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Introduction to Probability and Statistics 131B Lecture 6
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Title  Introduction to Probability and Statistics 131B Lecture 6 
Title of Series  Introduction to Probability and Statistics 131B 
Part Number  6 
Number of Parts  15 
Author 
Cranston, Michael C.

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. 
DOI  10.5446/13606 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Mathematics 
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Transcript
00:06
Art last time we talked a lot about back method moments estimators photo any idea was thought he was a true pleasure light industry still arise out some parameters that you want to move estimated but Hsub or 2 for example you might be sampling from under normal distribution with some unknown primer where squared beta what would be new data to a basic square In a gamma distribution you'd have to primers on estimates what but but but but was or what he where are the binomial
03:23
distribution in history isn't Salzar boxes to history and will and items and tape boxes of probability that the proportion of box when he
03:40
won box to so on video by John density here Art all now but the circulars slowing somewhat Williams products and numbers at some point so you're there OK not a K parameters that might be a story an Agassi finally maybe they want Islam underlying the decisions the sort of Trevor so loser examples were where prior might be just the right Swan tremors through the right band said and you want to make them typically for some time 7 what the upward these parameters are functions of the moment but but the 2 she said and the SO IT status of J. would some functions deaths of J of Our some number are the moments and we can approximated by a sample moments but she gave class he by this large numbers which says that 1 over and some I 0 1 2 and the case Favre XI to expected value of X 1 case which is new what they what's sets what it is not my vertical dyslexia upside down these like brought bigger 0 he oversaw site he the key that's that's Rivers a lot large numbers of probability that these sample case will it differs from the real case owned by Werner von close to 0 as well you might recall you might not or you might not have been in the past but it will charge church what sets us 1 arrest as I said source and 1 on 138 weirder exercise that said if he is continuous over the sector size grown somebody must remember continuous and star star Olds then he said it the to the a exercise there is obviously patina and in UK patents the new cable saying as is user definite of continuity for us similar similarly Walter Winchell case he teeth over for example that FJO there that Chase continuous as we have here all out to if for eateries
13:05
sample moments are converging through these moments fixed numbers in the sense back To the distance between this doctors
13:20
and a specter being the public at the disposing specter inside here and this factor is Europe's upside was 0 and because if that is use FJ that factor binds said this sector will grow 0 on sets cable azide what is a assessing a mail this is giving you the method of moments . estimator for primary figure day but all Vermont counseling red
14:37
worth phase Jr motion 2
14:58
so you get the primary and the distribution page 8 from this function of some moments where 1 estimated faded I that while we are still in the new highly amused reviews a sample 1
15:15
at we all we use the same function that delivers a sailor J. only put the moment in what would shown is that if the function the gives you the a parameter
15:30
from a moment's is continuous then they left the moments estimator
15:37
book birds to be true value the crap as is called consistency Persepolis Etheridge a set is of we call it it they were right all the details of more less we did shown that if the function FJ discontinuous where half days a function gives do these true the parameter from true values of these moments than hack OFJ of new onehalf new it is they consist of estimate of that's because the new tax Our consistent estimators of the newest and it is preserved under attack continuity of do Our that would go a 1 because they've days getting closer data so on Monday the probability that they is far from there the hats far from users and the probability are always going to 0 7 province closers going to 1 which is what we want close the thing yes rating with my problem said answer questions OK so early examples we had on Images Review 1 of them have a normal up so lawyer Express you so effects is normal With we New square recalled 1st moment vexed due to the 2nd lawyer Max and Express it Hill and similar where says functions