For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### Order Statistics

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Premise
- Example Question: How Tall Will the Tallest Student in My Next Semester's Probability Class Be?
- Setting
- Definition 1
- Definition 2
- Question: What are the Distributions & Densities?
- Formulas
- Example I: Distribution & Density Functions
- Example II: Distribution & Density Functions
- Example III: Mean & Variance
- Example IV: Distribution & Density Functions
- Example V: Find the Expected Time Until the Team's First Injury

- Intro 0:00
- Premise 0:11
- Example Question: How Tall Will the Tallest Student in My Next Semester's Probability Class Be?
- Setting
- Definition 1
- Definition 2
- Question: What are the Distributions & Densities?
- Formulas 4:47
- Distribution of Max
- Density of Max
- Distribution of Min
- Density of Min
- Example I: Distribution & Density Functions 8:29
- Example I: Distribution
- Example I: Density
- Example I: Summary
- Example II: Distribution & Density Functions 14:25
- Example II: Distribution
- Example II: Density
- Example II: Summary
- Example III: Mean & Variance 20:32
- Example III: Mean
- Example III: Variance
- Example III: Summary
- Example IV: Distribution & Density Functions 35:43
- Example IV: Distribution
- Example IV: Density
- Example IV: Summary
- Example V: Find the Expected Time Until the Team's First Injury 51:14
- Example V: Solution
- Example V: Summary

### Introduction to Probability Online Course

### Transcription: Order Statistics

*Hi, welcome back to the probability lectures here on www.educator.com, my name is Will Murray.*0000

*Today, we are going to talk about order statistics.*0005

*I have to tell you what order statistics means, let us jump into that.*0009

*I’m going to start out with an example question.*0013

*I teach at a university and I'm wondering next semester, I know that I’m going to be teaching probability.*0015

*I look at my class and I see all of that 30 people enrolled in my class next semester and I have never met any of them.*0022

*I just got 30 random names and never met any of them.*0029

*I wonder how tall is the tallest student in that class be?*0032

*That is the kind of question that we are going to be answering using order statistics.*0038

*I might wonder how tall is the tallest student in my class be, how tall will be the shortest student in my class be?*0047

*Let me try to connect that with some variables.*0053

*The idea here, we have N independent random variables with identical distributions.*0058

*Their distribution function, we are going to call F of Y.*0065

*Their density, we are going to call f of Y.*0068

*Of course, that is always the derivative of the distribution.*0071

*It is always the density, f is always F prime of Y.*0074

*To connect that backup with the example, imagine I'm looking in my class for next semester,*0080

*I’m saying I have 30 students in my class next semester.*0086

*Each student represents a random variable and that random variable represents how tall that student is.*0090

*I’m going to have 30 students in my class, that means I have Y1 through Y30, 30 different heights.*0097

*I’m wondering, how tall will the tallest student in the class be and how tall will the shortest student in the class be?*0103

*In order to study that, I'm going to define Y sub 1 to be the smallest or the minimum of the Y1 through YN.*0111

*YN is the largest or the maximum of Y1 through YN.*0122

*This notation is not very good, I’m following one of the standard textbooks in the field*0127

*but this notation can be quite confusing for students, because there are two different Y1’s here.*0133

*Let me quickly identify the difference here.*0139

*This Y1 with no parentheses, that is just the first random variable.*0143

*It is the first student that walks in my door, I’m going to call that person Y1.*0148

*Y1 with parentheses, that means that I look at all the random variables in I select out the smallest one.*0153

*I call that one (Y1), that is the smallest one.*0165

*It is like, if I'm talking about my students in my class, Y1 with no parentheses*0170

*is just the first student who walks in the door on the first day of the semester.*0177

*Y sub 1 with parentheses means I wait till all my students come into the class.*0181

*I ask them to all stand up, I look around, I find the shortest student in the class and I say you are Y sub 1.*0187

*Y sub N without parentheses, that is the last random variable that you look at.*0194

*That is kind of the last students that walks in the door of my classroom on the first day.*0202

*Y sub N in parentheses is the largest of all of them.*0208

*You look at all the variables and you pick whichever one is biggest.*0214

*In the classroom example, that means that I wait for all of my students to file in.*0220

*I ask them all to stand up, I see who is the tallest in the class and then I labeled that person as YN.*0226

*It is the largest of all the variables.*0234

*Make sure you do not get those mixed up.*0238

*The Y sub 1 without parenthesis and Y sub 1 with parentheses,*0242

*the Y sub N without parenthesis and Y sub N with parentheses.*0246

*The question that we are going to try to solve today is,*0249

*one of the distributions and densities of the minimum and maximum, Y sub 1 in parenthesis and Y sub N in parenthesis.*0252

*Remember, we know the distributions and densities of the individual variables, F and f.*0260

*F and f are known in terms of homework problems, they should be given to you.*0266

*You should know the F and f of R.*0271

*We are going to try to find distributions and densities of Y sub 1 and Y sub N in parenthesis.*0277

*Let me give you some formulas for this.*0287

*These formulas actually take a bit of work derive.*0288

*I’m not showing you the whole derivation here but the distribution,*0291

*it turns out that the simpler ones are for the maximum values.*0295

*These first ones that I'm going to teach you are for the maximum values.*0299

*And then, we will do the minimum one later because they are more complicated.*0305

*I’m not showing you all the derivation but the distribution function here,*0309

*the distribution is the probability that the maximum is less than some cutoff of Y.*0315

*What is the probability, maybe that your tallest student will still be less than 7 feet tall.*0322

*The way you get it is, you do F of Y ⁺N.*0327

*F of Y is the distribution of the individual Yi’s.*0331

*Little F of Y is the density function of the Y sub I, the density of Y sub i.*0344

