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Expected Values
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Title  Expected Values 
Title of Series  Introduction to Probability and Statistics 131A 
Part Number  5 
Number of Parts  16 
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  10.5446/12890 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Mathematics 
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00:05
The course Easy or Books Office Alright hired a hard Quevillon visuals of affirmation against die Greek So differed from a tortuous guess You but I guess that means that work will that are you are right we do have passed Friday at Wikipedia either ever countered said that that its phase of her life for which means the observation that life and because a solitary of baby Monday hand in the assignment otherwise danger agreed that a was that readers appointed you can hand Let's go get the option of having a product But it is which on Monday A But I Did The Were still in Chapter 3 discussing drug rebels basic general ideas is that up times random Berlusconi come in pairs They hide what size I ask where you're a size 14 sure bigger size 14 so you had to buy online all of the joy I know that my teenage son has size 14 sometimes by my life are but person is population has size 14 seems pretty low but the 2 sides of it or would you like to take new digital random you might capture human being measured his foot size and measures like these through random variables size and you'd expect some relationship between them shorter individual you might expect shorter foreign original might talk that What's going on with our history and rebels tried capture somehow in a joint distribution function the relationship between the 2 and sold 4 Rob can discrete random variables this just means that She for political There's some probability mouse function on Paris and the probably that the random variables X and Y which are somehow correlated are lasted little Aleksander lie respectively as this summer This would be a summer Laura at most probable number of points despite a reports to be floor in a continuous case But All But be dry just hit a scholar joint distribution function is killed by role some Well labeled as saying Capital capital why similarly here But truck day so it's continue of years examples of
05:44
EU A net order bond I guess not right this is We Aero miserable such prefers thanks The That's part of a sense of the pair X Y where X Y is little to reduce funding just Arie's are so Georgian in the fight while it sink back So where it remained to be from exerted interval in underlined But it For all of What is it means for forever will review foreign leaders it means something about the density of the structural means at the density is equal to 1 over the length of if See happens to be in that and 0 if not What was what But We don't see is constant under almost constant visit cast as determined by the requirement that the inner density What number 1 of the density over the whole line has to be 1 We read this function over the library get well Over most of liners No contribution from might favor a yourself contributions from each with leaders of contribution we just give a contribution to the rollback back on this but we could be entering a constant or where you can get the constant fans alike general that is 1 Not go back up here is something of a uniformly distributed on set watch its density Can't deny that such a 0 an Turner constant Right here last be 1 this Means that joint density for That is why I am But The we won over square get better verbose lasted or spurred on 0 if not Took Here's a circle radius r Center at the origin novice but the point corn exwife and thus look For a joint distribution function of this chair at the point of Alexa life of its defined to be that Ed Terms of formula race Since all the essay From eyes and renewed acts
11:27
From my Grenada which means all over This Regen oust plank Show And sold I was a baby talk mineral Well below the hard to work complicated but not too bad I don't think Do it for arbitrary X and Y had was considered by trying to find out what is distribution of acts and also to discover what his future wife using this So we get Actor won reply are squared on this region variable or choir squared shaded why don't I achieve this area over here Cabinet zeros or right over there and fancy 0 outside of this so that a role would be a honorable hire square on the set of a our let's try to find the probability that Cal X is less able to stop Or that it Or can we get that can we get this from there Other information So we do have this I would cap but The Lexus Cup quarters worldwide the Centergrove care we get rid of this Restrictions Now How be ready you X plus Maybe none world Look at the picture from an errant tally when It is When this is no no longer become are restriction by guests in other words take this value What if they protect this to be infinitely what region that such a region because Why photo video music was up here