of a new 1 so we know that this will be enough to hire moments where are these functions of what's the function in this case it is just knew 1 right and Sigma squared all of well it's due to minus new 1 square cracked it's a 2nd homer by lawyers for an F1 OF 2 are their book on all meals in the rural serve both continuous 2 Burton G what were down want at base at or but for more that she was there is have to talk but he too old is Kansas less makers go through a load of the force on or am I find the functions they give you these parameters moments are continuous the system as Richard music and as the number of samples gets large as mayor gets closer to be true value of the premise but the course noise comes up how close and the price of our forces and involves some probably stand there and the distribution of but the key but but but if at all so ranks of yes there is a measure of the accuracy of the estimate they need to know what the variance mirrors that means you need to know the distribution of it all so in their
25:08
normal example of what this vision of new hat have but but that but a His well let's look at new hat areas where there are signs that signs are IID normal mean squared random variables and you half its the sample letters on Iran's sexpot what's a distribution of that's part of the actors are this what's a some what's the of some of normal quotes means what's expected by the well water and expected orders on everyone around some of the expected values it's expected values bring it them Soviet and you divide by and we had to do so this is normal would mean you whereby but the various things Grover and because variants of Osama veteran variables as some of the variances 1st of all you wanna Red Square the some of and where some of the bases for each time we get a square here some hot air those rich and Sigma spurred by by integrating things word over and over the deletion of this estimated normal means the standard barrier auto maker but as a low problematic What is a stand there for sampling to find out what muon sick are so can we say with the standard here are no so what we do we estimated the Sigma where we're using our estimate we don't know look for was where the resort show half over and Saturday and what's a decision of figures kept squares well off I believe it's ominous and times they have where over squared is squared with and want freedom as a result from chapter sticks 4 or 5 or is 6 sex they have written this Assam directs the sigh on fast the 1st period last time the 1st day of class of a 2nd such a part so what EDS Standard area here we need the various they try square what's evasive and square with entities that are problem are said easy probably yes yeah comes in so Doc but but stood and degrees of freedom and the will distract back Midas 1 OK Kasal Hi squared with them degrees of free or rice they are the sums of squares of independent standard all of the talk trail so collapse old xxx where what the various erect iii E which is S what's expected ibex square poets what's expected acts the value of this summer the summit expected values what's expected by V 1 scored well see what is normal means Iran's 1 so Specter but wants wears a very easy ones which will be 1 such writers as 1 drug that was 1 all these is 1 of them expected by summer's end expected by year's end where we get and squared sodium and squared Ellis tho source Of the key a phase in Iowa square over some and so are you the final terms and off title terms the diagonal like that none terms all qualify backs where 5 through this
34:16
kind of competition 73 Earl 4 times by now slowly
34:35
identically distributed Wigan and terms you're there the same as comrade firms are in the South End Times and minus 1 Missouri resilient squared terms but often there will be many here and here we get the the expected by value ZEI squared times Lindsay spurned ZTA squared are independent venerable so we expect Illuzzi iceberg VGA spurred expected eyes desperate times about squared and recognize iceberg and Picozzi days where the variances of laws and rules which will be 1 expected ICS Bertuzzi taste birders what carrying dealt with some of its area Xu she this is because the EIA squares ETA Sparta independent Beloit busy its troubled means Iran's what expected ICS there is a very races EIA is 1 as wonders of wants a what so odd years and fans and wrote do it by yourself cause behind with fire this all OK because that somebody firms are in the summer but stood isn't to step I agree with this step from year of economic terms are in the summer armored forces other and comet reserve for and so therefore there and squared terms in the south of the leaders of parts terms are here and parents were altogether so there must be this place