*F of Y and f of Y should be given to us.*0354

*And then, we just drop them into these formulas f of Y is the derivative of F of Y.*0360

*We can either remember the formula for it or we can work it out from the function for F.*0369

*If you take the derivative of this then you get N × F of Y ⁺N-1 × its derivative, that is the chain rule kicking in there.*0378

*The derivative of F is f.*0390

*That is why we get that f that you have to tack on the end there.*0392

*We are using the power rule and the chain rule there.*0399

*For the minimum, it is a little more complicated.*0402

*These formulas correspond to the minimum value, the smallest or the shortest student in my class.*0405

*Right here, I forgot to mention that this is the density of the max.*0417

*Here we are going to find the distribution of the minimum.*0429

*The way you find it, I’m skipping the derivation here but the probability that the minimum will be less than Y.*0439

*What is the probability that I will have a student below 5 feet tall in my class?*0446

*What is the chance of having a student less than 5 feet tall?*0450

*The way you do it is you use the distribution from your original Y and you can drop into this formula.*0453

*And then, you take the derivative of that to get the density of the minimum.*0461

*If we take the derivative of this, this initial 1 drops out because it is a constant.*0472

*And then, we have by the power rule N × 1 - F of Y ⁺N -1.*0479

*And then, we have the derivative of the inside stuff which is f of Y.*0486

*That is of the formula that we are going to use to find the density function, for the minimum variable.*0494

*That is a lot of formulas, I think it is good if we jump right into some examples and*0500

*we practice those formulas and see how they work out.*0506

*Let us start with example 1, we have 24 students in a class.*0511

*Each one is writing a term paper and I guess the teacher said that paper can be any length*0514

*from 0 to 7 pages, which is a little bit artificial.*0519

*Usually, a teacher will say it is got to be 5 to 7 pages, but we are going to keep it simple.*0522

*We are going to go 0 to 7 pages long.*0527

*We want to find the density and the distribution functions for the length of the longest paper.*0530

*Maybe, you are the teacher and you are wondering how much you are going to have to grade.*0536

*You are wondering, what is the long paper that I’m going to read here?*0540

*Let me start out by identifying the fact that we have a uniform distribution here.*0545

*I need to find the density and distribution functions for the uniform distribution.*0551

*We had a whole lecture on the uniform distribution.*0559

*I will remind you what formulas we got for that.*0565

*The density function for a uniform distribution on the interval from θ1 to θ2, the density is just 1/θ 2 - θ 1.*0569

*In this case, it is 1/7 -0, just 1/7.*0588

*The distribution function is F of Y, you integrate that from 0 to Y.*0593

*You get, just the integral of that is just Y/7.*0601

*That is, as Y goes from 0 to 7.*0604

*I now have a f and F.*0610

*It is easy now to use the formulas from the previous side to solve the rest of it.*0615

*We are looking at the length of the longest paper, that F sub Y sub N,*0621

*that is the maximum one, the max length of the paper, of Y.*0628

*I will use the formula from the previous slide, it is F of Y ⁺N.*0635

*In this case, N is 24 and F is Y/7 ⁺24.*0641

*That is it for the distribution function, that is the distribution.*0659

*The density function is f sub Y sub N of Y.*0668

*You can take the derivative of the term above but you can also use the formula.*0675

*I’m going to use the formula just to practice that.*0681

*N × F of Y ⁺N-1 × f of Y.*0684

*N is 24, F of Y is Y/7, N – 1 is 23, f of Y is 1/7*0690

*It looks like this could simplify, combine the terms a little bit.*0703

*24 × Y ⁺23 and now I have 1/7 ⁺23 in the denominator and one more factor of 7.*0707

*I got 7 ⁺24, there, you could have gotten by taking the derivative of the distribution function.*0717

*My range is still the same as before, Y goes from 0 to 7.*0726

*That is what I have for my density function.*0735

*Those are the distribution and the density functions for the length of the longest term paper,*0746

*that will be submitted.*0752

*Let me review the steps there.*0754

*I first identified that we had a uniform distribution, I looked up my density function*0757

*and my distribution functions for the uniform distribution.*0764

*We have a whole lecture on the uniform distribution, earlier on this series.*0767

*You can scroll back up and see it.*0771

*I think it was the first continuous distribution that we study, it is the easiest one.*0773

*The uniform distribution, the density function is always constant that is why it is uniform.*0778

*It is 1/7 because our range is from 0 to 7.*0784

*The distribution function, you integrate that from 0 to Y.*0788

*If you integrate that from 0 to Y, you get Y/7.*0793

*I'm using the two formulas for the distribution and density of the max value.*0797

*That was on the previous side where I gave you the formulas for order statistics.*0804

*F of Y ⁺N is Y/7 ⁺N, our N is 24, I forgot to put a nice box around that because that was my final answer for the distribution.*0811

*My f of Y, if I just drop in N = 24, my F of Y and my f of Y,*0825

*simplify that down I get the density function for the maximum value there.*0834

*That is the end of example 1, I am going to reuse the setting for examples 2 and 3.*0842

*I want to make sure that you understand this,*0849

*in example 2 we are going to find the density and distribution for the length of the shortest term paper.*0852

*Make sure that you remember what we came up with here, we will reuse these in examples 2 and 3.*0858

*In example 2 here, this is a follow-up to example 1, the setting is the same.*0867

*With 24 students in a class, they are each going to write a term paper.*0873

*The papers length can be anywhere from 0 to 7 pages.*0877

*Maybe, if a student has a really profound thought, the student could express her amazing thought in half of the page.*0880