right And every thing tools left out This line that would be watching region The bay All this factor all I need is widely revered are but But they were seeking with wife and there were world every left Acts as the extra 6 been feathery Herbert we integrate 20 sanity here He is And there we get Rhodhiss richer boxer for cap elects godmother Gamberger distinction between Kappa waters and This work whatever you have Surely the surrender urine might find distribution of X and you know the joint decision Becton white despair rate the wider all from eyes proving that gives it for acts how'd you get the decision function for y In a great yet take still extra Vidalia words in a rapist from an area next from mines Kennedy can such distribution function of why they by this That's is taking extra familiar What's This says if you know of a joint decision of X and Y you can retrieve the single the decisions and would give that those are named Carlos marginal Filipino just joint decision Wrexham like you could retrieve individual distributions of exwife of marginal Using or other variables so here
17:07
Last we would integrate Einarsson Trinity's 2 x mineral biased and they have X Y But The new rules TV radio And I think we can Say kind what that is it's 0 death axes less than minus At its prime movers line over the left of that point but were courses point miners are 0 4 acts or you're probably being less that it would be 0 because what's the value density over here but Bowden once I get above my eyes are starting new stuff A little different to the left of 0 than to the right I when axis of the left of 0 but bigger than my Assad The cake Was Exs right here what we're better order properly cap collects being West I be this area divided by higher in that area by our fires bird argue it will find that area Sri computer 1st leave it to you I know I can do it but would you like me to do it up so started to act I can do it easily when x is 0 with the probability that capitalizes lasted 4 0 With Well it's the proportion of the area that this occupies in the whole thing Half OK we deceit to compute that area but me turn it On its side is circle radius Arias and we won't find theory above a certain height of us Hero Our what's this court orders this point here at lawyer here because height That's R This is a test and so that this quite Oreos what corner Are squared squared 0 and this point would have 4 minus square of are squared my assessed squared and 0 and Oriana a finely area what what what Syria this retained a region that Look up S times twice a strike It's a sea legs times a base that rectangle Hockey you get Tania blue area Malaria is Jesse Mineral Of this function From here to here Areas area under a graft rappers in row of the function of Row minus screwed are squared wiser squared plus are squared squared and what's that function
22:57
Well Chris is Circle radius are said origins so expert was wise burgers Osbert so why is part of our expert And body that is wrong We Substitution are X equal are ordered it's up to him you might the square root make substitutions of moving inside a square Quips Oh signed as a good choice and NDX would be Are my are signed data playing and so this would be done before I make sea change of to this I don't I And then there's becomes twice with the substitutions are squared minus X squared Becomes part Signed data right What are spurred further backdrop we are that's 1 minus school science burned On my signs where sites signed signed here but are that I get minus Of bias are squared another sign signs where data Decatur now on the moon tax went from Theory here so electors Um What is stated when is a 0 waters they have to be so say this picture must be saying Ira too Quite Texas 0 downsized sat back at gets its X 0 they aspire to comes on tour this anglers that angle be well leader They this side of outsiders Cosigner so cosigner data when you're here this angle would have cosine 38 less or are so figures are cosine so you go from 0 to ah cosigned SRI They audio resigned square It was at site sorry us Yeah you're right side a Siberia sites where at what you forget it 1 is 1 half by a surplus areas might work
27:55
You can check by putting in illegal 0 2 0 years when here Figures 0 area onetoone my so we put Why Myers foresight to fear The Parents so that happened at 2 Cancel this gives of later and then Harold Koh sign is Signed Bowyer 1 half of a fright And so it becomes minus are squared times 1st evaluate the data between them The top limit the bottom against oxide As and nothing for 0 and Emilio side of choice Oxide he or she says Plus But what data 0 and data 0 sigh of a to 0 I heard Now if review It If this were Signing You're not Toys R signed or 42 You have to use a addition formula City as We But sign of past me She Start with signed off on or how The strike Yes it up I always have to check this 1 So the real pioneers score sign a course I B sign a sign that said cosine magic part of side a and me I don't think it's signed a cosine being passed cosign a sign of be alright so we could barely it signed up to a A sign of a a Co Saturday Of a Soviet sign of this thing that only take that we get as Overall R and Whitacre course That pedophile stuff is fighting We what right And that gets through the half and that you have to rewrite As we get the same terms and now the cosine of the oxide are
