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Introduction to Probability and Statistics 131B Lecture 7
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Title  Introduction to Probability and Statistics 131B Lecture 7 
Title of Series  Introduction to Probability and Statistics 131B 
Part Number  7 
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  10.5446/13652 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Mathematics 
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Transcript
00:05
OK so let me give a brief review From Emily's it true he to what but he the report as but case so on sample sound values and you
01:25
assume that these values are instances of what kind of variables that have a joint density of this form and there's some
01:35
unknown parameters later in this of interest to you so are for example you might be working for a mn OEMs can company and period in the wake of the Yemenis want them out very consistent distribution they should be about time under way but be something and it's very much from that something about be waving their members probably something like a normal random variable with some means data are slated to be doctor could be some means and there lots things to be small
02:25
because different in their size it should be uniform 31 estimated what they mean and variance are the eminent well we are you do it lightly maximum likelihood
02:42
is you take the likelihood function for why should say log function at back a new user value of data I I think that maximizes caliphate or case so in this way the set up small Exs are observed data so their data that you actually received from somebody or you obtain yourself by measuring and evidence we also sometimes right at Soto do it too yeah open the T catalexis indicate that there are were a number of roles a observations and there are libraries around rebels of the strike I don't think so when you see little Lexus that means observer status at random Ozark rent for gay and so
05:45
on have once again just CL works so let's go through example the normal which is important Q.
06:22
Chile's key no not center of of uh IID normal mean new racing squared and so about problems like dormitory members you still formerly of the joint density like Eminem might be creating their rates independent somewhere and you might also assume that weights are probably history because of things such settlement for or my wife my soul a lot of problems than normal this year then so be it density but what it 2 parameters UN sigma square which are unknown anyone determined we know it looks like when there independent medical history about this factor 1 over a spread and time because rejoined density of IID venerable was a product of their individual densities and individual that's all hell I ran a rare would have that in its regular the sum but were not Vermont applies and terms of minus X 1 finds new squared over where is minus X minus 2 squared over squared minus X and New bird or more altogether it is 1 Ian and Co summoned the expo track and of course we killers directly with do Walker raft affect them OK well there's a recast as varied as far as muons Sigma square are concerned some dozens on them for example well in this case would be won over route to fight to the death at 1 over route but I'm lot of that led to a lot of this like a lot of 1 of route the maximizes functions May later from maximizing somewhere with French and cynical 0 and the part that's possible away so I don't care what they look like but the parts of depend and dancing scrutiny or what I take a lot of this thing that for now this is the what valley went to the Enright right they because of the things where they screwed that your prayers the thing is they hear race that we and our roots and denominator article or 1 over segment of the and our logs water 1 or a minus log also a lot of 8 and that and log they the becomes miners and loggers this term they are here from bias and amusing to rules about lava Reyes minus logging lover of data being a father and then I take lot of exponential whether we use irritating natural by alarming catalog the lottery veteran burst functions like but some might want to and XI view where over through where they Oh maximize yes doesn't change our Macs might find the Sigma Nu make
11:59
this as large as possible given the observations about what fluorite soul winners where do more demands cultures of ballplayers dimensions attained their heads same values radiant must be 0 0 0 electorate sold for 1 of Sigma world listing and the gradient of that mainland stigma might be PLT PO music and video said at both these tourists have to be 0 or redefined the music were both of these fresh said that bugs OK so looking back over here to L with respect to welders suspected near 0 after that respect user of get something here earlier through coming out I meals and the derivative my a new is of 1 that boxes are we get some Michael oneday and to XII new over to a whisper urges to adhere to their a cancel parents of the parallel expect a new is summation 1 and excise biased me over stigma square they inquest that are now is a alloy respect to Tsinghua up this says 0 as it this would have what Her logs or right of trees so you might as ever said and then what class right these last from there story now that way and were a wary WikiProject this took on cell making it passed to what they like minus 3 times as thanks minus 2 0 Sigma street the but as I have already with the things that we can't solve and there's a gradient of loglikelihood function that size given by the the UN sigma that make the Spanish and both quartet or was at once so To make the the first one managed up to and it's what diaries of segment news newly that role the case OAO say when you have satisfied but the multiply twoway Sigma squared so this just becomes a condition O'Neill and many of and times so this summer excise might send you a 0 or some size and times meal so are new hat that's would indicate parameters they're back so much the parameter is the less farm meaning 1 over and they knew that protect
