For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### Moment-Generating Functions

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
- Moments
- Moment-Generating Functions (MGFs)
- Moment-Generating Functions for the Discrete Distributions
- Moment-Generating Functions for Binomial Distribution
- Moment-Generating Functions for Geometric Distribution
- Moment-Generating Functions for Negative Binomial Distribution
- Moment-Generating Functions for Hypergeometric Distribution
- Moment-Generating Functions for Poisson Distribution
- Moment-Generating Functions for the Continuous Distributions
- Moment-Generating Functions for the Uniform Distributions
- Moment-Generating Functions for the Normal Distributions
- Moment-Generating Functions for the Gamma Distributions
- Moment-Generating Functions for the Exponential Distributions
- Moment-Generating Functions for the Chi-square Distributions
- Moment-Generating Functions for the Beta Distributions
- Useful Formulas with Moment-Generating Functions
- Useful Formulas with Moment-Generating Functions 1
- Useful Formulas with Moment-Generating Functions 2
- Example I: Moment-Generating Function for the Binomial Distribution
- Example II: Use the MGF for the Binomial Distribution to Find the Mean of the Distribution
- Example III: Find the Moment Generating Function for the Poisson Distribution
- Example IV: Use the MGF for Poisson Distribution to Find the Mean and Variance of the Distribution
- Example V: Find the Moment-generating Function for the Uniform Distribution

- Intro 0:00
- Moments 0:30
- Definition of Moments
- Moment-Generating Functions (MGFs) 3:53
- Moment-Generating Functions
- Using the MGF to Calculate the Moments
- Moment-Generating Functions for the Discrete Distributions 8:22
- Moment-Generating Functions for Binomial Distribution
- Moment-Generating Functions for Geometric Distribution
- Moment-Generating Functions for Negative Binomial Distribution
- Moment-Generating Functions for Hypergeometric Distribution
- Moment-Generating Functions for Poisson Distribution
- Moment-Generating Functions for the Continuous Distributions 11:34
- Moment-Generating Functions for the Uniform Distributions
- Moment-Generating Functions for the Normal Distributions
- Moment-Generating Functions for the Gamma Distributions
- Moment-Generating Functions for the Exponential Distributions
- Moment-Generating Functions for the Chi-square Distributions
- Moment-Generating Functions for the Beta Distributions
- Useful Formulas with Moment-Generating Functions 15:02
- Useful Formulas with Moment-Generating Functions 1
- Useful Formulas with Moment-Generating Functions 2
- Example I: Moment-Generating Function for the Binomial Distribution 17:33
- Example II: Use the MGF for the Binomial Distribution to Find the Mean of the Distribution 24:40
- Example III: Find the Moment Generating Function for the Poisson Distribution 29:28
- Example IV: Use the MGF for Poisson Distribution to Find the Mean and Variance of the Distribution 36:27
- Example V: Find the Moment-generating Function for the Uniform Distribution 44:47

### Introduction to Probability Online Course

### Transcription: Moment-Generating Functions

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

*We are going to talk today about moment generating functions.*0005

*Moment generating functions are one of the most confusing topics that people encounter in probability.*0010

*I'm going to try to walk you through them and show you what they are used for.*0017

*You might prepare yourself to be a little confused at first because every time I taught it,*0021

*it is my students who always found them to be a little confusing.*0026

*I will try to show you how it works.*0029

*The initial idea I want to talk about is a moments.*0032

*We start with a random variable and it can be discreet or continuous.*0037

*We will talk about moment generating functions for all of the distributions that we have been studying, *0040

*all of the discrete ones, binomial, geometric, and so on, and all of the continuous distributions, uniform and normal, and so on.*0046

*We can talk about moments and we can talk about moment generating functions for all of these distributions.*0055

*The first definition is the K ^{th} moment of Y taken around the mean.*0063

*Let me highlight that.*0069

*The K ^{th} moment of Y taken around the mean is just the expected value of Y ⁺K.*0070

*The K ^{th} there can be 1, 2, 3, and it can be 0, although people do not usually need to look*0079

*at 0 as moment because that is not very illuminating.*0084

*I said mean but I meant to say R gen.*0088

*We are also going to talk about moments around the mean.*0091

*But, it is important here that we are talking about the moments around the origin.*0093

*There is some notation that is sometimes used for this which is ν K prime.*0100

*That is really not obvious why we would use the notation ν K prime.*0106

*I’m not going to use that notation in this lecture, but if you are following along *0111

*in your own probability course or in your own probability book, you might see the notation ν K prime.*0115

*What that means is the expected value of Y ⁺K.*0121

*Those mean the same thing.*0126

*There is another notation that you might see in your book which is that idea of central moments.*0128

*Instead of taking the moment around the origin, we will talk about taking the moment about the mean.*0138

*Which means, instead of talking about Y ⁺K, you do Y – μ ⁺K where μ is the mean of the original distribution.*0144

*And that is called μ sub K and that is why we have to use μ sub K prime for the one that we are studying.*0155