33:04
Rodhos The We just ain't the going a site overall are back What's a sorry sight Court of our squared minus So the cosigner singles is my by that and so we get miners are squared oxide over are bias as over Times Square of our square by square over and that's for all but then we had to do what This area move out to subtract green area of Final answer it's 4 x lesson 0 and vigor
34:13
My Sr But for also by necessary yes Which rose 2 0 s wherever Hours where squared Syria Green member air end up as saying This is seen as a singer and bombing of broadcasting these are set a nest West might minus escort of our squared minus squared minus oxide or are and now there's been assigned mistake made somewhere no positive number here so there was there's a minus sign plus sign somewhere here but I still wanna go looking for it Maybe Have They they Are we all they'll come into play because of differing from year to here Are leaders in the area from that part don't that CIA O'Leary from this point to this Tough August source I'm sorry there's some sign carrier somewhere beginning a positive number But I'm shirts Educational who got the calculus and were but I just be babies more partners What you might remember evaluate signed squared and you might remember how to deal with both sides of our science But a way this is you would have to be bottled There's always Rex lessons 0 of Brecht's Wigan 0 gets viewer with different because 1 axes Bergen 0 and you're starting to look at areas like that but the needed just actually subtract 1 might say something like that which we just itself that right answer here But we just under are attempted do is find out I'm sorry there's 1 last I think that I didn't Here I divide by pious where its fourspeed a relative area and say they here So it is fondly area this can you use that defined Areas like this With what we did satisfying the unsavory right So this area will be oneliners that 1 competition that's so Are at a loss about where the money is signed That's about integrating enough on Monday on A masseuse somewhere near Jerusalem minus signs that it was right everyone of Belle all of the notes and maybe but put up a Web Press he From Uh on sex with origins He For back So On the virus Yeah Then At that time we were I mean computer that probability of a that's a half life death That World War so well the model probability the bacteria the viaduct piracy were so that I think requires finding area . 0 somehow had some stuff here Probably list here Up I wonder
39:55
I I suspect that the The minus sign came in here and probably should never ever ask them what I'm looking for for my assigned to build sewer That's prop pour over there Well It might been that these limits immigration should have been reversed And that was after my a case so Bomb again not retrieved the But King the old thought I But what will But it Was was all over Are Or or or or or each she put or but Show For how As she is It is tipped off to tee but he she is full Eritrea the Meyersville decision functions where you get My result fracture the Asian bond racks industry cases you take that Joint PMF probably mass functions and overall they 2nd variables but In the 1st variables only up to act Or and the continuous Casey period out 2nd verbal which is why 1 and in the 1st River area Any reverse this for us of capital The scan also gives the marginal probably mask function of acts at all or why in the this week case All or be marginal densities of X or Y In the continuous case here you get me in a discreet case The marginal PMF for capital X is just some or all we have of ever This is being joint PMF B and wore white uses some overall you only half of you why where this is joy came out for for X and Y and a continuous case against for the 1st ran a rare will is gotten by entering the density of the joint rebels respect for the 2nd quarter fixing the 1st best
45:54
For why as got my AARP Xray were so but said a look at an example arm At bay Wasn't where continuous their breath our honorable and their joint density core is given by this shift Exs Between 0 and why This is the value of the density in all the cases the density is 0 F f that's fine me module lessees 4 4 4 4 4 4 0 But balked at all Of But All 4 Q Fish Fish so let's NewWave switcheroo 1st Exor wife extra playing so we do this J. Gallo on I guess we could start by saying that Maazel density 0 variables by that but still This is still the right formula for what this function when X is negative Going to this 0 so let's do navigation here when X as fast as we know advancers Texas negative ads are not always ask When is this density different from 0 and the question is in the thievery Alexis Suspect Poseidon number for what values of