmly to we take start on this proportion times the so on what they also differ would Woodall strata at the same time 7 sites to get out will vary in other words a small now if the the rates is operations and or the population sizes begin such where this the big number compared to that this would be a big debates and were small and population was to make wouldn't sample many from that where the population reasonable size and the various brigade that that's driver would get
1:05:13
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
1:12:01
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
1:12:37
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
1:22:06
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 gammaray 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
1:31:37
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