*I want to emphasize that there are 2 different ideas here.*0162

*There is the moment around the origin and there is the moment around the mean.*0166

*In this lecture, in the probability lectures here on www.educator.com, *0170

*I'm just going to look at the moment taken around the origin.*0175

*I have got some more common one and it is easier to understand the ideas for that one.*0179

*I'm not going to talk anymore about the central moment.*0186

*We will talk about the moment around the mean.*0189

*I just mention that, in case you see it in your course, you know what the difference is.*0192

*You do not really need to study both of them, you can figure it out.*0198

*If you know one, you can figure out the other one just by doing some computations.*0202

*It is not necessary to study both of them.*0208

*You pick a system and then you follow that, and you can find all the information you need within one system.*0210

*The system we are going to use is the moments around the origin.*0216

*I would not talk anymore about moments about the mean.*0220

*I just included it, in case you see it in your course.*0224

*This has told us what the moment generating function is, let me jump onto the next slide and show you what that is.*0227

*The moment generating function for Y is M sub Y of T.*0235

*That := means it is defined to be.*0241

*It is defined to be the expected value of E ⁺TY.*0244

*That is a very illuminating definition, I do want to highlight it here*0249

*because it is probably the most important definition we are going to have in this whole lecture.*0255

*It is not obvious what it means right now, and I'm not going to clarify it right away .*0262

*I’m just going to throw the definition at you and then we will practice using it to solve some problems.*0268

*MY of T remember is defined to be the expected value of E ⁺TY, that is E like the exponential function.*0274

*The important things that you need to remember right now is, *0284

*the first one is that the moment generating function is a function of T not of Y.*0289

*When you calculate the moment generating function for distribution, you should have a function of T.*0303

*You should see a T in your answer.*0309

*By the time you simplify it down, you will not see a Y.*0310

*We will do some examples and you will see how it works out.*0314

*The Y always disappear, you always end up with a function of T.*0317

*Here is how you use the moment generating function.*0323

*Once you know it, this first line is kind of trivial but I included because it will make the other lines make more sense.*0326

*The expected value of Y⁰ is equal to the moment generating function with 0 plugged in for T.*0337

*The expected value of Y⁰, Y⁰ is always 1 because anything to the 0 is 1.*0347

*That is the expected value of 1 which of course will be 1.*0355

*It is not like we are really learning anything much from the moment generating function,*0359

*because we already knew that the expected value of Y⁰ is 1.*0363

*In the next line, the moment generating function starts to become useful.*0367

*What you do is you take the derivative of the moment generating function.*0372

*And again, you plug in 0 for T and what that tells you is the expected value for your distribution.*0376

*Now, we have something useful, we fused the moment generating function to find the mean of the distribution.*0383

*In the second line, what we have done is we have take another derivative M prime prime.*0391

*We plug in T is equal to 0.*0398

*What that tells us is, the second moment of the distribution E of Y².*0401

*Why is that useful, the reason that is useful is because it helps us to find the variance of the distribution.*0407

*We can use this to find the variance.*0414

*Be careful here, the variance is not the expected value of Y².*0419

*Let me remind you how we calculate the variance.*0424

*We calculate the variance as σ² is equal to the expected value of Y² - the expected value of (Y)².*0428

*If we can figure out these 2 moments using the moment generating function,*0440

*what we can do is drop in the expected value of Y² here from the MGF.*0446

*We can use the MGF, the moment generating function, to calculate the expected value of Y².*0455

*We can also use the moment generating function to calculate the expected value of Y MGF.*0464

*Both of these ingredients, they go into calculating the variance come from the moment generating function.*0473

*That is how we use the moment generating function, is to find these two ingredients to calculate the variance.*0480

*There are other uses for the moment generating function, later on in statistics*0487

*but I'm not going to get into them right away in this lecture.*0492

*Instead, what I want to do is show you some of the moment generating functions for our favorite distributions.*0496

*We will start with the discrete distribution.*0503

*We have here, all our favorite discreet distributions, binomial, geometric, *0506

*negative binomial, hypergeometric, and Poisson distribution.*0511

*Here are what the moment generating functions turn out to be.*0517

*For binomial, it is PE ⁺T + 1 - (P) ⁺n.*0521

*By the way, the binomial distribution, we often define Q to be 1 – P.*0527

*That term of 1 – P, people often write that as Q, and they simplify the way *0535

*to write the moment generating function somewhat.*0543

*For the geometric distribution, PE ⁺T/1 - (1 - P) E ⁺T.*0547

*Again, there is a Q in there, that is equal to 1 – P.*0554

*If you want to simplify this down, you can write this as PE ⁺T/1 - Q × E ⁺T.*0557

*A little bit simpler to write at the expense of having one more variable.*0565

*Negative binomial distribution is almost the same thing, except on there is an R in the exponent.*0569

*Almost the same as the geometric distribution.*0577

*Again, you can put in a Q for 1 – P, if you like that.*0579

*The hypergeometric distribution has no closed form moment generating function.*0583