again this summer that because an end to this project and square so it's not these are all the terms on the diagonal entitled terms since Iran's were the and spurred by center and times and rest and this is a during sodas becomes and times and lines 1 and that really use us to computing the 4th moment of a normal which and so bad don't wave much of it would be that the auto then that would a sailor says because of emigrating isn't even function so also the writer 0 twice in a row over the line and a fire multiplied food tires on a route but there were 2 underwrite and I'm really make a substitution equal to act where over to lend to it it is not d is XTX playing and number 4 substitute in these values that was a liberation of aging no if no figure or goes here for the day why I had to borrow an ex I think that it's away from here hours all we solid execute In express that terms of wine so let's all this equation for acts in terms of white we're just because of that sorry so what's XQO do this will show but Salam there but that to order that 3 hours solid here and widen the 3 yeah can you do do that for but they want Sapporo of
41:27
2 lawyers has to have 2 3 hours I think that's who squared or for at K 2 take and why in in a world where 3 ounces 5 as biased 1 year Eric Brewer case as part of the gamma with OMB over 1 Elf equal 5 pounds students for over High Times of 5 or Sagamihara I guess right now and again function 5 pounds it's 5 have style is a gamble 3 half time don't function at onehalf he 1 half replied so Books 6 I forgot that time doing well at sea 6 say 6 bigger 6 they were worried Porter said Joe up there photo said OK that's expected by Red Square us back over here but but scifi Beth at present at the end of the the rehab Oh yes I'm decisive 3 of their with 3 and a half so this is 3 . 2 hours this is not 1 here about but there would get 3 outs and then 1 half deviled onehalf and that would be 3 of us becomes 3 pairs event squares that subtracts away a castle that there were 7 3 and so it's too and variants of Camelot and 1 degrees of real where about cigarettes were then what's the rates of Sigma have squared while race of this saying where they are but area twostep so far your are fault line by 1 of the killers or called was part of both squared and on his ties further and minus 1 degrees of freedom so it's variants is too who'd end Midas what she saw lot the ball so what these standard for the key areas he say let's work but 2 and
48:19
back OK so on grim booths or look at similar arrest for what's Gamma get some idea but better moments of test measures you have to look at these
48:45
distribution of Our and tough use a S S meter for this standard there because of that we use and their evolves on the unknown parameters his that later background but it's so let's take a break and come back and do another method for for estimating called the facts of the case well I made a mistake in the 1st period I forgot To put
49:21
a rude and here was not that I guess it should be spared here to its fruit and it's a Mosquero renders the various that er that would be extremely rare signaled a rude and I swear k plant south for a different type of later To now he can or will T back he can ever be she told reporters the Bible but but it all over he called OK so are examples policies are favor of song from whatever Bender joint density we think of it as being a dicey given that from their land every express that way work sponsor and are they are not new villagers these are the possible values the probability that that's what little X 1 X 2 with little X 2 centers Aleksei sold there may be some argument that tells us that these will be persona and rebels for example you might satisfy the conditions we call the law rare events that led us to model a number of alpha particles emitted by substance In an hour these days beside venerable where the number of successful seeds for Mapletree hurdle elm trees beside venerable so there the radical arguments that would lead us to think that the thing were sampling is an example of a son Brenda verbal and that he would we we want know what's the weather like this sort of talk somewhere else mainlanders from Serbia feared dry density of arts and air
54:42
annihilated at X want to excess I have I to do we might
55:18
know the world's instances of a novel or honorable Kitty density joint density of these random rebels could be expresses way of forgiven and say do what which consists Tech 4 we do expresses product K 2 Q. like that so these 2 examples head independent and rules I go back here I could also expressed as well as a product before just like that but we don't always look at not everybody wants another example might be X through acts of the or am are Mofidul mail 2 2 4 or what now priority 1 Ms. Casey joint density would depend on Amber plus 1 parameters key user probabilities as the number of optics place and they sink and in this case couple
