Add to Watchlist
Introduction to Probability and Statistics 131B Lecture 2
23 views
Citation of segment
Embed Code
Formal Metadata
Title  Introduction to Probability and Statistics 131B Lecture 2 
Title of Series  Introduction to Probability and Statistics 131B 
Part Number  2 
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  10.5446/13214 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Mathematics 
Series
Annotations
Transcript
00:05
carried each day will start Chapter 7 NBA title with this survey sampling it not be the house where went but it but if so there's a population of size capital and so the individuals in the population are measured in some ways and results in the following list of measurements 1st Digital has measurement X warned elects on 2nd because of the lack to 3rd wares elect Sarah sold what are some examples of individuals population could be used university and distributing heights of students at the University of height the 1st and the 2nd 3rd or it it could be their SAT scores this could be the SAT score 1 these parts with that or this could be a lot of evidence proves I think of and this could be the weight of the firstever them the to sector 1 when the current or could be citizens of Irvine this could be the income of the 1st wanted the 2nd OK
02:45
so we always measure populations like this and we want Johnson inferences about what these measurements might be in reality and finally won last example might be on once used every election and these could be Mr. sturdy
03:05
gobbler preferences so these Kabila so far ones and zeros so 1 would mean that the Our 1st individual favorite of candidate a his were of candidate be needed a bunch of zeros and ones or you could ever zeros and ones by splitting people into 2 groups may be diabetic or not diabetic be 1 of the individual diabetics will not be 1 of the individual diabetic 0 not to him so were interested in assessing some information or making some inferences about these numbers so 1 . com summary of those numbers as the average as a scholar the population so in the 1st example was a B average height of a student at UCI In the above example but whether God these people are they retained a year not number 1 showed up every time the individual their candidate a 0 when they were not this than would be the portion of the population that the favor of candidates so I public icon case all and then that I can't case difficulties this area and as a proportion of the population that has the property at Let there the value to you want so in the case coming diabetic excise 1 with a person's I don't diabetic and 0 if not a severe force before so the population 5 and sometimes a population total is interesting so 1 example might be this is somebody cars individual 1 somebody cars individual cars and visual don't you
06:10
would then be the average number of cars per individual work of household that you might be interested in comedy cars are told his called the population follow but and this is just
06:37
capital in Taiwan's populist average but Scott population talk sometimes when assessing the other side data you might be interested also in half are on average these diaries vary from their average value Escobar population rates but is a measure of what the buzzing off RXI from squared in the new take average distance in manufacturing emblems you want us to be big or small small needed almost all their millions of the same size Farrell exactly same size is be 0 so let's to a lot of manufacturing you want to be rather uniform so want this to be small so this is something that you may want to analyze or you may want to know what's what's what's size now on you might imagine this event a Cosco or Albertson's and many other places even stores itself following candy that there are a lot of em Williams this number is rearing so do you think
08:51
you'd be able know what it would take a lot of effort to know what this is exactly you have to measure every Emily everywhere where every or would you know what this is not so but you do want know what is actually exact knowledge of this quality or the quantity remnants of these are very difficult would that be so difficult that you would be will make a profit selling of rooms if you went to the effort of measuring so what you do is you try to take some representative sample of the events they may be a handful of them more a basketball and measure that would and that's what we do in statistics is called sampling take some smaller subset of the whole population and try make inferences about what is and what this is using that some also on
09:52
this happens every election cycle you hear public opinion polls about who's gonna win elections what they do the Iast every person every voter in their states what they're going to that would be an enormous effort and very costly rather than Asbury border with public company like a survey company like Gallup for Reuters will call maybe 3 thousand people instead of 3 million or 13 million and they'll be turning about what this troupe of forces of don't favor of candidate and maybe even have some idea but the variation of it solely for talk like that me a right answer me to so readily evidence of population of sites capital and these measurements of individuals and that population there may be repetitions in this list for example it our ask everybody in this room are right here the number 19 more than once and the number 20 more number 21 more than 1 or a fiery measure students the university I might add to the year since I might be 6 feet more than what some might see 5 foot 8 more those numbers Llanymynech so instead so what we do is a tiny condenses into a list of distinct numbers that appear here whoa story was ship but she could not is or half bad just missed but a soap I will let these musings values appear in the list of and a completely means every numbers the year here sir once a lawyer so for example um if until we suppose that the height of the students in this room are on the tried to use about 60 inches start 5 4 64 inches over the 6 5 6 5 6 5 7 8 it was a good kid I suspect 70 70 70 70 selling to 72 . 7 3 3 4 7 4 7 4 force of 70 a