*If you try to calculate the moment generating function of a hypergeometric distribution, it just blows up in your face.*0589

*There is no reason to go there, we would not go there.*0595

*The Poisson distribution much more well behaved, it is E ⁺λ × E ⁺T – 1.*0597

*Couple of things I want to mention about all of these, one is you might be wondering where these come from, *0605

*how do you calculate these moment generating functions.*0610

*Stay tuned, I will tell you because we will work out a couple of these in the examples.*0612

*Or you can just scroll down right now, if you are bursting with curiosity.*0619

*Check out example 1 and 3, I think, we are going to do the binomial distribution.*0622

*We will calculate the moment generating function.*0631

*For example 3, we are going to take the Poisson distribution and calculate moment generating function.*0633

*You will be able to see where these come from.*0639

*Another thing that I want to point out about this is, that you notice that nowhere on here do you see the variable Y.*0641

*All of these are functions of T, you see T everywhere here.*0650

*The moment generating function is always a function of T not Y, it is a function of T.*0664

*If you are calculating a moment generating function, if you still have Y on your paper*0672

*then you need to keep going until you can get rid of the Y, and try to simplify it down into a function of T.*0677

*These are just the discreet distributions, we also have a number of continuous distributions.*0688

*Let us go ahead and look at those.*0693

*Here, our favorite continuous distributions, uniform, normal, gamma, exponential, Chi square, and the β distribution.*0696

*The uniform distribution is a very simple distribution.*0704

*It has a surprisingly complicated moment generating function, E ⁺T θ2 – E ⁺T θ1 ÷ T × θ2 – θ1.*0707

*I keep saying my θ in the wrong order.*0721

*We are going to calculate that one out by hand, I think that is example 5.*0724

*If you want, you can scroll down and take a look at example 5.*0732

*You will see how we calculate the uniform distribution.*0737

*The others are more difficult, I did not put them into examples.*0740

*The normal distribution E ⁺ν T + T² σ²/2.*0745

*All of this is in the exponent of the E.*0750

*There is a lot in the exponent there.*0754

*The gamma distribution is 1 - β T ⁻α.*0757

*The next two distributions, remember are actually special cases of the gamma distribution.*0764

*The exponential distribution is just the gamma distribution where we take α equal to 1.*0770

*If you look at the gamma distribution, the moment generating function, and just plug in α = 1,*0778

*you get the moment generating function for the exponential distribution.*0786

*It is quite nice and simple.*0789

*The Chi square distribution is the gamma distribution with α defined to be ν/2.*0791

*Ν is the number of degrees of freedom and β is equal to 2.*0801

*If you take the gamma distribution and you plug in α is equal to ν/2 and β is equal to 2, *0808

*you get the moment generating function for the Chi square distribution, 1 -2T ⁻ν/2.*0818

*The β distribution, if you try to calculate the moment generating function, *0829

*you will get into a horrible mess and it just blows up in your face.*0834

*We say that there is no closed formula in moment generating function for the β distribution.*0839

*By the way, if you are a little rusty on what all these words mean, uniform, normal, gamma, *0844

*exponential, chi square, β, we have separate lectures about each one of these distributions.*0849

*You can go back and you can read up on the uniform distribution.*0854

*You can practice the normal distribution.*0857

*You can study the gamma distribution.*0859

*Of course, the exponential and chi square distribution, those are special cases of gamma distribution.*0862

*You will find those in the lecture on gamma distribution.*0867

*Just scroll up here and you will see the lecture on gamma distribution.*0871

*You will get the exponential and Chi square thrown in there as a bonus.*0874

*There is also a lecture on the β distribution, you can read up all about that.*0878

*The only things that are not in those lectures are the moment generating functions.*0884

*That is what I'm telling you about right now.*0889

*Let us go ahead and jump into some examples, and see how we actually derive*0892

*these moment generating functions, and then see how we can use them to calculate some means and some variances.*0894

*I see we have one more slide before I talk about the examples.*0905

*A couple of useful formulas for the moment generating functions.*0908

*If you have one known random variable Y and you do a linear change of variables.*0913

*If you define Z to be AY + B, := means defined to be.*0920

*If you define Z to be AY + B, then the moment generating function for Z is related *0927

*to the moment generating function for Y, except that there is an A missing in there.*0935

*Let me just go ahead and write that A in there.*0945

*It is just MY of AT and then E × E ⁺BT.*0951

*That is how you get from the moment generating function of Y to the moment generating function of Z.*0959

*Very useful, by the way, when you are converting normal distributions.*0966

*When you convert to a standard normal variable, you are doing exactly this kind of variable change.*0970

*This is quite useful, when you want to calculate the moment generating function.*0978

*Second useful formula, when Y1 and Y2 are independent variables.*0982

*Z is Y1 + Y2, there you are defining Z to be Y1 + Y2.*0987

*This only works for independent variables.*0995

*But when they are not independent, you can say that the moment generating function for Z *0997

*is the moment generating function for Y × the moment generating function for Y2.*1002