17:58
U.S. that's where the 1st thing that it is not database simultaneously so you have to use that knew that you had in in the 2nd quarter and at the same place they they now Norris said the result is not new here I sorry signal doesn't involves a well but the but faxing or his 2nd term vanished we
18:30
set back always their says Vermont also somewhat stretch swept is a place where the 1st forest that story that new hat and solve this pleasure OK so likable they over a lot by a Bible scifi Sigma squared and I find that it's the estimator where would be 1 or a sum equal 1 2 in excise minus new hat I read if that is at the let's Barbaro right this way that's the
19:46
MLE from you have U.N
19:50
all but the but I he it was the was in the 3rd quarter but of day no we did this already figures may be the most support sample if so this case shut but but for 4 4 4 coffee tea or but they could call or cash she 2 the case of a sampling descriptions the To estimators like before taking any measurements in all that your new half will be there average what you got and that's squared will be the combination of nor the function of what you observed but before you make observations which replaces random variables and so they knew had that's were render revels visions are not again be used to find sea ice for mu and sigma square tho so Seoul what are Shiites covers of England be cheaper faster than cost OK so our is now we might be tempted to ex parte directly but that's normal with what Mubarak the excise are from EU I mean new embrace Mosquero and doing or service bird so quickly some mail and revisits Doc she at court for dead this ratio is 80 venerable and reason free of this goes back to Traffic Burroughs by verbally or prove that look 6 OK sparked an answer independent ground rules about ex parte forces normal guesswork squared will fight squared 1 degrees of Israel and when you divide a normal binding and vise versa take show so if you want the 1 a hundred times for myself corporate governance formula at that was good for all of us where y fine there are also the probably that this is on myself now what about this vision out of this order but that is no at its nectar sparked origins a sector at a rare will mean a 0 this this Caribbean alright means their own they squared off and recovered the what about this thank asked it's a positive trend in rural abstract art is desperately tries promote by pirates its positive independent Sawyer shows racial where these factors but yours it was the fixed number it would be a sense about up this distinction is that your that's soaked bad bad means or he but whereby if it's he admires 1 . fell off a number for which probability that capital to and want us to do little to ease my on myself when GETS status witches and spot at the purity and minus 1 random variable This is the number that makes this probability equal 1 myself just like with the North there were Uzi well over 2 hours developed nor and verbal whether those evil for me it's a problem capitals is a movie about my Saudi find
28:42
that the table their tables in the back effort they will back for the sea venerable and it backs its like that they will the normal and that and if that table for so if we wanted
29:27
say a 95 per cent coverage there will overtake L 38 . 0 5 a a that
29:55
would mean that what really want to be Ben Myers 1 of health or 2 . 0 5 or 2 2 for Sabine might sort of
30:13
Q White 0 2 5 Sawyer what would mean that the probability that are she ran rare will it would be quite . 9 7 5 please 1 might say this is no His over to see what we find that well it the degree of freedom here depends sample size and the degrees of freedom are in need love and they 2 wrote last infinitely available here it jumps from 120 and so when reviews infinity and it's something I 5 thousand some sample and uses but here's a sample sizes of what both because it goes consecutively Wanda 13 and 14 and 16 and 120 minutes so Our sample size were unknown because samples event is a thousand years infinitely that president and Sybila bottom role here and it and supper says she thought this is where are the notation changes over from cover intervals table what did yes the other team members from asymmetric and news cable labeling of the numbers he sub he meaning probability of being too left at this point is very under the that's where is this notation of Alpha means there tour up was 1 myself Louis translation water we want we want find where the probability of a thriller and so for an equal to a thousand nearly and the 3rd 3rd call from right it's 1 next itself would fight also like the normal thing that's in renters Apollo's Anderson so that's I thought we put here in the new face of the table where we doing here we're fighting where area as the density of the left of this point it quite beds of 5 sets of cheese what better part of a let's try another value and how what 30 they will have 30 years found bottom and that's I want to do for the robot every word the 3rd call from the right . 0 4 0 0 5 areas 31 here but I'm sure 30 Reebok the same visit my swan would you rose 32 30 this is 30 vendors 32 were 13 and my astrology 31 29 disuse say OK so now eyes this site find these ideas once you from those values was the conference was to Pungoteague abilities with the conference we'd assault for me as it e could about I'd have to put their root end Over here by by rude and multiply by us and then add new thinking she he er West sort of the year they were before as a subtle is here are well maybe sample size and they will be change notation from contested for covers and will be condemned now Vado a covers a lot for in Africa that the disease on where we can include the probability that it's exactly on me and there are so few like this is going to have it all