is ever vaccine different from 0 well has beer to X sorted the Laura Eleven evaluation here X for relatives of U.S. annexed this is 0 here from next to infinity Everex meet that range with me As bigger next That's the value of the density of and the square where you're Biedermeier slammed V do me They and this is easy if that's minus 1 or OMB UMC 11 hasn't played a minor slammed a is my Aslam be evaluated between accent in giving checked out by differentiating this thing back here The value of this at infinity as well somewhere fast Said The city is 0 as part around but there's a minus sign in front of it and that acts was abducted I've got to get my under EU demise slammed the next select distribution this except excellent Award from parameter land The model that distribution of capital exponential with tremors that Dallas Skippy marginal distinction for why we do that by carrying out the other variables But With or So on again this is 0 FY wise negative theory All of this formula that it's the interval of the joint as image I guess would be 0 wives negative and device positive Wider able area the Xray and always ask ourselves is different from 0 paralyzed fixed so it's a question of For values of you is from wiser explores the number and that is different from 0 in this case when the the other variables between 0 and wife other
52:59
Between lot So this roast beef from where where 0 2 1 And what survive the density In that case when this weird Biedermeier slammed a white why why this is Everview Y FOU y Latin square What they were in Iranian These folks feel very a In this as independent news employ about so the where we get Biedermeier demise slander wine in a row from 0 YDU is Aaron this I believe is being gamma density But Alfie that to land The gamma so you treat this as a brick formulas for the Basel densities also find something called the yard conditional as a in a continuous basis which means they density of Cadillac's if you know the value of capital They should think Ibiza probably cap Alexis There's given capitalizes on TV that I What's the probability that aid was a probably a given me it's probably a year sect being over probability of being so in this case you fervor probably be years
55:36
While a wouldbe Catholics is the Lexus is capitalized the wire what saprobic tablets is Alexine capitalize the the life of a it's not exactly correct But This is what that would mean the dance the joint density of the appear divided by 3 density before why sorry this is probably cap Exs elects capitalize alike byebye probably capitalize a lot better way for a condition I acts get the same thing in the numerator but now we divide by fancy for capital access the floor this week case It's the same except we used Kiefer Ali mask function I guess it's a joint probably mass function divided by being marginal ones and going the other way For what At So let's returned Example would just before the break that's a continuous computer be conditional densities But Well I told you his but also asked what does the variable Why do A water X my range over and they began this week values continuous in industry case there would be taking and only a finite or Conable said about maybe the natural numbers axillary because the natural numbers are written confining extra widening like a fine etc That No bacon range over a life that means continuous but formerly the but there was indeed Sweden and continuous redefining models discreet you take a sample and continues to to think and you know what the yearold signed was a sample of the letter S or so let's see Has it right . com earache done the model best here Or as close A blender the square Why he slammed Y and wireless parts I I think that's a week found Salas finally a conditional Betsy I hope But But OK so on when all this we differ from 0 4 hours 1st all the definition says it's this as a abiding by this we should make sure that the Lexus positive I suppose VAXes negative vote would be conditional that's of acts given divided capital idea X kept low let's get sick on negative body You know by capital OK And now for a year I guess we assume that be acts between 0 my otherwise is 0 Why is this case true could act Cirby beer wine no face so the rainy case of 1 and writers here is given by that formula Landis squared Biedermeier slammed Y and the denominator is our Lambert you my Icelander a X and this is lavender each the miners slammed a why sex
1:01:51