*It could be potentially 1/2 page term paper or can run as long as 7 pages.*0889

*We want to find the distribution and density functions for the length of the shortest paper.*0894

*We did already start figuring out some useful facts about this problem back in example 1.*0899

*We figured out that this was a uniform distribution.*0905

*I went back and looked up the density and distribution functions for the uniform distribution.*0908

*What I figured out was that, my F of, Y my distribution function was Y/7.*0917

*My f of Y was just a constant value 1/7.*0926

*That is going to be very useful, now I'm going to invoke the formulas for, the shortest paper means Y sub 1.*0933

*F sub Y sub 1 of Y, it is a little more complicated than what we had for the max values.*0943

*The min value, its distribution function is 1 - (1-F of Y) ⁺nth.*0952

*N is the number of variables that we are looking at.*0968

*Here we have 24 students, N is 24.*0971

*I'm going to plug in what I know here.*0976

*By the way, this formula came from the formula in the introductory slide.*0978

*I think it was the second slide of this lecture.*0984

*Just go back and check to see the formulas, that is where this formula is coming from.*0986

*That 1 - (1 -, my F of Y is Y/7, 1 - Y/7) ⁺nth or 24 here.*0991

*That is 1 -, if I want to put those over a common denominator, it is not absolutely necessary but it will be 7 – Y/7 ⁺24.*1009

*I do not think that is going to get any better.*1023

*That is my distribution function that I just found, for the minimum value.*1026

*Let me box that up because I'm going to submit that as my answer to the first part of the problem.*1035

*Let us figure out the density, my f sub Y1 of Y.*1042

*I could take the derivative of what I just figure out or I could use the formula from earlier on in a lecture.*1049

*I think I’m going to practice using the formula.*1055

*Let me remind you what that was.*1057

*It was N × the (1 -F of Y), looking at myself here, to the N -1 × f of Y.*1059

*Let me just plug in everything here, the N is 24, 1 - F of Y is 1 – y/7 ⁺23, my f of Y is 1/7.*1071

*This will simplify a little bit, 24 ×, I can put over a common denominator.*1085

*I get 7 - Y/7 ⁺23 × 1/7.*1092

*Now, I can write this as 24, in the denominator I have7 ⁺24.*1099

*I will have a 7 –Y ⁺23.*1107

*What is my range, I forgot to mention the range on this.*1113

*The Y is still going to go from 0 to 7, that is because the original distribution was Y going from 0 through 7 .*1117

*That is the density function, I’m done with that problem.*1129

*Let me review the steps and then I can move on.*1141

*We have a uniform distribution, we already identified that back in example 1.*1144

*We identify the distribution function F of Y is Y/7.*1149

*Our density function was 1/7 and N = 24 because we are talking about 24 students here.*1153

*And then, I used the two formulas that I gave you back on the second slide of this lecture.*1161

*You can scroll back, you can see the distribution function in terms of F of Y and the density function for Y1,*1167

*because we are talking about the shortest paper that is why we are looking at Y1.*1178

*Those are the two formulas and then I just dropped in what my F of Y is, Y/7, dropped that into the formulas.*1183

*F of Y is 1/7 and then I simplified it down to get a distribution function and a density function.*1190

*These are both representing the density function and the distribution function for Y1, which is the shortest paper.*1199

*If I'm wondering ahead of time, how long will the shortest paper be that this teacher is going to have to grade,*1207

*then I would follow these density and distribution functions to calculate those probabilities.*1214

*We are going to use this term paper example for one more problem here.*1221

*Make sure you still understand the uniform distribution, as we move on to example 3.*1227

*In example 3, just following up from example 1.*1233

*We got 24 students in a class, each writing a term paper.*1237

*The term papers lengths are uniformly distributed from 0 to 7 pages.*1241

*Not a totally realistic example, but maybe some students are kind of lazy and returning very short term papers.*1245

*Some students are very industrious and they are writing 7 good pages.*1251

*We want to find the mean and variance for the length of the longest paper.*1255

*The longest paper means the maximum value.*1259

*We are going to be looking at Y sub N, the maximum value, when in parenthesis that is the maximum value.*1262

*We are going to use some of the answers that we derived in example 1.*1272

*Let me remind you of what we figured out in example 1.*1277

*In example 1, we figured out the density and distribution functions.*1284

*In particular, the density, that is what we are going to use.*1288

*F sub Y, that is not a Y sub 1, that is F sub YN.*1291

*I will remind you what we found in example 1, that was 24 × Y ⁺23 divided by 7 ⁺24.*1298

*That is coming from example 1, if you are have not watched example 1 in the past few minutes,*1312

*you might want to go back and just review that.*1318

*Make sure you know where that is coming from, so it is not a total mystery in solving example 3.*1319

*That is the density function for the longest paper.*1326

*We want to find it is mean and variance.*1332

*Let me walk you through that.*1335

*The mean is the expected value of Y sub N.*1336

*What we do, to find the expected value, you just do the integral of Y × the density function DY.*1341

*We integrate over the full range of Y.*1355

*In this case, this is the integral of Y × that density function we just figure out.*1358

*Y ⁺24 now, I bumped up the power by 1 because I had that one extra Y and 7 ⁺24 in the denominator DY.*1368

*I have to integrate this from 0 to 7, Y = 0 to 7.*1377

*That integral is actually not that bad, the numbers are a little ugly, 24/7 ⁺24.*1382

*I want to integrate Y ⁺24, this just a power rule.*1390

*I get Y ⁺25/25 and if I evaluate that from Y= 0 to Y = 7 then what I get is 24 × 7 ⁺25, that is going to cancel, × 25.*1393