14:02
lot of people here Mr. B this is the list of here these numbers we list of measurements of the individuals in the popular vote with this be at different
14:20
ideas so where 61 is 64 who is 65 areas 67 or 68 5 69 cities of 60 70 cities 7 72 of 73 seats of 9 74 so our last time I said these Greek letters for 1 and 1 is that as a Greek letter better Seattle I don't think that's what course like that they go across my desk I'm not sure what corresponds that's a great but it see now in this example on the heights wicked this this add up all those numbers and abide by population size and population size is coming picture 20 25 cell in an example of Cabela's 25 that would be the sample means is 1 over 25 times 64 64 must beware assault on 65 secure but all 2 for sending of look like that that would be all right on the population in terms of access the but also do it using these numbers it's also equaled 1 or 25 25 Army Times a 64 years might be good time see 1 right and then 5 appeared 3 tired of 3 times see to fast past 4 times before so I do here I take a number that 1st number the year's biggest most wanted multiplied by the number times appears on the list that take the next most 1 multiply times it appears on said it sold its insisting what these numbers are defined and subject to be the number of indices I or individuals or populations such that the value assigned to the individual is so in this example and 1 to cover times 1 hears overtime 64 years on Lewis's through and through only individuals have 65 might be 3 and for would be pop up right of entry entry that survives 67 only 1 individual that and for I want to trading 5 people have 68 etc psych every right this summer well I this 1 overcapitalize again and summation I 1 and some today but right is and onetime C 1 and 2 currency to as measure work off surgery as far as the total lumber distinct numbers on the list South America over here at all sir rightly population means that basis this collapse this summer mutate all the numbers of agree with backs 1 group them together there and 1 of them their covered values C 1 my life and 1 times you want and you go the next guy you have that you haven't or use the group although once you have this common value there and 2 of them like get lonely going alert more more and tab level be bigger than or equal to 0 are everywhere by the end May in the back case that's where the excise er 0 wonder what's are hungry distinct values took to sit back and soak see 1 would be 0 2 would be 1 stars to end war and want an end to the be expressed the Times is 1 of Peronist us if we know this we tell into its Miss Over here would be the summer here is well In terms of air and water into this is 1 calmly times and 2 times this is 1 and 2 times in 0 although the times where would we get what this it is 2 here so he is and 2 over and capital and sent some 2 years NYTimes capital right hinders the sum would be ends of 2 is this term is 1 hungry times and twotime at work and is the number of times 1 appears on the list and other times this 0 so we had bowlers numbers we just get 1 every time there is a value will ones or otherwise would get somebody wants and to 1 the to so he is and to over and so to his times and avoid this do it these add up to what number capital and Sosa severe capital and Midas cabin for 1 as he as you know it
22:57
also on a toe but China expressing variants using these numbers the seats that specify appellation Reyes using suffers To that me 1st um simplify that show us I expand the square at this end of August but the summit the sum of its eyes wears some of his staff and the some the that on the 2nd song neither the video nor the 2 and that I 2nd pole let out and here this doesn't depend on Iet off the same every time I am that many times sony's I this time that Kaplan Times New Square but then we divide by Kaplan solicitous becomes a square I know what is this thing here but at that that's new as new it never knew newer and 2 times meal for the myositis as is minus Q New Square plus new squared is my my where they OK was 2 thank you bye so is this is another way of writing me population various business is like art the variants of something is the 2nd moment might be 1st moment square that solves same notice when we already expressed using numbers and I see where about this well calmly times is XI square here on the coming times 1 these numbers appear less well newsy sector so using those numbers prison beat both 1st cap wanna capital the sum J. Eagle want are we get the numbers CJ squared times over times appears on the list and minus new squared video what over capped with song J. will want to alarm cities of