*What moment generating functions do is they convert sums into products.*1008

*That is really not surprising, that is essentially based on the fact that E ⁺X + Y is equal to E ⁺X × E ⁺Y.*1013

*Remember, our initial definition of moment generating function was in terms of the expected value of an exponential.*1022

*The fact that moment generating functions convert sums of variables into products of functions,*1030

*converts addition into multiplication, is really not very surprising.*1038

*But, you do have to check that you are talking about independent variables.*1042

*Let us go on and talk about some examples where we will actually calculate some moment generating functions.*1047

*In example 1, we want to find the moment generating function for the binomial distribution.*1055

*Let me remind you of the probability function for the binomial distribution.*1061

*It is been awhile since we studied that.*1067

*If you do not know what the binomial distribution is at all, just check back in the list up above, *1069

*you will see a whole lecture on the binomial distribution.*1075

*The take away from that lecturer right now, is that the probability of a value of Y is equal to N choose Y,*1079

*that is a binomial coefficient.*1087

*P ⁺Y Q ⁺N-Y, that is for Y ranging between 0 and N.*1089

*It represents the probability of getting Y heads when you flip a coin N ×.*1098

*Let us try to figure out the moment generating function for that distribution.*1106

*M sub Y of T, using that definition of moment generating function, defined to be the expected value of E ⁺TY.*1110

*How do you find the expected value of a function of Y?*1124

*Here is how you do it, I showed you this in a very early lecture.*1127

*It is the sum over all values of Y, of the probability of that particular Y, × that function of Y, E ⁺TY.*1133

*We need to expand that and figure it out.*1147

*What values of Y are we talking about?*1150

*I read from here that the range of values of Y is from my equal 0 to N.*1152

*The probability of each Y, I wrote that down right above.*1158

*It is N choose Y × P ⁺Y × Q ⁺N- Y.*1162

*Now, I have to multiply on this term E ⁺TY.*1169

*What can I do with this, remember I’m trying to simplify this into a function of T,*1175

*which means I'm trying to get rid of the Y, which means I have to do something clever.*1180

*Here is what I can do, I notice that I have P ⁺Y here.*1184

*Here, I have E ⁺TY which I can write as E ^(T) ⁺Y.*1189

*I can combine those two factors, that is what I'm going to do.*1196

*Y = 0 ⁺N of n choose Y of PE ⁺T ⁺Y × Q ⁺N-Y.*1200

*If you stare at this very heart, you are supposed to recognize something, to have a small epiphany, if you will.*1216

*In fact, you might want to stop the video right now and stare at this formula, *1223

*and go ahead and have that epiphany.*1228

*I will wait, did you come and have that epiphany?*1231

*I think it is worth staring at that equation because it is really fun to recognize something.*1235

*What you are supposed to recognize in this formula is the binomial theorem.*1239

*I will remind you what the binomial theorem says.*1244

*It says (A + B) ⁺n is equal to the sum from Y =0 to N of N choose Y A ⁺Y B ⁺N-Y.*1247

*You might have seen the binomial theorem used in a slightly different variable,*1262

*but it should be the same theorem because it is a universal truth.*1267

*What we have here is exactly that formula.*1271

*We are sort of reverse engineering the binomial theorem now, but my A is going to be PE ⁺T, my B is Q.*1275

*We have a perfect match of the binomial theorem.*1284

*It is A + B ⁺N, that is PE ⁺T + Q ⁺N.*1287

*Notice here that, we have a function of T.*1299

*T only, there are no Y left anymore.*1307

*The moment generating function is now a function of T, we have solved the problem.*1312

*If you do not like that Q, where did that Q come from.*1319

*You can always put it back in two terms of P.*1322

*You could write this as PE ⁺T, Q is 1 – P, all of that is still raised to the nth power.*1325

*I think that is the version that I gave you on the chart of moment generating functions a couple of slides ago.*1335

*Now you know how those two correspond to each other.*1341

*We are done with that example, we found the moment generating function for the binomial distribution.*1346

*Let me recap the steps we went through.*1352

*First of all, I have reminded myself of the probability function for the binomial distribution.*1354

*Here it is, N choose Y P ⁺Y Q ⁺N-Y.*1360

*Here is the range of Y values involved.*1364

*And then, I used the definition of the moment generating function, found on one of the earlier slides in this lecture.*1367

*It is the expected value of E ⁺TY.*1375

*The expected value of any function, the way you calculate it is you sum/Y.*1378

*This would be an integral, if you are in a continuous distribution.*1383

*But since binomial is discrete, we are using the sum.*1386

*The probability of Y × that function E ⁺TY, I expanded P of Y that is what I did here, I expand P of Y into that.*1389

*And then, I noticed that there is a P ⁺Y and E ⁺TY.*1403

*I can combine those, if I cleverly write E ⁺TY as E ⁺T ⁺Y.*1407

*I combined those together as PE ⁺T ⁺Y.*1413

*And then, I really had an epiphany, I said look, that is exactly the binomial theorem.*1417