*A lot of that is going to cancel, my mean or expected value is 24.*1415

*24 of those 7’s are going to cancel, leaving you with just 1/7 in the numerator and 25 in the denominator.*1425

*It looks like I'm done and there is going to be a lot of 24 and 25 in the variance, it is going to get even worse.*1439

*Let me write this in terms of N, I think it will be a little more meaningful if I just use generic N here.*1445

*This is 7N divided by N + 1 because remember the N here was the number students in the class, that is N = 24.*1451

*7N/N + 1 is the expected value of the longest paper that I'm going to turn in.*1461

*Notice by the way, if N is very large then the limit of that, as N gets very large is 7.*1469

*It gets closer and closer to 7, as N gets larger and larger.*1477

*That means that, the more students we have in this class, the more likely it is*1481

*that someone is going to turn in the paper that is exactly or very close to 7 pages.*1486

*The more students we have, the more likely it is that the longest paper will be about 7 pages.*1492

*That kind of makes sense, if you have more students in a class, meaning N gets bigger,*1498

*it is more likely that the longest paper is close to 7 pages.*1513

*That really makes sense with your intuition.*1527

*If you have just have two students in the class, might not be very likely that one is going to produce a 7 page paper.*1530

*But, if you have 100,000 students in a class and their lengths are uniformly distributed from 0 to 7 pages,*1536

*chances are you going to get 1 that is pretty close to 7 just because there is so many students.*1544

*Variance is a lot messier here, let us calculate out the variance.*1549

*Remember to find the variance of a variable, what you do is you do*1554

*the expected value of Y² - the expected value of (Y)².*1560

*What I'm going to do here is find the expected value of Y² first.*1568

*E of Y² is the integral of Y² F of Y DY.*1573

*I’m going to use Y² where I used Y before.*1583

*I’m going to start using, the integral of 24 Y ⁺25, because I have Y ⁺23 and I bumped it up by 2 powers.*1589

*We still have7 ⁺24, then I got my DY here.*1601

*If I integrate that, I saw a 24/7 ⁺24, now Y ⁺26/26 by the power rule.*1610

*This gives me a good form of old fashioned calculus here.*1620

*Y = 0 to Y = 7 and I get 24/7 ⁺24 × 7 ⁺26/26, and that simplifies to,*1624

*Because I can cancel a lot of the 7’s, 24 × 7²/26 there.*1644

*Again, we have lots of 25, 24, 26, I think they are all coming from that original 24.*1653

*I think it is useful to write this in terms N.*1659

*What I’m getting here is 7² × N divided by N + 2.*1663

*I’m going to bring in the expected value of Y, that part of the formula.*1676

*My sigma² or my variance is E of Y² so 7² × N/N + 2 - the expected value of (Y)².*1681

*The expected value of Y, I already worked out here.*1694

*7/(N + 1)², that is going to get a little messy.*1697

*7² N/N + 2 - 7² N²/N + 1².*1705

*I think I can put those over common denominator.*1717

*It actually gets a little messy and then it simplifies really nicely.*1720

*Stick with me and I will show you how it works out because it is kind of fun when it simplifies.*1723

*I have a 7² everywhere and then, I'm going to have N × N + 1² for the first term.*1728

*My common denominator, I’m planning ahead is going to be N + 2 × N + 1².*1736

*I did multiply that first term by N + 1², the second term have to multiply by N + 2, N² × N + 2.*1743

*That looks pretty horrific but that numerator actually simplifies nicely.*1751

*It did, when I work this out before.*1756

*In fact, I can factor an N out of everything here.*1758

*7² × N, and N + 1² is N² + 2N + 1 - N × N + 2 is N² + 2N.*1760

*That is because I factored one of the N outside, there is 1N outside.*1775

*I still have that same denominator, maybe I will write that down.*1780

*Miracles, a lots of terms cancel, in fact the N² + 2N cancel.*1787

*We are just left with the 1 in that numerator there.*1794

*That simplifies down to 7² divided by, I still have that denominator,*1797

*7² × N divided by the denominator N + 1² × N + 2.*1804

*Of course, I was shooting for a number here, let me go ahead and fill in the number there.*1813

*That 7² ×, my N was 24, I’m not going to multiply this out.*1817

*N + 1² will be 25² and N + 2 is 26, that is my variance.*1825

*Not the most illuminating answer in the world, not nearly as easily verified,*1835

*or it does not easily conform to your intuition the same way the mean did.*1844

*But it is a number, we have solved the problem.*1849

*There is the mean and there is the variance of the longest paper.*1854

*Let me review the steps there, I'm using the density function for the longest paper,*1857

*for the maximum value that I figure out in example 1.*1863

*If this part of the solution came as a total surprise to you, go back and watch example 1 and*1866

*you will see where this comes from, 24Y³/7 ⁺24.*1873

*And then, to find the mean, remember our original definition of the mean many lectures ago,*1878

*was you just integrate Y × the density function.*1885

*We are integrating Y × the density function, Y × Y ⁺23 gives me Y ⁺24, that is where that 24 came from.*1890

*I bumped that power by 1 because of that extra Y there.*1899

*When I did that integral, it turn out to be a pretty easy integral, just use the power rule.*1903

*I dropped in my values, my range was Y goes from 0 to 7 that is*1908

*because we are given this uniform distribution from 0 to 7 pages.*1915

*That is where those limits came in, and when I dropped in the range of values Y goes from 0 to 7, I get 7 × 24/25.*1919

*I noticed that, that was 7N/N + 1, I’m kind of like thinking of it in terms of N because,*1931

*if you notice that when N goes to infinity, that gets very close to 7.*1937

*The limit is 7 but when you plug in bigger and bigger values of N, it gets closer and closer to 7.*1943