38:05
of it now Florida's we'll use don't try square she said half with the
39:01
chiefs Mrs. ties where would I lawyers wanted reason yet the sub yes they were easily all right now producing my have get out of the way start and then missed a seed spurred rules are what on squares In normalcy of visionary not respect or in venerable bigger than 0 sum the density for this world of something like that British and will look for hard and awnings you're always stock high scripts but the so Decena and my son these for you as the general shape this number by our convention means the place where the area under the density the writer places helpful over 2 Solari here will be how far too means area to the right this spot is 1 might itself over to and that would mean the area under the density to the left of the spot would be helpful sigh means Erion here is what want myself in a hammock fourmonth conference in all 5 shy making those 2 numbers are burdened by 1 or 2 kinds minus 1 1 myself over it and that face the probability that I swear and minus 1 1 myself or to room and have square Over segments where last time square miles 1 over this would equal 1 minus L does this venerable has the stenciling and probability line between these 2 values would be on myself all right so we need to solve this fall before square I saw I guess you food 1st everything went invert they squared over and they had squared less wriggled 1 over and minus 1 squared it 1 myself or to be able to 1 over where minus 1 or 2 and then multiplied by and they have where Soviet and not worried over want for 2 2 and a half square Over priced and my 1 where 1 myself or 2 is a 100 1 myself per cent CI for about life things here is of great little different cells are find these numbers table and find the
44:33
cards per table stable 3 of us can be able to his normal table 3 years the price they will force so table looks like
44:42
that of poverty in America but was like this that crap like that and again
44:50
have to fee the degrees of freedom and table also has now in did rotation is about the trip there so it's so over what 90 per
45:11
cent covers edible neck a sample would be well we the study . 9 selfless quite 1 were reviewing for things of the former Elf over 2 Bell Florida would be Points of Light and 1 went itself over 2 would be quite right but and the yet Reebok B sample size laptop From degrees of freedom is as DDS degrees of freedom and ending the endemic of stable goes on to quell 16 consecutive doing their jobs 18 20 24 30 40 60 120 of whom but that say the sample sizes 25 and 1 would be 24 related is column has 24 and were looking for white notation the table too areas so left Mrs. called with key being looked sample size and across here you'll find it values where the area under the fence left I used he to settled in that notation Waterloo here would there it'll at point to be what of trial it is quite won out over 2 . 0 5 theories this To be . 95 so we want can't summer sub . 9 but the archives of 20 4 of . 0 so this notation means there leftist . 0 5 month means that absurd to the right of privacy incinerator left bank is what you read the table and find out it's a price of . 95 expert . 5 winners Thai script . 9 5 is the problem but on the 20 4th it's an 36 . 4 2 and stuff but the thing we want want this point and the price of price of 20 squared 0 . 95 would be tried 1st sub . 0 Lisby theory at this point my means of right where is organized by the same place and they also . 0 5 and for 24 x 13 . 8 5 or that so patients but to look in the right column column Sarette Rhode Rhode 24 and my body here they are so will reduce their finishes off with the final answer radical 24 and Elfrieda where once we get tough absurd 25 meters 25 they had where over price lights went over to
50:17
36 . 4 2 up to 25 vignettes squared over a 13 . are so that's how you would generally find Collins animals of being older sampling distribution of MLE as you do in this case we use the sampling distribution of Obama said have new phone not correctly to review but what we do for a new we use a fact that route acts are my view over asked what he and then forcing us where we use the same tight and went to that because we didn't know we were square so it's a break and then we'll see reviews of it or not OK now we consider the plant is a former chief of team who he was OK so is for piety samples and we don't know these sampling description of the amount in the previous example the normal we knew this sampling restrictions on up to calm resembles for both new and square this method requires large samples also sends a rely relies on a settlement but it at the center has been big enough for the settlement there approximation to thwart 4 but so for nite I need a sample the sure each of these will have of X and there's some parameters figure our no anyone it gets messed map there have been some likely have estimated the loglikelihood function as a font the rebels and why is that what's being joint density Iran March the product of individual ones they take a lot of the product worked as a to products bird right and Iowa State at busy are of elevator discuss that was
55:19