58:30
restrictions this is Eagle that went X 1 plus sex to
58:48
bust of a at In total at Dover objects written in these themselves must end and of course the condition of this being called for normal if put up all over somebody from bill is passed you want OK so those are
59:28
examples of these possible try densities were beginning to figure out what these parameters are from sample and so bomb well
59:47
defined we will but tell of the Travers to be log of we densely so the AP's these observations are Sexton this definition osha Sawyer some data set I'll values observed the LECs want to elect and so we you will order the status the and this expression and here has more or less a minute given the status of the probability that this happens but this happened now you might cobbler some of the they well take the point of view that likely things happen more often than unlikely things so this thing you observers probably a likely thing that means it has high probability under this set of status words the Raiders that made this observation the most probable you maximize this expression in the status and give you back select the best later all but a kid I am areyou our backs means the argument that maximizes the expression but saying Q since the doctor words functionalities expects that's call the Emily so it's double examples but let's go the ones we mentioned earlier strike force on 1st where the status here fathers only 1 day that's a lie so Fulton L will log likeliest overlord die before blog likelihood function so you're giving observed ideas now I have ever Frost only parameters lambda take a lot of this function at the observed ideas so I take a log about things and that gives me and the legs Our Lord lamb where is I come from this is this is an ex parte is not expires the sum divided by and so this is an times expired May of minus the larger of the denominator there the about that all right down but don't worry but assuming 6 concert for it Our maximizing arable land so what happens this week different Heidi Max Mosley different gone looks terrible but causes or trouble whatsoever Whitacre largest Jeremy minus and whether they it's scourge L L How are they FOR from so a different take this it for cash at With respect letters that that it 0 and the value of land we get Islam must be equal to expire right so we have excellent wrecks and are there and Robles Lambert had equal to expire bizarre estimator Emile busier volley of the true so what's the distribution of land that adding up an independent random variables so old handwriting by and what's what's the mean of the resulting rural what's Amin the individual Exs that's expected value of the honorable so the expected by Islam what is at stake that means this is and unbiased estimator of what else so what's various of ex parte our duty land way when the mean raised honors for the crown this so but they never ran out front so and meanwhile variants Dover and if we do what's acts are minors when over Obama over rude and a erode Labrador is it is approximately nor will mean 0 grades what Woods tells us about the sampling description of that sort the land here In
1:09:43
the back some likelihood as mayor of Atlanta for the best stance is Islam equal to the sample average do the but here are a function of enzymes to promoters likely loglikelihood function bans on 2 runners given the data let's use that bad former mayor because only dig a log or product words that become some of the Lord right
1:10:57
top the firstly Amie a slot same Will there be logged water resources my thoughts in area Song I will want it might have balked hand we had a minus sung by April 1 end XI my new squared over he through up at sea a lot like the function With observed a so let's find herders OK had um forward different to sector this goes away this goes away in here we get what through comes down a year water Sigma square outer front and Annex II bias new and when is 0 pull where do this some I get summit excise miners and New term and times so would be 0 knew his ex parte therefore Emily for New Year's you have legal bar and the sampling distribution formed new active is normal Remuera racing Mosquero over and that's MLE forcing was wary call federer Specter Sigma squared we have what this depends on where maybe I could die every writers it was great a half their departure are divide it respect a single which you refer when I saw this get actually lies and over to logs Is there in terms of his ruling minus 10 or 2 burned by different take boxing expects over Sigma squared so 1st part would be minors and over 2 segments square guru of this scar on different take this Sigma squared Texas is a constant over great back that was wary it is my aside constant overseeing the 4th very through suspect acts of 1 Araxes like this 1 over express so here a year plus Asian Article 1 2 and XI new squared it over quotas to their keep it before they nice at this equaled a 0 to Africa