*That is not surprising, because if you think about it, if I have a very small class, if there are 5 people in the class,*1949

*and I say we are all going to pick a length between 0 and 7 pages,*1957

*it is not that likely that anybody is going to get that close to 7.*1962

*But if you have a huge number of people in the class, if you have 5,000 people in your class*1966

*Maybe it is some huge online class and they all write papers, and you look at the longest one.*1972

*The chances are that the longest one, I picked term paper out of 5000 people,*1978

*it is going to get pretty close to 7 pages.*1981

*The more people you throw into the mix, the more likely it is that the longest one will get closer and closer to 7 pages.*1985

*That is sort of very assuring.*1992

*The variance does not work out quite nicely.*1996

*First, I want to find E of Y² and that is because I'm remembering this old formula for the variance, E of Y² - the mean².*1999

*To find E of Y², I did the same thing as with the expected value except we have Y²,*2008

*instead of Y which means we are integrating Y ⁺25 instead of Y ⁺24 before.*2014

*It is still an easy integral, plugging Y = 7 and I got 24⁷²/26.*2021

*That is anticipating, all these different values of 24 and 25 and 26.*2029

*I think it is easier to think of them in terms of N.*2034

*I just wrote that as 7² × N/N + 2, remember my N was 24, number of students in the class.*2037

*Using this formula, my variance is that E of Y² -, that 7/N + 1 that is coming from here.*2045

*That is what we got dropped in here.*2056

*And then, I got into some messy algebra.*2058

*I was just putting these two terms over common denominator, I can factor out a 7²*2063

*and I could also factor an N from both terms.*2070

*My common denominator was N + 2 × N + 1².*2073

*I’m going to multiply the first term by N + 1² and the second term by N + 2.*2077

*After I factor out that N from both terms and I expanded both terms,*2082

*I got something really cool because N + 1² gave me N² + 2N + 1.*2087

*N × N2 gave me N² + 2N.*2093

*All the terms dropped out and I’m just left with 1 there.*2097

*I just simplify down the 7² × N.*2101

*That is really nice, I’m going to translate it back into the N into 24 so I got an actual number for the variance.*2104

*I do not bother to put that into a calculator, it is not the most revealing of numbers.*2114

*You will notice when N gets very big, it goes to 0 which is kind of reassuring because with more and*2121

*more students in the class, there is going to be less variance in the length of the longest paper.*2126

*But beyond that, there is not too much to be insight to be gained from that expression.*2132

*It is just a number and we know that it is right.*2137

*Let us move on, in example 4, we have got a basketball team.*2142

*The team has 10 players, it is a women's basketball team.*2148

*Unfortunately, what happens with professional sports teams is that every so often somebody gets injured,*2154

*it is just a fact of life*2163

*If you are a coach, you do have to plan for there will be injuries from time to time.*2164

*If you have 10 players, you never know, sometime during the season one of them might get injured.*2170

*You have to worry about that and you have to plan for that.*2176

*That is what this example is all about.*2180

*We got 10 players, 5 of them will play at any given time, and then we got 5 reserves.*2182

*We have been studying the sport of basketball for a long time.*2188

*You have notice that over the long run, each player gets injured now and then.*2192

*It is very hard to predict but on average, each player gets injured about once every 5 years.*2198

*That is just an average, that certainly does not reflect individual player statistics.*2204

*There might be unlucky players who get injured almost every year.*2209

*There might be players who can go their whole careers without a major injury.*2214

*We certainly hope to find players in the latter category but its very hard to predict.*2219

*What you are trying to do as the coach of the team is, you got 10 healthy players right now,*2226

*you are worried about when the first injury is going to come.*2233

*What will be the first time that you have a player getting injured.*2237

*We want to find the distribution and density functions for that.*2242

*Then the following slide, in example 5, we are going to find the expected time,*2245

*how long will it be until we see our first injury.*2249

*Let us puzzle this out here.*2254

*First thing we notice here is that this is an exponential distribution.*2256

*That is actually very realistic description of real life here because a basketball injury*2260

*is not something that happens with any regularity.*2267

*It is every so often, it happens and it might happen twice in a month or it might never happen over the course of 10 years.*2274

*There is no predicting in it.*2283

*If you are safe for 5 years, it does not mean that you would not get injured tomorrow, unfortunately.*2285

*Let me remind you what the exponential distribution is all about.*2292

*It has density function f sub Y is 1/β × E ⁻Y/β, where Y goes from 0 to infinity.*2296

*By the way, the exponential distribution is part of the family of gamma distributions.*2310

*We have a lecture on the gamma distribution earlier on in the series,*2316

*it was in the chapter on continuous distributions.*2320

*If you do not remember the exponential distribution at all, it is a very good time now to go back and check it out,*2323

*and get yourself understanding the exponential distribution again.*2329

*One thing you learn is that the mean of the exponential distribution is this β.*2334

*That means, we have been given that the mean is 5 years, β here is 5.*2338

*Our f sub Y is 1/5 E ⁻Y/5.*2346

*Our F, that is the distribution function, what we just found before was the density function.*2353

*F is you will integrate the density function.*2358

*I’m going to integrate from 0 to Y of 1/5 E ⁻T /5.*2363

*I got to call it T, since my Y of U is elsewhere.*2369

*The integral of E ⁻T/5 is just -5 E ⁻T/5.*2375

*But the 5 and 1/5 cancel, it is -E ⁻T/5.*2381

*I did a u substitution there in my integration.*2387

*If you are not comfortable with that, you may work it out yourself.*2389

*I did u substitution, u = -T/5.*2392

*And then, I have worked out my DU as well.*2399

*You might want to fill in that step yourself.*2403

*I’m integrating this from T = 0 to T = Y.*2406

*If I plug in T= Y, I get -E ⁻Y/5 and if I plug in T = 0, I get E⁰.*2412

*It is negative, it is a +, it is – a negative so it is +.*2424

*E⁰ is 1, that is my distribution function, my F of Y.*2430

*That is kind of describing the individual × until injury for each one of the players on this team.*2437