times and subject Kansas where such that America if but OK so friendly and we get Sigma squared is 1 Nova and game she book at him end reversing a steps off that remained get back to this it is it born is like this going from the the 1st line of the but in reverse expected that stricter is actually con we go from there yeah we expanded this where I saw the are you for debt the same would be right what happens only add up this squared times cycle is a sister where is a way out of the CJ times ENJ replies to knew where would abide by them what a rent and here we get a plus where realized news possibly bird that might be which is what you get here I mean here West New script thing so maybe are just carry out what major I sure this is 2nd term rail times is the evaporated and the New squared times NJ Jaco want are wasn't whereby up and J. J. people want are earlier with the sum Jay Poland are NJ and
31:33
someone that's a number of times of the virus 1 appear analyst fight that's Kaplan and wonders the number of times I see what appears on Willis and those are the kind of guy seek to appears on Sosa's is counting on May times the numbers appear on the list so when what their Utah number in the population capital and so when
32:00
we add up new squared times ends of jade from year to year that just gives us capital in Times News where but I would divide by Kaplan that gives me a script OK and what's this thing here that's new again but so it they minus New square is to news past me skirt which is rare so here's a way to express these populations variants using the numbers CJD and aunt she got to they so as a population is a cobbler population parameters he is in court but if I and so we were part we ought measure exactly that by seasonal population the world this take a smaller part of the population and strike some using the data that we gather tired as to make new top singers but but or but but but but but but but but this might mean in a case of evidence we take a pocketful of Circle pressure like a N and measure them so we selected at random so let X 1 X 2 of excess BET result taking measurements but were up end individuals selected randomly from the population well replaced so that means stuff In the case of students to maybe go theater circle grabbed 50 students and measured but you don't grab the imminent release them and then grab another 1 as you might say way Oregon in this we'll sample different individuals they will accept the same individual twice that's what without replacement 2 reaching a grab but to students deal reached twice and so we treat eases when rebels randomness is in these procedures so we're this in the
37:06
room maybe I have 5 people at random maybe trip Richard for a job or Carlos show L Wright also Darren are all food I think there a case I by neck his acts gap want the tablets 5 or I might still measure Craig David I want Brent and Josh and noble the Eduardo Dick don't get by 5 differ around 5 deferments or might take 5 different people so the Rendon this isn't needs such a procedure
38:06
when I get here depends on which individuals as much like Internet random so a choice of an individual's is as likely as any other choice of those and individuals photo attitude they all get so are under that something what's probably adjacent to the July measure are the same guy in the measure or sample who has a measure went sees subjected home only ever different eyes there are that divided by the total number of about at a How about being joint distribution a keg changes OK here as for next the act OK sales for the quarter that Reginald but with the sequentially the are here this change this is just 1 of these you mean major you mean changes native land is we have on death of Rio where are distinct values that we have comedy of these values the spending so the probability that the individual example has to be just a fortunate that the values of those like this think XI XJR independent enough I don't think that we should try 2 computers using conditional probabilities right this is the definition more or less a conditional probability so it's I could give us a catheter warned turns out this is different the patio or Keillor the same and whether they're different let's consider the case where the same what's probably the slightest case if we know that this 1 is also it so you know they want 1 of individual the population of his values has been withdrawn from the population a ready Anaya sampling again what's probably it provided seek a future 1 individual have we are not remaining with the values day and were drawing from what size population Apple and when you see is picked dowry once he isn't operators 1 last and there's 1 less individual opposition to serve is probably now not getting it 2nd time never see and whether differed only see Kayseri Romania free picked out of CL & L's different of case but we are in a population of 1 has picked out Kaplan Midas what to say so to continue with a joint just vision or tell readers
43:40
computer but the young Mr. probably isn't what that probably there that's better the win over Apple and right so by despite Apple that but this is a case where Lazio so little Over would be and case over Kaplan early get
44:07