it's the place where it's the data were the maxim of this occurs
55:41
and I ask you for another
55:42
view taken Mecca arts classes that's still true all couple so you've probably seen argmax and found us know they won't be taking events see I took some academic courses that's right the Her back forte awaited
56:25
your arriving this method may be up say what the end this and that they had no repeat what it is for exorcist aerospace the curve that's the ran Redwood dispensary we take a lot of things different perspective Davis where they where are the following Merrell between OR and means roomier for maximum likelihood as as mayor and this agenda we have size IDJ soft but but now also but of all but project so and as a sample soft data not as a primer We want to discover match ahead had is we think we used to as make enough Chris attracted enough they and multiply by this factor the result will be approaching a normal ran a rare would mean 0 rates crack at 2 but e
1:00:17
plant exposed to get your attention today trying different techniques of case sewer let's assume this is true for more about how what we get conference interval for a this direct talks with the music according to right last year it real so we
1:00:58
proceed as before the public etc well we know that the probability that might evolved over to lessen able to root and higher data not times say that might not to USA was over to probably approximately to 1 might itself now I know I say a proxy here limiting result previously removed the example of the 1st great power with I said equal because we knew the exact stationed there during have these distinctions possibly normal means and busy is the theme from the normal and sell solving this for a safe d at but for now but of course Of the keg the same kind of thing we've before the Bible sites by this over rude and fuel for 2 hours later so rude and I not justified by rude and I'm not menu of bad tracks half from both sides of a multiply by a minus 1 from with this and there is a minor minor flaw here With that might not be using this as a copper before they not yet you don't know they are not sold what to do you say that you're statistician already don't locator not in the eye of fate and that he would say that for a after all it's estimating how they are not
1:04:33
enter the supplement their here that I will have nearly variants 1 right thing on the left side rude and my data not time said at they variants approaching 1 sort the rates Of that term without food and from what was evasive that might be enough To but for now if we for she for me for the fight
1:05:42
None of us dozens of too so the sample sizes bigger isn't too small and that various a small music they then that should be closed so is and I used to
1:06:06
continuous interface they do not in here was that boats but the he where he 2 she now wore but but 4 of the all so what
1:07:02
as I they look like specific example the you're just for special days E he but it turns out that it's not always easy to compute using that each she or but losing this fact like it's a little different doesn't not all white managers but but pursues at 1st and oysters jurors no square there's a square or that they would lose the rules there a time that say we take these people songs from her land world we don't know and the bell of land that is Media some I want end Aug at 12 XI do they lender F O X and Y under its landed they over at factorial slander practical 0 1 2 so what's the God raft of electors Lampe well it's acts of lavender bias slammed a miner's logger X factorial so l however would be bogged down some I don't want to venture capitalists my it's and land a bias on May 1 father of excise vial that up evaluate said XI is XI here and here and then you add up I'm going from 1 end where you where you at this up with excise here far from 1 and that US that would be here with and get where when you these would divert start but OK so I guess
1:12:00
prison look that's the likelihood function of what's what's whether yet it is and later computer cap
1:12:27
elects here across here we differentiate twice the specter land but but that was the what's suspect alarm where are the 1st Lord now catholics sigh going gets minus 1 juror but where it 0 and now tragedy again forget minus X over the square so the size where is on years of my assigned here's a minus sign cancel is expected value of X over land squared and expected and their words by by members spurred what's expected value of a post on program is 1 over all members 1 overlay so you're doing this for a was similarly for Sun River acts are was if could all I saw it to actually party that
1:14:25
it's briefly recall you you are
1:14:28
maximizes party backs a function of 70 the 0 but there are that Davis Over minus and artists or minus and and this does what new printers that goes away yet some of the excise overland that must be some most
1:15:04
equal Aaron means of their where the some midsize must be or land so what would you call Our coverage lawyer for Lambert using this technique he world too the Kyoto
1:15:59
where there is a farce might as well as you Belfour to not look better by now and near we of what were and I learned that is what land so are bad what so Everex part on here are plus leadoff overtones divided by and over at and 5 acres fractions and operators say can be made into inverse fractions of wary of ex parte minus L over to square root of while all ran X bar plus a belt or to wherever at spar over and using a subculture 1st
1:17:17
ever crew to T it forcing office use this formulation here there they are going is 1st let
1:17:53