*That is kind of describing the density and distribution functions for Y1 up to, I guess we got 10 players on this team.*2448

*What we are worried about as a coach is, how long it will be until we see the first injury on this team?*2457

*Right now, we got 10 healthy players, we would certainly like to keep them all healthy*2464

*but we know that is a sooner or later, we might have an injury.*2468

*We are worried about when that first injury will come.*2473

*The first injury will be the minimum value of the Y.*2477

*The first injury will be the minimum of the Yi, which remember is what we are calling Y sub 1 in parentheses.*2481

*We want to figure out the density and distribution functions for Y sub 1.*2498

*We have a formula for that, that formula I will remind you what it is.*2504

*But if you do not remember where you saw before, it was on the second slide of this lecture.*2511

*Just scroll back and you will see the second slide of this lecture.*2515

*The distribution function for the minimum value is 1 - (1-F of Y) ⁺N.*2519

*This work out fairly nicely, this is 1 -, it is going to be a little messy at first.*2531

*F of Y itself is 1 – E ⁻Y/5 ⁺nth.*2537

*Those 1 - in the inside cancel, we just get 1 - E ⁻Y/5.*2548

*N is the number of variables we have, that is 10.*2561

*This is 1 - E ^, I can multiply those exponents there.*2568

*E ⁻2Y, that is my distribution function for the minimum of those variables.*2573

*F sub Y1 is the density function, let me write that what I found there was the distribution function.*2584

*The density function, I could just take the derivative of the distribution function, that is what I just found.*2599

*It would be quite easy to do that.*2605

*What I like to do is practice the formula that I gave you earlier on in this lecture.*2608

*Probably, it would be faster to take the derivative.*2612

*I just want to remind you of the formula.*2614

*It is N × 1 - F of Y ⁺N-1 × f of Y.*2617

*If I fill in what those are, N is 10, 1 - F of Y, that is the same thing that I figure out before.*2628

*1 - (1 – E ⁻Y/5)⁹.*2635

*That f of Y is 1/5 E ⁻Y/5.*2642

*I see that I got a 10 × 1/5, that is 2.*2651

*Here that is E ⁻Y/5, it is getting a little messy there.*2655

*I can do better than that, here is –Y/5 ⁺10.*2664

*It is to the 9th power, but then I got one more E ⁻Y/5, that 1/5 combine with the 10.*2669

*This simplifies down to 2E ⁻Y/5 to the × 10 because we got E ⁻Y/5⁹ and E ⁻Y ⁺1st, × 10.*2679

*This is 2E ^-, 10/5 is 2, make 2E ^- 2Y, that is my density function.*2697

*As I mentioned before, the much quicker way to find that would be to have taken the derivative here.*2711

*This is just, if we done D by DY of the distribution function, the derivative of –E ⁻2Y is just positive 2E ⁻TY.*2720

*That would be much quicker way to do it.*2730

*I just want to practice the formula that I gave you on the opening slides of this lecture.*2732

*In case you want to find the density function immediately, without sort of detouring through the distribution function.*2738

*That answers our two questions here, we got the distribution and density functions for the time until injury.*2746

*I did not really tell you the range on that, but it is the same range on your initial Y.*2752

*Let me throw that in here, 0 less than Y less than infinity.*2757

*In the worst of luck, it could happen right there on the first day, somebody gets injured.*2763

*If we are very lucky, it could go forever without an injury.*2768

*Let me review the steps here. First thing I did was focus on the word exponential distribution.*2774

*The exponential distribution, as I learned when I studied the gamma distribution in the earlier lecture is,*2779

*its density function is 1/β E ⁻Y/β.*2787

*Its range is from 0 to infinity, in this case the mean is always β but we are given a mean of 5.*2792

*Our β is 5, in this case.*2799

*I just dropped in β = 5 here.*2802

*The exponential distribution is a very good model of this physical situation,*2804

*because an injury is something that you absolutely cannot predict.*2810

*It is not the kind of thing that, if you stay healthy all through this month,*2814

*it does not mean that you are more or less likely to get injured next month.*2821

*That is exactly the kind of behavior that the exponential distribution models.*2825

*In fact, I think that when we learned about the exponential distribution, I called it the memoryless distribution.*2831

*Meaning that, if you stay healthy all through 1 month, next month is you sort of do not remember*2836

*that you are healthy last month, you just get a fresh start in the second month.*2841

*That was the density function for the exponential distribution.*2847

*The distribution function there is, you take the integral of the density function.*2852

*I changed my variable to T because I want to integrate from 0 to Y.*2858

*Integrating that, I did a u substitution here, actually I did it my head.*2862

*I integrated that and I got 1 - E ⁻Y/5.*2869

*Remember here that you cannot ignore the T = 0 terms.*2874

*You got to include that T = 0 term because when you plug in 0, you did get the number 1 there.*2878

*You cannot drop that out, that is where that 1 came from.*2888

*The T = Y gave me the E ⁻Y/5, that is my distribution function.*2891

*Those all represented the density or distribution functions for individual players,*2899

*that represents the waiting time for an individual player to experience an injury.*2905

*What the problem was actually asking is, we have 10 of those players,*2913

*there are 10 different players that are all running around, hopefully, none of them gets injured.*2920