and Katie minus 1 over Haviland minus 1 times and case over it is else if taste well conditional probably exists in these probability that X CEOs in over Avalanche a case so are they independent if they were independent what would this be a this there were independent visibly and carry over countless times L over cap ranked 5th in the Is that what we see here no it's not it will go the act so they random sample the random variables in this random sample are not independent is a big difference between the rebels and Samper had to make adjustments for the fact that these are not in the Rev. if you have in the event rebels however sizes Hideo from us saying that these are but if X 1 the Razorbacks bar is what he said now for rates of a multiple world a random variable worried that multiple I swear and take out right and the race O'Sullivan the parables as a some lovely these basis to ease of use that independent identically distributed be saw these the
47:03
same distribution that's that line there doesn't depend on now that means they have the same decision for all I know is that they are not even if their independent and candidate they asserted that the summer be Singh squared furry I went up
47:21
at times and where abide by and we get over he we don't expect that for this case at will be the case for this type of Baidoa as is called simple random sample where you select individuals randomly and without replaced take a break come back and they say would very expects
48:12
warriors for simple random OK so let's try to Pewter variants of the X par for a simple random sample Lance what In could key key key Our blacks RBT Gypsy OK so again dude if now a from West structure we saw that generally the base of a sum was a summer score basis and Robert Cove rates of excise XJ is expected value times X J bias we expect size such special by year so we should compute these comments I no it's a little different when I dare say even when they're different source but this summer 2 songs 1 where Jerry call and if I pervasive XI XII that the variants among them are equal more work that is it my I did it all right so that could do the case buying realism anyone and to do so Pappas arising was you could do that but on the cover days Azul but different so obvious but it's they so this is a sum differ from J. by J. going from 1 was like the diagonal and after I you think of pairs iconic Jay Friday Interco from somebody terms are in the summer and this summer and spread send because somebody terms were in this song and inspired right by Monday in Lagos 1 and so parents were terms here me and spread terms still there and their hands were but or deck now will use this burst into very select next the but but it however I destroyed although there already again but but so pursuing was expected that diaries that I have a E but may I done that he effort XI again as a some or all of possible values that XI there's so much as a beta Caesar J. Anthony what YCJ becomes a problem that size equal the subject an orders a Siegel and if I put the water ran outside as my church coach Is that what is at that's new expected IOX ideas or if tech ever yet and why would be because this is Furnia identically distributed all have the same expected trade him to but that's also of file or added to favorites of XI auto yet yeah I know that they have won the death of a child a bevy of appears to have a hundred 100 people you'd say double the average value because no randomness is 100 individuals that it's a sign of the values divide by 100 so
56:42
now expectations means what the randomness selection procedure so you you randomness isn't selecting these individuals will get what abut the
56:57
base of XI bad for the people of the but chou hard all 4 or 2 Bridges found expected of excise where In where we do we had all viruses overall possible buyers excites where have those well there's a lot times probably excise but and now we just put in end of day over Kapolei near our right this way yeah what that Hey look back also waste you'll see it relates the story arrears baby still here is still here I am areas but this is Neil as news where my friends what this would be the 1st so this is where the rates of excise is squared by side OK by so its return auto applications so this is new this is new so we need to compute back it she but did with the solute tutor fresh here we are trying to reinvent Robles the Joint Distribution assert their expected by product would be His the 2 buyers that they could have comes a problem take on those 2 guys we are right now if you look up there and see that things are different when cases equally well and when it is not equal or something here so much we do with this summer up into case for on saying order different because we're going for subsea different values for this and depending on whether players are not so it's but this and the 2 thumbs and when they're equal a probably is a B C K squared who he or and then we will write down the someone when they're different she added today the novel story you're With that all on Friday British yeah I do here but this is where the same sort of thing yeah fixed their effects Mayorga some alright injury
1:01:54
member who wants a a wealth forward where some more levers various
1:02:03