*I reminded myself of the binomial theorem here.*1423

*I noticed how this fits that pattern and this is exactly PE ⁺T + Q ⁺nth.*1427

*Notice that, it is a function of T, there are no more Y left in this.*1435

*If you do not like the Q, you could always expand it out into 1 – P.*1441

*That was the role that Q played in the binomial distribution.*1445

*Hang onto this moment generating function because we have not really used it for anything yet.*1449

*We just figured out what it was.*1455

*I just justified this formula on the chart at the beginning of this lecture, but I have not used it for anything yet.*1457

*What I'm going to do in the next example is, we will use this formula to calculate the mean of the binomial distribution.*1465

*We will see for the first time what MGF can be good for.*1473

*Do no forget this formula, we are going to use it again right away in example 2.*1477

*In example 2, we are going to use the MGF for the binomial distribution to find the mean of the distribution.*1482

*We calculated the moment generating function for the binomial distribution in the previous example, example 1.*1489

*If you did not just watch example 1, maybe go back and watch it right now.*1496

*What you will find out is that the moment generating function, this is what we calculated in example 1,*1500

*turned out to be PE ⁺T + Q ⁺nth.*1507

*What is that mean? I have no idea.*1514

*But let me show you how we can use it.*1516

*Remember that, we can calculate the mean of the distribution, the expected value of Y.*1519

*The way you calculate that, the way you calculate it now that we have the event Scientific Technology *1527

*of the moment generating function is to take M sub Y prime of T at T =0.*1533

*You take its derivative and then you plug in T = 0.*1545

*This is something that we learned in the second slide, I think, of this lecture.*1549

*If you scroll back a few slides and look at that, you will see where this comes from.*1554

*Let us figure out what the derivative of this is, PE ⁺T + Q ⁺N.*1559

*Remember, T is my variable, everything else is a constant P, Q, E, N, those are all constants.*1565

*N is an exponent, I'm going to use the power rule.*1572

*It is time to review your calculus 1.*1575

*The derivative of something to the nth is N × all that stuff, PE ⁺T + Q ⁺N-1 × the derivative of this stuff inside.*1578

*That is the chain rule, I have to do PE ⁺T, and Q is a constant, I do not have to do anything about that.*1591

*That is the chain rule that I had to write PE ⁺T on the outside there.*1599

*At T = 0, I got to plug in T = 0.*1603

*If I plug in T = 0, it is N × P × E ⁺T is just 1 + Q ⁺N -1 × P × E⁰ is just 1.*1609

*In the parentheses there, I see that I have P + Q.*1625

*Remember that, Q is 1 – P, that means P + Q is equal to 1.*1628

*I have got N × 1, that P + Q magically simplifies into 1.*1635

*1 ⁺N-1 × P × 1.*1642

*1 ⁺N- 1 is just 1, and I have got N × P.*1648

*That is the mean of the binomial distribution, you can call it the expected value or the mean, *1655

*I do not care which one you use because they both mean the same thing.*1660

*This is something that we did know years and years ago, when we study the binomial distribution.*1664

*But, it is nice to have the moment generating function to confirm it.*1672

*The mean of the binomial distribution is N × P.*1676

*Let me recap the steps there.*1680

*I started off with the moment generating function that I calculated back in example 1.*1682

*That comes from example 1, if you did not just watched example 1 then you are missing out *1688

*because you would not know how we derived that.*1693

*Maybe you go back and watch example 1 to see where that came from.*1696

*To find the expected value of any distribution, what you do is you can take*1701

*the moment generating function take its derivative, and then plug in T = 0.*1707

*We took its derivative, a little bit of calculus 1 coming here.*1713

*We got the power rule N × PE ⁺T + Q ⁺N -1.*1716

*The chain rule means you have to multiply on the derivative with the stuff inside.*1721

*That is where the PE ⁺T came from, and the Q just goes away because it is a constant.*1725

*And then, I plug in T = 0 that is why I got E ⁺T is 1 here.*1731

*P + Q turn into 1, and that all simplifies down 1 ⁺N-1 just turns into 1.*1738

*That simplifies down to NP, now, I know what the mean of the binomial distribution is.*1746

*We are going to do this again, something similar with the Poisson distribution.*1755

*If this still does not make sense then you got a chance to see the same kind of process with the Poisson distribution.*1760

*Stick around for examples 3 and 4.*1766

*In example 3, we are going to find the moment generating function for the Poisson distribution.*1770

*It is kind of working from scratch there.*1775

*Let me remind you, first of all, the probability function for the Poisson distribution.*1777

*The probability function for the Poisson distribution, there was a λ parameter in there.*1782

*It is λ ⁺Y/Y! × E⁻λ.*1787

*The possible values of Y there could be anything from 0 up to, it is unbounded.*1795

*That is the probability function for the Poisson distribution.*1803

*If you do not remember that, if it looks like I just completely brought that in from that field.*1807

*Maybe, what you want to do is re-watch the video about the Poisson distribution which can be found in the same set of lectures.*1813