*What we are worried about is that each one of them has a possibility of getting injured.*2924

*I’m wondering about how long we are going to have to wait until the first one gets injured?*2931

*As soon as we have one injured player, it is really is going to change our coaching strategy on the team.*2936

*The first injury that means the shortest time which one of those players is going to get injured in the shortest time.*2942

*That is the minimum of the Yi, that is exactly Y sub 1, that is the one in parentheses.*2950

*I’m going to use my formulas to calculate the distribution and density for Y sub 1.*2955

*My formulas tell me, these are the formulas from the second slide of this lecture.*2961

*Just scroll back, there is the formula for the distribution and there is the formula for the density.*2966

*Then, I just go through and wherever I see a F, I fill in this.*2972

*Wherever I see a f, there is a f right there, I fill in the density function for the exponential distribution.*2978

*The nice thing is there is a lot of 1 – F, 1 - F actually simplify down really nicely into E ⁻Y/5.*2987

*Both ×, I had a 1 – F and simplify down into E ⁻Y/5.*3001

*When I put in my exponent N = 10, that was because there were 10 players right there.*3008

*We got a simpler form for the distribution function.*3014

*After some work, I got a simpler form for the density function.*3018

*Of course, if I wanted to save time and really use every resource, I would not have use this formula for the density function.*3021

*I would just started with the distribution function and taken its derivatives.*3030

*If you take its derivative, you get the density function very quickly to be 2E ⁻2Y.*3034

*I want you to really make sure you understand these answers from example 4,*3042

*because in example 5, we are going to revisit this example.*3048

*We are going to take it a little farther, we are going to calculate the expected time until the first injury.*3052

*We will use these answers, I will not derive them again from scratch.*3059

*We will figure out the expected time until the first injury.*3062

*If you understand these answers, it will make example 5 make a lot more sense.*3067

*Let us go ahead and take a look at that.*3073

*Example 5 here is a follow-up to example 4.*3076

*If you have not just watched example 4, you really need to understand example 4*3080

*before you can make some sense of example 5.*3084

*The situation back then, you have a basketball team with 10 players.*3087

*Each player, we are worried about how long it would be until she experiences some kind of injury.*3093

*Because that is unfortunately what happens with basketball team, every now and then a player gets injured.*3098

*We are interested in finding the expected time until the team's first injury.*3104

*Let me remind you what we figure out in example 4.*3111

*We figured out that the time until the team's first injury.*3115

*The first injury means, we are looking at Y sub 1, the minimum time until an injury among all those players.*3121

*We figured out in example 4, the density function for F of Y sub 1 which I will remind you was 2E ⁻2Y.*3132

*2E ⁻2Y, that is the density function for Y sub 1 and that came from quite a bit of work in example 4.*3148

*I’m not going to repeat that but if you think that that is coming out of left field,*3158

*I’m mixing my sports metaphors, this is a basketball team and nothing should come out of the field.*3164

*Go back and look at example 4, that should all make sense to you.*3170

*I want to find the expected value of Y1.*3174

*What I do, there is a quick way to do this and I’m going to hold off from that.*3180

*I’m going to do it using the definition, and then we will go back and*3184

*see how we could have found the expected value very quickly.*3188

*Let me find the expected value, sort of not using any special cleverness here.*3192

*Remember, it is the integral of Y × F of Y DY.*3197

*In this case, it is the integral of Y × my F of Y is 2E ⁻2Y DY.*3204

*I need to do a little integration by parts to make that work.*3214

*Let me go ahead and integrate by parts.*3217

*Here is tabular integration because I'm feeling lazy.*3222

*E⁻² Y, take derivatives on the left 2Y, the derivative of Y is 2, the derivative of the constant is 0.*3226

*The integral of E ⁻2Y is -1/2 E ⁻2Y.*3237

*The integral of that is +1/4 E ⁻2Y + -, the integral there is, 2Y × -1/2 is –Y E ⁻2Y – 2/4 that is ½ E ⁻2Y.*3243

*I’m dividing this over the whole range of Y, which I neglected to mention before, Y goes from 0 to infinity.*3270

*Of course, you can figure that out by looking at the exponential distribution or*3278

*by sort of understanding the physical setup of the problem.*3282

*The time until the first injury could be as small as 0, if you get an injury right away.*3287

*Or it could that be arbitrarily long, if you are lucky, your basketball team will play for 50 years without ever getting injured.*3292

*It is unlikely but you can certainly have that.*3299

*I’m integrating this from Y = 0 or evaluating this from Y =0 to infinity.*3302

*If we plug in Y = infinity, these exponential terms are going to drag everything to 0.*3309

*You can do a Patel’s rule on that or you can just know that exponential terms in the denominator will always be polynomials.*3317

*I'm not even going to worry about my exponential terms, they are both 0.*3325

*If I plug in Y = 0, I get 0 for the first term.*3331

*In the second term, I get + because it is - a negative, + ½ E⁰ + ½ E⁰ which is ½.*3335

*What that means is that, if you are this basketball team coach and*3347

*you are wondering how long is it going to be until I see an injury in one of my players.*3353

*The expected time, you hope not to injure anyone ever,*3360

*but the expected time until your first player gets injured is 1/2 a year, 6 months, that is not so good.*3369

*Dangerous sport, stay away from it.*3377

*If you got 10 people on the court playing at the same time and you keep them playing hard,*3381

*chances are in about 6 months, you are going to see your first injury.*3386

*That is the unfortunate consequence of your probability here of having 10 people play basketball at once.*3392

*I mention that there was actually a quick way to figure this out.*3400

*Let me show you now how we could have done that, it could have saved a lot of work here,*3405