over X fire which what or trying to get better they are this year's with Butte all these things so this is the beginning of the competition we did this unless they're both however computers then was we do what computers with fact new squared and then there's some overall I so the 1st step will be while the 1st of this thing that's what it so let's let's do or say what were to really to which it substituted in the a but when they're meanwhile way and minus 1 randomized Kaplan Westland events of Laura Capel all zip for but when they're too different forget the line there and so the case over Kabolin minus 1 time then Al Capone but it's a drum roll bars hallway a sub super giant abilities dialog table these terms I have this Ed Kaplan minus 1 and also a not only the capital inside the fact capital wireless want I know this is just a just buyer we haven't subtracted this case of completing the spot the key the key now all are this term and this term are little different course here different 15 on the same but this 1 has a minus 1 and 1 doesn't separate that out a separate out part has a mind is what the on younger right at 1st 2 and then Evan NK over Kapolei right this 1 this guy came out here this part is here but minus 1 times and over so this is the part with a minus 1 here they also have better when we got the Midas 1 but now when I don't have Midas 1 there what are these still where the terms in songs look like they're the same Ivory combined them Fox like that because here the virus raising required KDB Alor catered be a around ice but up and do some work is allocated as well as l I get K C WCK squared and times and and times and would taste from well it's exactly that Member their when I acted up Britain's breakthrough sigh separated numerator here into a minus 1 hand and put the rest of it with this summer 1 could argue those I factored I factory this fall a ought to hear this part came down here and the rest of it now I came here they show that all right sir all what's a saying 5th South ship the cops who says right Edward says this is new Square so this is a singer squared plus new Square Mr. Singh was where past New squared miserly Myerson was weird when 0 OK now what don't you like you want this expression or what who what would you like better here this is nice that's what that's a probability this is not a probability it's lacking something missing something was lead leads a capital letter but we want to keep our specially although start with swords led off by Kaplan are here How O'Reilly and so what we have their way out of a sound that looks like this son was like that following a is CTK and carryover Kaplan a Elise Cl a mail or capital and throw some little situs AK times a L K available for Monday and that's equal to that is that like every rewrite this table 1 2 a case times some equals 1 2 are AL and how do not apply to sounds together yet not quite all the possible
1:11:12
pairs together from the 1 from the 1st one from the 2nd 1 from the 1st what would have been a case it was the 2nd there would be a alright so get some Ocate sale so what's on that looks like there's there's a square of Osub this is the sum
1:11:37
paid 1 to bar C K & K over capital Eds to work square thank you back so what is a certainty and so was that some Neil right but chief she is key so right OK while I've to new squared terms such combined them what was the toll call Fisher New Square here that Kaplan like 1 dividing line capital and minus 1 Kaplan writes warned by like everyone most wanted 1 so hot howitzer by will account for the New Square just give me a square and for the same us where we are Miners squared over and and to recall what is that that's expected value of excise taxes they what worries ties still has looked into history here trying to us and we spoke up like that this was squared and we decide that substitute in the values now the first one is minus Sigma square over Kaplan minus 1 last that's what matters Over there and this is new as New York Times News squared Swiss track New Square His words cancel we conclude the comments Of XI XJ is Elias must wear over capital and what it the key and know what do we decide the various of XI West Bank so let's say about this from all quarters as means of XIX year independent what's the covariance 0 0 is an expression of and they are Our policy call rates depend on the population size and population size gets bigger that Grace gets smaller so become more like independent random rebels when a big part of it that doesn't want have no lies in negative you have to pull 1 value isn't likely that is more likely less you you that value again that's likely so can negatively correlated more so long as more the big smaller population the more dependent on their thing if its capital is big this number will be small while not perfect picture Sigma squared perfect sizing was where the population is bigger the debate is exciting resist number gets what with capital and that they want and so it's like this I suppose you're on you're growing salmon know they put barges are the ocean and music Jesus Stalin are good farmed salmon they have a bunch of salmon big Our are its was you want a measured the sighs of typical salmon the art of popular capelin movie The number of salmon and he pick out 10 salmon from their without replacement on that there be some correlation between their sizes will be there relates to various enough hobby population now that sampling replaced now suppose we do this with 7 the ocean this is a very big number was considered a resident sampling will replace and and sampling without replacement so what would sampling with replaced me you can't fish catches salmon measured so back and now you can't another what how likely is it Texas fairly