Salt freeing charter capped Robison 1 kid That Sir The only thing I think that covers every now wants father conditional density well this is 0 if phone wires lesson 0 For why is bigger than Was earlier if Arabia could capture stories they by saying here from 0 Leicester x Lester goal novel In a case where this is not true of the diocese Iraq Ajai Kenisha lessee otherwise Fred L I'm sorry it's my dyslexia comeback arms was divided by what here why so I have of capital wildlife and what sets of capital why areas and the squares y You might slammed a wine which is 1 over y 0 less than acts on 1 or wife and here byebye is were would abide by March on the backs of the numerator Islam the EU while the denominator Islam that you might slam acts Doing the formula for our others where lambda Utah miners slammed Y and so let's try to interpret these select see distribution of tablets of winnow capitalize It's that what is at present as a function lacks the suspects so why fixed here This does not mean that the lesson 0 a bigger than life density constant under from 0 work France is a wonder why 0 off general didn't see what kind ran rural given Capitalizing Cabello acts uniform decision on theater will work to Iraq capitalize what about
1:06:20
This is the capital why given the like Apple acts well if Bill x 0 what is this Alexa 0 is there for excellent so we would say this is a test for accidents on from that to infinity So What is it is it was is with the backs its Excellent so complexes next century history venerable capital why given value cap exponential on me in a row from extant Freddy so its interpretation here Think of light bulbs tablets is a lifetime of love life like has let the solicitor parameter lambda it burns out sometimes the moment when result orders little acts and you're told that now what happens we normally replace like Bob with another 1 and what's lifetime Well exponential when there's a burn not at the moment Y but what would a decision be well it's not 40 into the socket bomb attacks in and this is the time after that would be when the 2nd Why would be the kind of looks like is Lee Some of 2 end 2 exponential and gas for The terms of Al actually with some serious Arizona a problem with y 0 but with densities on you don't have to be careful about whether these airstrip or Not strict inequalities because nearly 40 do you integrate these functions and contains a value of functional point that won't change Does that contribute 0 area so I will be very careful what was bigger than Aubrey equal to my dance all or when you have a massive be very careful because our taking sums and you might be putting a positive value over Soviet yet be careful with These inequalities When you're doing here maps discrete but not when you're doing that to example would a discreet case would violate disarray ran a rare whilst 2 Now At crop It is an urn and 3 coins in and a Europe accounted point random and tosses and qualities of the head sites The North but Frank tales and a each resident probability of pets so Let's 1 hour The scholars point acts that scholars y Danza scoreless Cawein z or Cool Will pull out of the way but Indeed Urgent That's was Villa following PMF for all of And he the It OK do All those numbers add up to Warmth That's a relief say this adds up to onethird This adds up to Adds All of that There were these probabilities means of weird calling out and then tossed probability throughout Klein y and you got ahead Thanks This is probably group going Zenyatta tale But So it's try to give me modules decisions So it through on ground rules here as Echoing drawn but takes on values But for the LECs little idols E N Y will be the weather heads or tails so I will take on values vary 8 so what amount for lacks puts a probability that it's Quite quality effects they acts able to sell or he amassed over the other variables were the Goebbels AT & T fits the is going to have access and we some on the secondary will take BOX AT XT Adam often where we get on onethird a problem requiring the Lexus drawn on what about these marginal for capital while we fix the the little white observed
1:15:35
This is still a variable camp Alexis what's the probability that b . y We fix the line we some over the other 2 terms early have why Y 8 why at a point 6 plus 6 that's library want There were 3 marginal probabilities picking quite easy by their bravery cassettes only remaining thing but if you have 1 knife tonight you was this Aziz marginal out for Expert whereby the library So for the Xray will we some across this way For reliable will 68 and some there 1st So what's that Marginal density for wife of being ahead Bobby acts us feel age 1st year of disease which would be 112 for sponsors and what set equal to 30 and of The show a month to All of which For the marginal probabilities tale that's the key and some over the And here I think we should credit for your very 6 can new verify that proved so these are the Model S 4 . for the better Taylor They wear what they can Nichelle 4 It was probably a bad if we knew we drew drove point X at City of joint PMF divided by a marginal for X the joint probabilities went 12 And was a probability that we do quite exits onethird 1 4 what's approbative tales given we drew quite relaxed order to autumn Ezechiel's problem And onefourth volatile us to be fit for farmers tells us what The probabilities are 4 point x finances according the man as bubbly onefourth of heads and reports of failed to hit some loading yet they modulus visions are the initial this visions giving coin wire or coin Z I wandered of student but studio where aren't Why What's a probability that it's quite tossing If