*which is to look back at that distribution that we had there.*3409

*Notice that, this is another exponential distribution.*3420

*This is exponential, let me remind you of the form of an exponential distribution.*3425

*It has density function F of Y is equal to 1/β E ⁻Y/β.*3432

*That is the density function for an exponential distribution.*3440

*What we have is something that exactly matches that, if we take our β equal to ½ because 1/½ is exactly 2.*3445

*We have an exponential distribution here.*3455

*The expected value of the exponential distribution, the mean of the exponential distribution is β.*3464

*This is a property that we learned about the exponential distribution long ago,*3471

*when we are studying the gamma distribution.*3475

*That was because the exponential distribution was a special case of the gamma family.*3477

*Μ = β is ½ and that is really all we needed to do, if we had noticed that.*3482

*We could have saved ourselves from walking through that long integration by parts.*3491

*I guess it was not that bad, but it is really useful to recognize a distribution, if it does fall into one of your known families.*3495

*And, to be able to draw some conclusions about it right away.*3502

*Let me generalize this a little bit, we already have our answer here.*3508

*We know that it is going to be about 6 months until our team's first injury, that is our expected time.*3512

*Let me mention where that 2 came from.*3518

*If you look back in example 4, where that 2 came from was, it just came from the 10 divided by the 5.*3521

*10 was the number of players on the team and 5 was the mean of the original distribution.*3528

*That ½ in turn came from the 5 divided by the 10, which in turn the 5 was the β,*3539

*the mean of the original distribution.*3549

*And, 10 was the number of players so β /N.*3551

*I'm going to try to write this in general here, without making reference to specific numbers.*3556

*In general, if Y1 through YN are exponential with mean β then, let me write that β a little more clear.*3571

*Then, Y1 the minimum of the Yi’s, the minimum of Y1 through YN is exponential.*3585

*We can say what the mean of that is with the new mean is going to be β/N.*3607

*And that is not too surprising, if you kind of think about this basketball player example.*3617

*If I have 10 players, on the average each one gets injured every 5 years.*3624

*If I’m sitting around waiting for one of them to get injured, which I hope I'm not.*3630

*Maybe, I’m the team doctor and I'm wondering when I'm going to have a job to do.*3634

*If there is 10 people, each one getting injured every 5 years,*3639

*on the average you are going to have one getting injured every half year.*3644

*That is really not surprising, that is just 5 divided by 10.*3647

*If you start out with an exponential distribution with mean β and cobble together N of them,*3653

*and look at the minimum then it is exponential with mean β/N.*3660

*That is not too surprising, if you can think about the basketball players.*3666

*That wraps up example 5, let me review the steps here.*3672

*The key step here came from example 4, where we identified Y1 is the minimum of these Yi’s.*3676

*Yi is the time for each player to get injured, which hopefully is long.*3685

*Y1 is the first player to get injured, Y1 is what you are worried about.*3690

*As a coach, you do not want any of your players to get injured.*3695

*You worry about when it will be until your first player gets injured and forces you to change your team strategy.*3698

*We calculated the minimum, and back in example 4,*3704

*we calculated the density function for the minimum as 2E ^- 2Y.*3708

*The long way to find the expected value of that is to use the definition of expected value.*3715

*Definition of expected values says you integrate Y × the density function.*3722

*That is what I did there, I dropped in the density function.*3727

*I did the integral using integration by parts.*3730

*If that tabular integration was unfamiliar to you, I covered that in my calculus 2 lectures here on www.educator.com.*3733

*You can look at the calculus 2 lectures and you can find a section on integration by parts.*3740

*It shows you a little short hand trick there.*3749

*When I plug in the infinity, they all dropped out.*3751

*When I plug in 0, I got exactly ½, my expected time is ½ a year which I translated into 6 months.*3754

*That was the long way, the short way is to look back at this density function and recognize that as an exponential distribution.*3763

*It is just an exponential distribution with a new β, β is ½.*3774

*If you know your exponential distribution, you know the mean is just β.*3781

*I could have immediately jump to my answer there of 1/2 a year,*3786

*without ever having to do that integral and all that integration by parts.*3793

*And then, I extrapolated that a little bit because I figure out that that ½ came from 5/10.*3797

*If you go back and look at example 4, you will see that that 2 came from 10/5.*3805

*And then, the ½ came from 1/2*3810

*In turn, that comes from 5/10.*3814

*The 5 and 10 comes from the original problem.*3816

*The 5 was β and the 10 was the N.*3821

*There is the N and there is the β.*3824

*The ½ came from β/N.*3827

*We have an exponential distribution with the new β is the old β divided by the old N.*3830

*That is a general principle there, if you have N exponential variables with the mean of β,*3838

*then their minimum will be exponential with mean β/N.*3845

*That is a very useful property and it kind of explains this idea,*3850

*if you are wondering when your first player will get injured, every player gets injured, on average once every 5 years.*3855

*If you got 10 players, you are going to have people getting injured once every 6 months on average.*3862

*That is just 5 years divided by 10, gives you the 6 months, that gives you 1/2 year.*3867

*That kind of explains and sort of justifies and reassures that all of our mathematics is correct there.*3873

*That wraps up our lecture here on order statistics.*3882

*This is part of the larger series on probability here on www.educator.com.*3885

*I'm your host along the way, my name is Will Murray.*3891

*I thank you very much for joining me today, bye now.*3893

## Start Learning Now

Our free lessons will get you started (Adobe Flash

Sign up for Educator.com^{®}required).Get immediate access to our entire library.

## Membership Overview

Unlimited access to our entire library of courses.Learn at your own pace... anytime, anywhere!