1:16:18
and when solving for sigma squared the book forget that but Lemieux his boss be replaced by a new hat so our Maxwell likelihood of her 1st it was where does 1 some light will want to the legs I minus elected and so on terms the ran red given bring back and the randomness or but that 1 way but still None independent case Of them Malta I know he to each or of OK In order to maximize the specter that values given me that observer elects consulates and and so will administer this causes any trouble on maximizing in the values of these look like terrible expressions but different states specter of very will be eyeing cast hours of Our maximizes function but remember there is a constraint and raise size In there used to be what the user to do their dead after 1 the virus piece of Iife add up to 1 remove maximization with a constraint that involves grant choirs so a constraint functions this must be 1 GE must be 1 story maximize given 2 years what the smokers where the greatest lineup so a true vectors lineup of 1 as a model the other ballots a grade of C eyes warden of the great Jews do do GDP I continue is this 1 of the great Jesus Becker composer ones from whatever by around Archer composed of land everywhere about the grace of Allah no easier he's there here Madrid rose suspect to our obtain would be XGA time 0 this suspect piece of jade want or these days of days component of the great BXJ over T.J this must be landed this must be this whoever so they all must be equal the same things so photo each other and I'm not pulled up is equal the sum
1:21:50
value what is lavender all arco with fighting for our reserve where also was a some of these wants the slender what's some writes a number of excise there were things and sell I am ratings the readers distributed and the and so some sizes 11 so some sizes and so what's left Flanders and so what's what's what's arrests mater of he so our our sore back smokers at the chicks is is is he is due to take there was water there such food these just portion of things in the box of their portion of things about next month and what's so given a random sample oratory X 1 direct centers random variables With Vermont unaware of this vision the the cost of the each would be all this consider but these are unknown as 5th so what's assembly distribution of what's is resume XI the Czechs want tourists and Emma are multinomial what's this reserve X 1 XI is binomial from a N and P I it this way hero really care about
1:25:35
whether some reason box iron aren't so you have in the new number now the other 1 sites so it's a
1:25:49
probably XI is number J court all the elitist competition to and choose J. he lied to the and the 1 might To the end Midas but gives a this way you want they also oxide then my history other places where you have to know you're going experiment giving a box of all embarks J His successor Canadian there's failure was probably success he I probably injury successes that BPI J but and the others have been failures everyone I think stating that had to tables among succeed with her trials at it and see what that had been truck that binomial so that sampling distribution MLE is finally by the by is so that health what to decide what Standard areas know using or will need to know the answer distribution of the estimators near fears dressmaker XI Iran's that's binomial with primary and then unknown value he so by play so of what's a varied so binomial center the says variance X ideas PRI times 1 my so the standard error of I hat is what's racist get me this Over and squares so the senator would that I want my over and know what's a problem of using that as the standard error while you don't know what it is Aziz a defaulting using something you don't know where we'll replace the PRI by every day I have weather all of the I'm sorry and right now is suing Bernoulli and they think it's obvious that we divide by and so so we Greece divide despite and squares the arduous this Over a committee screwed thank you that's script he I want my Over the race of a meals and times times but Over and you're sure variants of this what happens with 1 run becomes 1 over and where this by 1 Red Square to get the periods of acts I over and but know that we don't know he still use caveat that's something with computers have the data you can give how of this idea by this estimate is for the true value by OK so Amir remarked that the the address these like this if there visually smooth and Maxwell likely U.S. mayors are consistent American system that means they converged 2 the things arrest me as a number of samples gets larger converted arrest parents think that's a world stock today you get 7 minutes it to think about and time is money aside this giving you some money