them back not arbitrary a wearable computer I the top supplies easier for that where the odds to think giving computer by using the other what which would suffer it for this is the original definition of ice Theda's also Trevor which Adidas right wrist where that were well we get the about 3 terms like this bird over members by way was what I just saw 33 competitions public discovered just what will be easier to get that the Class E S Y Sacha head or what's a sequel but that called the density right was rated density get 1 day that's placed the but Sega step further what a difference a disrespect board any fault there are OK but if that was nice enough to make it but that make exchange your different it inside and then there are right if FSIS that's the end of the day it's ice freighters exchange order a great difference over here day but that would say so I was at Sarah because this sis just that all right bye bye that's it but but I would say that respect Bader Al R the acts and that at the time of the accident and Dana yet there are their respective Bader there a lot well a whereby different take this again with different if function 0 French Council on the theory of a suspected fade of the old because of the
1:23:21
different differentiation inside when will the differentiation side we have 2 more data to revive this time acting this time to relax and this is 0 it list of a different inside it up as nice not functioning in that a E OK now rule on on a from Member when I went from year to year do with this I multiplied by often divided by I've ever do that again this is still 0 I just thought fight Nevada by ass and this is a journal the Vologda left right that's their log about Saigon what's this this rule over right and there is a event so we get their long after their luggage sister all of us must be 0 now on effect says density at Rolex and data expected died of a function of Axis can be expressed shock and mineral little legs times at next invaded been back strength is expected by foreign Roble of notes and so were taking a function of X here's what I'm saying earrings so what is this in terms of the capital acts as the expectation of what Mr. resists Jeter's is a far from or where what this what Naji is as squared some of these 2 Wesbay 0 this 1 is so I must be minus that thing so why is it too because in a density is 1 Mike trains story about man who died of a stub talk of distilled badly great pain hopping around tripped over Bruce until well broke his neck and back orgasm sub top OK so moved that Chu almost there let me do this settlement there next time stuff stopped because it will take more than 5 minutes to describe a
00:00
Explosion
Mathematics
Sampling (statistics)
01:22
Frequency
Arithmetic mean
Population density
Maximum likelihood
Lecture/Conference
Multiplication sign
Distribution (mathematics)
Parameter (computer programming)
Variable (mathematics)
Bilinear form
02:42
Expected value
Maxima and minima
Logarithm
Randomization
LAN party
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Likelihood function
Number
06:01
Trail
Musical ensemble
Divisor
Multiplication sign
Routing
Water vapor
Parameter (computer programming)
Mereology
Weight
Rule of inference
Likelihood function
Fraction (mathematics)
Maxima and minima
Mathematics
Population density
Root
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Square number
Cuboid
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Condition number
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Sigmaalgebra
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Gradient
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Hausdorff dimension
Normal (geometry)
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17:57
Summation
Sigmaalgebra
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Term (mathematics)
Forest
Resultant
Estimator
19:41
Free group
Sigmaalgebra
Divisor
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Forcing (mathematics)
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Independence (probability theory)
3 (number)
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28:41
Lecture/Conference
Table (information)
30:08
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Root
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Infinity
3 (number)
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38:37
Area
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Number
44:41
Metre
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50:13
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1 (number)
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55:17
Divisor
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Subtraction
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1:00:15
Trail
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Resultant
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1:04:33
Musical ensemble
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1:06:00
Lecture/Conference
Multiplication sign
Interface (chemistry)
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1:11:47
Computer programming
Sign (mathematics)
Spherical cap
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Prisoner's dilemma
Square number
Units of measurement
Likelihood function
1:14:19
Maxima and minima
Arithmetic mean
Oval
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Equaliser (mathematics)
Functional (mathematics)
1:15:28
Fraction (mathematics)
Inclusion map
Fermat's Last Theorem
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Operator (mathematics)
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Maxima and minima
1:17:49
Sequel
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Subtraction
Social class
1:23:20
Group action
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