*Just scroll up, you will see a whole video on the Poisson distribution.*1821

*In particular, you will see this formula in there, you will see where it comes from.*1825

*Now, I want to find the moment generating function for the Poisson distribution, N sub Y of T. *1829

*By definition, this is the definition I gave you earlier in this lecture.*1838

*I highlighted it, you really would not miss it.*1843

*It is the expected value of E ⁺T × Y.*1845

*How will I calculate the expected value?*1852

*For a discreet distribution, you take the sum overall possible values of Y,*1855

*the probability of each of those values × the function that you are calculating E ⁺TY.*1862

*If this were a continuous distribution, it would be almost the same, except, instead of the sum, *1868

*we would have an integral.*1874

*Also, instead of the P we have an F.*1876

*But it would still be the same basic format, just you might want to get comfortable switching back and forth *1879

*between sums and integrals in your mind, because they really play the same role.*1886

*One for discrete distributions and one is for continuous distributions.*1890

*I'm going to plug in what P of Y is, it is the sum on Y.*1897

*I guess Y is equal to 0 to infinity, that is coming from this range on Y here.*1902

*P of Y is λ ⁺Y/Y! × E ⁻λ.*1907

*I also have this term of E ⁺TY, what can I do with that.*1914

*One thing I notice is that E ⁻λ is not really doing anything.*1918

*Because it does not have a Y in it, that means it is constant, I can pull that outside.*1924

*E ⁻λ × the sum from Y = 0 to infinity.*1928

*Λ ⁺Y and E ⁺TY, I can combine those.*1935

*E ⁺TY is the same as E ⁺T ⁺Y.*1940

*This is λ E ⁺T ⁺Y/Y!.*1946

*I do not need to write the E ⁻λ because I wrote it outside, and that accounts for all terms here.*1954

*Again, I'm going to pause and let you stare at this for a moment or 2, *1962

*and have an epiphany because there really is a revelation to be made with this formula.*1967

*Do you see the revelation that we had at this formula, just stare at it, there is something really good.*1975

*As a hint, I will remind you of the old Taylor series for E ⁺X.*1982

*The Taylor series for E ⁺X is the sum from N = 0 to infinity of X ⁺N/N!.*1987

*Look at this, we have got the same formula here except that in place of N, we have got Y.*1997

*In place of X, we got λ E ⁺T.*2004

*What we really have here, of course we still got E ⁻λ, is E ⁺λ E ⁺T, very nice and simple.*2009

*By the way, notice now, that we have gotten rid of the Y.*2022

*We got it down to a function of T, that is very convenient because that is*2025

*what a moment generating function is supposed to be.*2031

*It is supposed to be a function T not of Y.*2033

*That is essentially the mean right now, I will do a little algebra to simplify it but we have done the hard part.*2038

*I can combine these E ⁺λ E ⁺T – λ.*2044

*E ⁺λ, if I factor that out × E ⁺T-1.*2050

*That is the moment generating function for the Poisson distribution.*2056

*We are done with that problem.*2061

*To recap the steps there, in case anybody is a little confused.*2070

*Poisson distribution is one we studied earlier, there is another video lecture on the Poisson distribution.*2074

*Just scroll up and you will see it.*2079

*In particular, you will see the probability function for the Poisson distribution.*2081

*There it is right there, λ ⁺Y/Y! × E ⁻λ.*2085

*Λ is the parameter that comes in for the Poisson distribution, that you sort of fix ahead of time, it is a constant on.*2090

*There is the range of Y, 0 to infinity.*2098

*To find the moment generating function, we take the expected value of E ⁺TY which means we sum the Y, *2101

*of the probability of Y × E ⁺TY.*2109

*And then, I just dropped the probability function in there.*2112

*There is the probability function, I sum of all the ranges of Y that we are interested in, that came from right here.*2115

*This E ⁺TY, I discovered that I can write it as E ⁺T ⁺Y.*2126

*I can combine it with λ ⁺Y.*2132

*I factored out E ⁻λ, I can factor that out because there is no Y in there, it is a constant.*2135

*What I realized here is that, this exactly matches my Taylor series formula for E ⁺X.*2141

*What I get here is E ⁺λ E ⁺T.*2148

*And then, I did a little algebra to clean that up into E ⁺λ × E ⁺T – 1.*2152

*Hang onto this moment generating function, we are going to use it again in the next example.*2160

*We are going to find the mean and the variance of the Poisson distribution,*2165

*using the moment generating function.*2169

*Make sure you understand this, and when you are pretty confident with it, go ahead and work on example 4.*2173

*You will see how we use this moment generating function to find the mean and the variance.*2181

*In example 4, we are going to use the moment generating function for the Poisson distribution,*2188

*to find the mean and the variance of the distribution.*2194

*We just calculated in example 3, the moment generating function MY of T is E ⁺λ × E ⁺T-1.*2198

*That was the moment generating function.*2211

*If you do not remember how we did that, it means you did not just watched example 3.*2213

*Go back and watch examples 3, that should make sense.*2217

*There was an earlier fact that I gave you earlier in this lecture which is that E of Y is always the moment generating function.*2222