00:00
Binomial heap
Gammaverteilung
Multiplication sign
Normal distribution
Moment (mathematics)
Square number
Parameter (computer programming)
Estimator
03:21
Point (geometry)
Musical ensemble
Product (category theory)
Moment (mathematics)
Multiplication sign
Decision theory
Distribution (mathematics)
Moment (mathematics)
Sampling (statistics)
Set (mathematics)
Parameter (computer programming)
Functional (mathematics)
Grothendieck topology
Number
Population density
Manysorted logic
Lecture/Conference
Cuboid
Right angle
Analytic continuation
Social class
Directed graph
13:04
Divisor
Moment (mathematics)
Sampling (statistics)
Set (mathematics)
Figurate number
Distance
Number
Estimator
14:16
Lecture/Conference
Phase transition
Distribution (mathematics)
Moment (mathematics)
Sampling (statistics)
Parameter (computer programming)
Functional (mathematics)
15:30
Musical ensemble
Group action
Consistency
Structural load
Forcing (mathematics)
Moment (mathematics)
Expression
Distribution (mathematics)
Sampling (statistics)
Variance
Mortality rate
Set (mathematics)
Parameter (computer programming)
Functional (mathematics)
Measurement
Estimator
Number
Lecture/Conference
Noise
Ranking
Right angle
Analytic continuation
Physical system
24:55
Trail
Free group
Vapor barrier
Diagonal
Multiplication sign
Decision theory
Distribution (mathematics)
1 (number)
Mereology
Frequency
Sign (mathematics)
Degrees of freedom (physics and chemistry)
Term (mathematics)
Square number
Euklidischer Ring
Social class
Random variable
Area
Standard deviation
Military base
Sampling (statistics)
Variance
Independence (probability theory)
Variable (mathematics)
Estimator
Field extension
Arithmetic mean
Summation
Order (biology)
Phase transition
Normal (geometry)
Hill differential equation
Figurate number
Resultant
Directed graph
34:13
Area
Multiplication sign
Forcing (mathematics)
Projective plane
Physical law
Moment (mathematics)
Independence (probability theory)
Variance
Routing
Line (geometry)
Mereology
Rule of inference
Substitute good
Hand fan
Number
Arithmetic mean
Wave
Lecture/Conference
Term (mathematics)
Square number
Equation
Figurate number
Sinc function
41:25
Area
Statistical hypothesis testing
Standard deviation
Presentation of a group
Sigmaalgebra
Gamma function
Multiplication sign
Moment (mathematics)
Similarity (geometry)
Line (geometry)
Mortality rate
Student's ttest
Mereology
Event horizon
Functional (mathematics)
Measurement
Degrees of freedom (physics and chemistry)
Lecture/Conference
Square number
48:42
Metre
Standard deviation
Distribution (mathematics)
Physical law
Parameter (computer programming)
Event horizon
Number
Explosion
Frequency
Population density
Manysorted logic
Lecture/Conference
Network topology
Condition number
54:36
Population density
Product (category theory)
Lecture/Conference
Optics
Parameter (computer programming)
Rule of inference
Number
58:27
Population density
Lecture/Conference
Sampling (statistics)
Total S.A.
Object (grammar)
Parameter (computer programming)
Condition number
59:42
Point (geometry)
Multiplication sign
Distribution (mathematics)
1 (number)
Parameter (computer programming)
Likelihood function
Expected value
Fraction (mathematics)
Manysorted logic
Lecture/Conference
Average
Lie group
Subtraction
Descriptive statistics
Product (category theory)
Forcing (mathematics)
Expression
Gradient
Sampling (statistics)
Independence (probability theory)
Set (mathematics)
Functional (mathematics)
Estimator
Summation
Arithmetic mean
Doubling the cube
Right angle
1:10:41
Interior (topology)
State of matter
Multiplication sign
Connectivity (graph theory)
Distribution (mathematics)
1 (number)
Water vapor
Mereology
Rule of inference
Likelihood function
Maxima and minima
Casting (performing arts)
Lecture/Conference
Term (mathematics)
Square number
Maxwell's equations
Subtraction
Area
Constraint (mathematics)
Sigmaalgebra
Expression
Gradient
Independence (probability theory)
Mathematical model
Axialer Vektor
Functional (mathematics)
Summation
Order (biology)
Normal (geometry)
1:21:39
Direction (geometry)
Distribution (mathematics)
Sampling (statistics)
Cuboid
Water vapor
Mortality rate
Biegeknickung
Random variable
Number
1:25:31
Area
Standard error
Standard deviation
Multiplication sign
Distribution (mathematics)
Sampling (statistics)
Variance
Division (mathematics)
Grothendieck topology
Number
Estimator
Frequency
Lecture/Conference
Square number
Cuboid
Nearring
Physical system