unlikely right so maybe sentence so sampling With a place with our placement become very similar when the population size is enormous and this is kind of expressing their if you're sampling with replacement there the writer rebels are independent B 0 throughout replacement there's a slight dependents bans on but if you consider maybe something like sampling sound from the ocean for some other population where the numbers are enormous setting with or without replacement fresh nearly the same but the population sizes a couple hundred bucks could be some this could make some difference but so now let's try to recall what were trying to fight the various events are so we have to but if the I here true she ordered will or we have to be back quickly e a another execs ideas singer squares this is a native of what but not all them so well we have put in the comments of River valuing at Kaesong River next step will be to reduce output thing so I did so throughout the basin expires this nice but this up when I the same with a different beast things have different values when I Siegel a I give the basement side when I from jagged for different substitute in the life of this is was greater public times I that up in times my life and squared BC over and that's 1st secular on areas where here all these things are these same thing all those terms as a lighter Sigler squared over capital and wears what Army Times and that in and spurred Wednesday because there were and squared terms altogether to begin with and my separated out In Lansberg the land left is simplified there's a stigma squared tagging factor out I get 1 or both here I did us Water minus 1 over this might my aside when the land the land minus 1 over if I think you get batters are each this comes here this is the land times the land 1 by 1 part cancels with but squared on here now moved back Luo and their 1 YES my 1 or Kappler writes work OK so it's very select spa is the population variance Over less than 1 might as well and when 1 or traveling writes 1 you could also combined this winter 1 fraction the denominator would be capital and minus 1 which reads expresses this capital and minus 1 over Kaplan writes 1 of them track this West tractor mice when it becomes plus 1 Kaplan Lisle and that's another way to express Bates far cases for simple random sampling what did we get when not the axes were independent we got this is what you get when they excise independent when their from simple random sampling there's this factor again if France the sample size over population size is really really small suppose that were on baby expos Earth 30 million both say they're hungry borders in the review board president of 50 million 60 million no area who actually vote 50 million 60 million and of population solar suppose you sample of thousand voters further opinion will a thousand Kaplan a 60 million over 60 million shares as of 10 3 over 6 times in the 6th Serbia 1 1 over 6 thousand bats 1 1 minus 1 over 6 thousand is about what . 9 9 for magnetic 9 is 9 9 of review visibly almost 1 times it ignore this if this fraction is really small but sometimes destruction isn't all that small yet do think sets evasive X parte said that so this number be toaster 1 action song that numbers of bucket so now we wanna do things like buying confidence intervals From sought are you limit their new source review that for a while but for us did eat tree all now it is a settlement there X bar is what see Aborigines out of X want to accent divide by and subtract means abide by this factor your prospering normal now this factor single related and it is a square the Razorbacks of are busy dedicates right In the independent case expires at soda other words we rewrite physics my new all over Beirut other various of ex parte is approximately normally distributed with large and says he saw large enough applesized size the same thing is true over here in this case it went capital and when the letter Kaplan a small Central but there was still Wang that but he for people but as proof or if is today now but soap fuzzier Belfast is being number which satisfies look at he key for Oprah he too much this relation probably normal random rare Meza responses is bigger than that physical Elf other words very under the graph of the density of the normal means race won the right of is Alpha then had a sore toe all 4 4 4 4 this probability equal us is what's probably being here at what Syria under the graph what Syria to the right of this point out over to what's data left of this point else Florida so what's area outside of edible Alpha so the of JTV shares 1 myself so stop there are also To so excited but what's happened we want a fine we really is ex parte estimated new now we know that this but his nearly normal but do we know what this this is what is what is vague next bar like evasive Rex Pa where we don't this this we don't know that so we have to use a sample of vastly so users sent over there but we can put this in there because we don't know what it is to get you something you don't know can you sometimes but the of but updates that find way decimate find an unbiased estimator for parents of ex parte and they will put that in
00:00
Universe (mathematics)
Archaeological field survey
3 (number)
Student's ttest
Weight
Mereology
Measurement
Resultant
02:45
Inference
Heegaard splitting
Category of being
Average
Multiplication sign
1 (number)
Total S.A.