you go through this procedure new 4 FedEx a just 112 I'm ages 13 or 36 Eliza's 3 over 13 and make it also do probability that It's while you're tossing given that you BY HSV1 Sex and the probably They had pianist 13 over 36 services 6 over 13 They and then finally Watson is a probability that It's scorned you selected if you get ahead and and finally what would be the problem that point z given the impression and it's Remaining up for over 30 There were 13 per se key X given Wyo Z UNH his former That's what they saw looked like learning A discreet at what it Knudsen something really hard motivated me when I was a student the teacher would save for did harder this sector size that I knew you 100 want what I would get my attention or it's a patent is really hard for some time that listen more something that twoyear tensions were removed something very important now well of this says on news but Which go all the but Degarmo decision comes up a lot and statistical problems You probably use Tripoli from time to time the reports or were after exam Eaglewood years for sometimes releases the statistics and you see a histogram of scores on the test and what
1:23:42
They generally see apparently what's It Turns 80 have well ITC that acts like photo refugee someplace is a duty extraordinary bad girls newsy see this is a normal distribution he also fewer Plott Heights of individuals see something like that so People studied art correlations to look for maybe some hidden cause light Smoking is bad But you might Asked about individuals A small of lung cancer and they find some correlation of and they start and see whether some cars out on doing over listed distribution it was mentioned in the book With Our In the case of Karen rebels the 1st element in the pair was father's Sutton was tireless and individually for normal This revisions and they want something called a joint nor distribution So what's the enjoying normal distribution X Y Group as a joint normal distribution sows continuous Eireann a rare loss for all of us Both to off all over but it did All but 4 4 4 4 fish she coup arm move where 8 and John I Rights Estonia I'm The check and the right formula Amery room back that real danger Of 2 or 3 Each week Up to now his role an exact plays a role there Yes So he or He And this is for all acts of of role as a number bigger than 0 Sigma acts and singer wine will be numbers bigger than 0 new acts and could be any number rollers somebody indicates the Connection or degree of dependents between X lot of Yard these are rose as robust growth and a of it and It's not a memory that Density that means it can read this overall are to you get 1 Orbit so was sleek that therefore for moment Arturo and checked for something called attendance is what about an extension of the idea in Events to the rebels What each scored for both 4 0 4 4 4 4 4 4 4 4 day for terror reverie roles are independent Give their joint distribution function factors in the Product down their individual distribution function So let's stood couple example slump in this example Figaro Regula 0
1:31:20
What was Then becomes and by way Utah If you have a density Whoa this become the density factors in Product of densities that beauty continuous case where but that is replaced the joint him out But factor and the of the individual him for In this example here with a growing 0 0 What does it become wheat won over troop I guess a Sigma will I get rid of this group because where inside their exponential by half Little X minus music capital acts Squared over to was the capital Cabot squared plus little wire minus the use of capital while where over Jerusalem as capitalized word whatever the next advantages And I can write this 1 over square root through Sigma sub x squared Being the minus 1 have blue minus music capital Over Oh not to hear and not to hear The fingers of capital X squared times 1 over square root Chubais said most of y squared is 1 have now might news of capital squared over over life Some dance joked as factors in the product of 2 functions what property please to function that All the extra rules out to be 1 Oliver Hawaii rules have to be in the other Nobody independent and this gives marginal density for capital axis gives a marvelous 84 Capital wife and their independent so in this case X and And Open Case End up any case arose from 0 this for factor and there will be detail What's the their But Skocilich case of young ahead missteps But it got ISS density of of highly independent In this stance may be factored in the Product of a function of X a function wife Well that's sort think from Omega about what Consequences this might have up what conditional distribution and density when Exon wire independent This is the definition If their independent this factor says the product of the marginal densities And that would just give the fiercely for tablets in other words given information about the value of capitalizing on that change the statistical behavior of cap elects not at all from a sure why did you nothing no information that true