*You take its derivative and then you plug in 0.*2234

*We will use that to find the mean E of Y² is the second derivative of the moment generating function.*2237

*You plug in 0, that is not the variance directly but you can use that very quickly to find the variance.*2247

*We are going to take the second derivative of this moment generating function.*2253

*It is going to get a little messy but it is not too bad, especially after we plug in 0, it is really not bad.*2259

*Y prime of T is equal to, we have an exponential function, it is just E to all that same stuff.*2265

*E ⁺λ × E ⁺T-1 ×, chain rule coming in here, the derivative of all that stuff in the exponent.*2274

*That is λ × E ⁺T - λ × 1.*2282

*Λ × 1 is constant, its derivative just goes away.*2288

*That is it, let me go ahead and take the second derivative while I'm at it.*2292

*N double prime of T, this is going to be nasty.*2298

*We are going to have to use the product rule for it.*2309

*It is not that bad, it is just kind of basic calculus 1 stuff.*2312

*Let me factor out the λ because that is a constant, I factor that right now.*2316

*The first × the derivative of the second.*2320

*The first function is E ⁺λ × E ⁺T-1, × the second one is E ⁺T.*2322

*I'm ignoring this λ now because I have pulled that to the outside.*2334

*That was the first × the derivative of the second.*2342

*The derivative of E ⁺T is E ⁺T.*2344

*The second function × the derivative of the first one is little a messier.*2347

*The second function is E ⁺T, the derivative of the first one is E ⁺λ × E ⁺T-1 × its derivative which by the chain rule is λ × E ⁺T.*2351

*All of that multiplied by a λ.*2365

*I could have simplify that but I do not think it is worth doing.*2367

*Instead, what I'm going to do is plug in 0 to each of these functions.*2373

*Let me go back above and Y prime of 0 is E ⁺λ ×, E ⁺T is E⁰, E⁰ is 1.*2377

*So 1-1 is 0, it is E ⁺λ × 0 × λ × E⁰ × 1.*2392

*That E⁰ is 1 is just λ.*2401

*N double prime of 0, go through here and plug in 0 everywhere I see a T.*2406

*Λ × E ⁺λ, E ⁺T is E⁰ which is 1.*2414

*E ⁺λ × 0, E ⁺T is 1 + E⁰ is 1, E ⁺λ × 0 × λ × E⁰ is 1.*2422

*Let us simplify this down.*2440

*This is λ × E⁰ is 1 + I see another one × λ, 1 + λ.*2441

*This simplifies down to λ + λ².*2451

*How are we to use all this information?*2456

*Remember, the expected value of Y is M prime of Y, M prime of 0.*2459

*The expected value of Y is MY prime of 0 which we figure out was λ.*2466

*That is λ right there, and that is that mean.*2480

*We figured out the mean of our distribution is λ, very nice to know.*2484

*To find the variance, it is a little more complicated.*2491

*Sigma² is not just M double prime, it is the expected value of (Y)² - the expected value of Y².*2497

*This is N double prime is E of Y², that is λ + λ² -, E of Y², the E of Y we figure out was λ, λ².*2506

*This is very nice, the λ² cancel.*2528

*For the variance, we also get λ, how convenient.*2531

*What we have done is we have calculated the mean and variance of the Poisson distribution,*2537

*based solely on the moment generating function.*2543

*Once you understand the moment generating function, you can find the mean and variance of the distribution.*2545

*Let me show you the steps there, again*2553

*We calculated, first of all the moment generating function, that came from example 3.*2556

*The work here was all done in example 3, and there was some work to be done there.*2562

*And then, we took its derivative which was kind of, no product rule in that but there was a chain rule.*2567

*You took its second derivative and there was a big product rule, and lots of little chain rules coming in.*2574

*It got a little messy, but when we plunged in 0 then all the E⁰ turned into 1, that simplify a lot there.*2581

*M single prime turn into λ, the M double prime, all the 0 turn into 1.*2591

*It simplified down to λ + λ².*2600

*Here is how we use those, remember, I told you on the 2nd slide of this lecture, that M prime is E of Y.*2603

*M prime gives you E of Y which right away is the mean of the distribution, that λ is coming from there.*2613

*The M double prime is the E of Y² which is not the variance yet but it factors into calculating the variance, *2625

*because the variance is E of Y² – E of (Y)².*2634

*That λ + λ² is where we got that λ + λ².*2640

*The E of Y also came from up here.*2645

*We plug in that λ in there, we got λ² which canceled off the λ² from E of Y².*2654

*It just reduced down to the variance of the Poisson distribution is λ.*2660

*Of course, those answers agree with what I told you several lectures ago, when we talked about the Poisson distribution.*2667

*It is really reassuring to have those agree with what we had previously suspected there.*2674

*In example 5, we are going to find the moment generating function for the uniform distribution.*2689

*This is kind of nice because the other examples were both discreet distributions.*2693