Student's ttest
Measurement
Local Group
Number
06:09
Average
Mortality rate
Distance
Measurement
Event horizon
Number
08:50
Statistics
State of matter
Forcing (mathematics)
Sampling (statistics)
Archaeological field survey
Student's ttest
Measurement
Event horizon
Grothendieck topology
Subset
Number
Inference
Voting
Universe (mathematics)
Cycle (graph theory)
Bounded variation
Condensation
14:01
Area
Multiplication sign
Correspondence (mathematics)
1 (number)
Sampling (statistics)
Basis (linear algebra)
Water vapor
Total S.A.
Price index
Surgery
Measurement
Local Group
Number
Flow separation
Summation
Voting
Term (mathematics)
Energy level
Right angle
22:56
Multiplication sign
Prisoner's dilemma
Moment (mathematics)
Staff (military)
Line (geometry)
Number
Summation
Spherical cap
Term (mathematics)
Square number
Theory of everything
Cycle (graph theory)
Game theory
31:32
Randomization
Multiplication sign
Sampling (statistics)
Parameter (computer programming)
Student's ttest
Mereology
Measurement
Number
Square number
Circle
Pressure
Subtraction
Resultant
37:04
Axiom of choice
Randomization
Process (computing)
Euler angles
Multiplication sign
Distribution (mathematics)
Sampling (statistics)
Independence (probability theory)
Measurement
Number
Mathematics
Operator (mathematics)
Subtraction
Conditional probability
43:37
Spherical cap
Lecture/Conference
Multiplication sign
Basis (linear algebra)
Sampling (statistics)
Mortality rate
Event horizon
Subtraction
Random variable
47:01
Covering space
Randomization
Beta function
Decision theory
Multiplication sign
Distribution (mathematics)
Sampling (statistics)
Basis (linear algebra)
Algebraic structure
Water vapor
Line (geometry)
Mortality rate
Summation
Sign (mathematics)
Lecture/Conference
Average
Term (mathematics)
Order (biology)
Directed graph
56:40
Area
Randomization
Group action
Product (category theory)
Multiplication sign
Distribution (mathematics)
Mortality rate
Cartesian coordinate system
Expected value
Manysorted logic
Lecture/Conference
Selectivity (electronic)
Right angle
Subtraction
1:01:54
Multiplication sign
Expression
Line (geometry)
Mereology
Event horizon
Numeral (linguistics)
Lecture/Conference
Term (mathematics)
Factory (trading post)
Square number
Right angle
Moving average
Table (information)
5 (number)
1:11:11
Musical ensemble
Group action
Randomization
Confidence interval
Multiplication sign
Water vapor
Function (mathematics)
Mereology
Fraction (mathematics)
Square number
Decimal
Algebra
Area
Theory of relativity
Sigmaalgebra
Sampling (statistics)
Physicalism
Substitute good
Proof theory
Arithmetic mean
Normal (geometry)
Right angle
Directed graph
Point (geometry)
Trail
Perfect group
Divisor
Event horizon
Number
Centralizer and normalizer
Crosscorrelation
Population density
Term (mathematics)
Reduction of order
Subtraction
Euklidischer Ring
Alpha (investment)
Graph (mathematics)
Autocovariance
Independence (probability theory)
Variance
Division (mathematics)
Line (geometry)
Set (mathematics)
Mortality rate
Estimator
Summation
Voting
Network topology
Dependent and independent variables
Theory of everything