here I know some of your wife doesn't tell meaning what acts answer Hornets are I tell you why I use on um 1 Myers epsilon off or bigger than 1 might accept wires 1 epsilon are or maybe even if you like epsilon on my stepson are of light myself so I would be something like that that tell but acts She is As outlined its course posted a 0 this line was higher where when and where is actually X ranges from year to year so the son of the year Act gives more German deterrent Kosovo's remove the value 0 final why is really big denied no be China Moss's agony information about exercise So acts independent and fact this function capped factor drew product of a function of X As a function So do more independence later well actually a lot more so ever have before Member of this I'm inside and tell Monday enough without the backs assignment
00:00
Point (geometry)
Randomization
Product (category theory)
Multiplication sign
Distribution (mathematics)
Measurement
Functional (mathematics)
Open set
Number
Explosion
Probability distribution
Lecture/Conference
Mathematics
Phase transition
Lie group
Random variable
05:44
Point (geometry)
Distribution (mathematics)
Line (geometry)
Mereology
Weight
Functional (mathematics)
Number
Hand fan
Arithmetic mean
Population density
Radius
Lecture/Conference
Wellformed formula
Term (mathematics)
Order (biology)
Circle
11:24
Point (geometry)
Musical ensemble
Divisor
INTEGRAL
Decision theory
Multiplication sign
Distribution (mathematics)
Water vapor
Rule of inference
Theory
Expected value
Population density
Spherical cap
Lecture/Conference
Square number
Circle
Area
Statistical hypothesis testing
Set (mathematics)
Line (geometry)
Mortality rate
Variable (mathematics)
Cartesian coordinate system
Rectangle
Functional (mathematics)
Radius
Order (biology)
Right angle
Marginal distribution
Thomas Bayes
22:49
Axiom of choice
Area
Water vapor
Mereology
Grothendieck topology
Substitute good
Theory
Sign (mathematics)
Goodness of fit
Mathematics
Radius
Root
Angle
Lecture/Conference
Square number
Circle
Figurate number
27:54
Axiom of choice
Area
Addition
Greatest element
Sine
Multiplication sign
Limit (category theory)
Mereology
Trigonometric functions
Grothendieck topology
Sign (mathematics)
Wellformed formula
Term (mathematics)
Green's function
Square number
Subtraction
34:13
Point (geometry)
Building
Line (geometry)
Multiplication sign
Decision theory
Insertion loss
Mereology
Frequency
Sign (mathematics)
Population density
Moving average
Analytic continuation
Subtraction
Area
Calculus
Mathematical model
Limit (category theory)
Variable (mathematics)
Functional (mathematics)
Charge carrier
Right angle
Marginal distribution
Wahrscheinlichkeitsfunktion
Resultant
Directed graph
45:37
Area
Addition
Shift operator
Theory of relativity
Range (statistics)
Distribution (mathematics)
Exponentiation
Infinity
Mathematical model
Parameter (computer programming)
Average
Variable (mathematics)
Functional (mathematics)
Theory
Number
Population density
Wellformed formula
Negative number
Square number
Marginal distribution
Subtraction
Position operator
Thomas Bayes
52:58
Gamma function
Latin square
Range (statistics)
Basis (linear algebra)
1 (number)
Sampling (statistics)
Water vapor
Mathematical model
Mereology
Functional (mathematics)
Fraction (mathematics)
Population density
Voting
Numeral (linguistics)
Spherical cap
Wellformed formula
Natural number
Negative number
Square number
Analytic continuation
Wahrscheinlichkeitsfunktion
Condition number
1:00:54
Area
Logical constant
Decision theory
Distribution (mathematics)
Set (mathematics)
Functional (mathematics)
Fraction (mathematics)
Population density
Numeral (linguistics)
Lecture/Conference
Wellformed formula
Square number
Uniform space
Condition number
1:06:13
Point (geometry)
Complex (psychology)
Randomization
Group action
Histogram
Statistics
Decision theory
Multiplication sign
Parameter (computer programming)
Student's ttest
Inequality (mathematics)
Rule of inference
Grothendieck topology
Number
Explosion
Mathematics
Population density
Spherical cap
Lecture/Conference
Term (mathematics)
Cuboid
Taylor series
Area
Statistical hypothesis testing
Exponentiation
Moment (mathematics)
Line (geometry)
Mathematical model
Variable (mathematics)
Knudsen number
Functional (mathematics)
Local Group
Summation
Order (biology)
Module (mathematics)
Marginal distribution
Resultant
Wahrscheinlichkeitsfunktion
Relief
1:23:33
Statistics
Musical ensemble
Divisor
Multiplication sign
Distribution (mathematics)
Insertion loss
Event horizon
Rule of inference
Open set
Number
Mathematics
Population density
Crosscorrelation
Causality
Manysorted logic
Spherical cap
Root
Wellformed formula
Square number
Analytic continuation
Condition number
Product (category theory)
Normal distribution
Element (mathematics)
Moment (mathematics)
Independence (probability theory)
Multivariate Normalverteilung
Line (geometry)
Cartesian coordinate system
Functional (mathematics)
Local Group
Orbit
Connected space
Field extension
Category of being
Right angle
Marginal distribution
Limit of a function