*This is the only continuous distribution we are going to calculate.*2699

*The others are kind of messy.*2703

*Even doing this for the uniform distribution, it is a little messy than you might expect,*2705

*considering that the uniform distribution is so simple.*2710

*Let me remind you what the uniform distribution is.*2714

*The density function for the uniform distribution is F of Y is always equal to, *2718

*those three lines mean it is constantly equal to 1/θ2 – θ1, where Y ranges between θ1 and θ2.*2723

*It is just the constant distribution, that is why it is called uniform.*2733

*Let us find the moment generating function.*2737

*By definition, the moment generating function is := means defined to be, the expected value of E ⁺T × Y.*2739

*The way you calculate the expected value of a function is, with the discreet distribution we are studying before was the sum.*2752

*For a continuous distribution, it is an integral.*2761

*The integral, this is also a definition of expected value.*2765

*It is the integral of the density function F of Y × whatever function you are trying to find the expected value of, *2769

*in this case E ⁺TY DY.*2777

*And then, you integrate that over your whole range for Y, which in this case is θ1 to θ2.*2780

*Now, we just have to do some calculus.*2788

*This is the integral from θ1 to θ2.*2792

*F of Y is 1/θ2 – θ1, that is just a constant there.*2795

*E ⁺TY DY, not such a bad integral, really not too bad.*2802

*The answer is 1/θ2 – θ1, that is a constant, I can pull it out.*2807

*What is the integral of the E ⁺TY, remember here, our variable is Y.*2813

*We are integrating with respect to Y.*2818

*The integral of E ⁺TY, if you do a little substitution there, let me go ahead and do it in my head.*2824

*It is just E ⁺TY × 1/T, that is because we are thinking of T as being constant here.*2830

*Y is the variable of integration, it is just 1/T.*2839

*If you take the derivative of that with respect to Y, you get back to E ⁺TY.*2844

*We want to evaluate that from Y is equal to θ1 to Y is equal to θ2.*2848

*We get, I will combine the T in the θ2 – θ1.*2857

*We are plugging in these values for Y.*2865

*E ⁺θ2 × T – E ⁺θ1 × T, I need parentheses here.*2871

*I could write that over a common denominator, E ⁺θ2t –e ⁺θ1t.*2882

*We divide that by T × (θ2- θ1).*2894

*That is my moment generating function for the uniform distribution.*2901

*Notice that, this is a function of T now, there are no Y anywhere.*2906

*That is what is supposed to happen with a moment generating function.*2913

*It should always be a function of T, it should not have any Y anywhere in there.*2917

*This is my complete answer here and I'm done with that example, except for a quick recap of the steps there.*2923

*Just to remind you, we have a whole lecture on the uniform distribution.*2935

*If you do not remember the basic premise of the uniform distribution, you can go back and do a quick review there.*2939

*The density function is 1/θ2 – θ1.*2946

*In particular, it is constant that is why I have three lines here to show that is always equal to.*2950

*The range goes from θ1 to θ2.*2955

*The moment generating function, by definition, we learned that in this lecture, it is the expected value of E ⁺TY.*2958

*The expected value of any function is the integral of the density function × that function.*2967

*If this were discreet, we have the sigma sign summation, instead of an integral, *2974

*and we have a probability function P, instead of a density function F.*2981

*It is really the same idea, when you look at these formulas, if you kind of blur your eyes a little bit, *2985

*you should see how they are really the same idea.*2990

*Integrals, like adding things up, the probability function is kind of the analogue of the density function.*2993

*Instead of the summation of P of Y, we have the integral of F of Y, and then, we still have E ⁺TY.*3001

*F of Y from above is just 1/θ2 – θ1, that comes from up above.*3007

*We will pull that out, since it is a constant.*3014

*Now, we have to integrate E ⁺TY, I did a u substitution.*3016

*My u was TY, my DU was T DY, DY was 1/T DU.*3021

*That is where I got that 1/T on the outside there.*3033

*It is the opposite of the chain rule or a substitution.*3037

*We still have E ⁺TY that is because we are integrating with respect to Y, not with respect to T.*3042

*The range on Y goes from θ1 to θ2, I plug those in and I still had 1/T × θ2 – θ1.*3046

*It is still quite complicated considering that it is a uniform distribution, *3057

*you might expect something simpler for the uniform distribution.*3062

*But you end up with this function of T that does represent the moment generating function for the uniform distribution.*3067

*I’m not going to take this one any farther, but if you want to, you could use this *3074

*to find the mean and the variance of the uniform distribution.*3078

*The same that we did in example 4, with the Poisson distribution.*3084

*You can calculate those out, it gets a little messy so I'm not going to do it here.*3087

*Instead, I'm going to wrap up this lecture here on moment generating functions.*3092

*This is part of the probability lecture series here on www.educator.com.*3097

*Next up, we are going to talk about by Bivariate distribution, we will have a Y1 and Y2.*3102

*That is another whole chapter of excitement, I hope you will stick around for that.*3107

*You are watching probability lectures on www.educator.com, my name is Will Murray, thank you very much for joining me, bye.*3112

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