For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### Expected Value (Mean)

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 0:00
- Definition of Expected Value 0:20
- Expected Value of a (Discrete) Random Variable or Mean
- Indicator Variables 3:03
- Indicator Variable
- Linearity of Expectation 4:36
- Linearity of Expectation for Random Variables
- Expected Value of a Function 6:03
- Expected Value of a Function
- Example I: Expected Value 7:30
- Example II: Expected Value 14:14
- Example III: Expected Value of Flipping a Coin 21:42
- Example III: Part A
- Example III: Part B
- Example IV: Semester Average 36:39
- Example V: Expected Value of a Function of a Random Variable 41:28

### Introduction to Probability Online Course

### Transcription: Expected Value (Mean)

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

*Today, we are going to talk about the expected value of a random variable which is also known as the mean of a random variable.*0006

*Those two words or phrases mean exactly the same thing.*0013

*Mean and expected value are used interchangeably.*0016

*Let us learn what those mean.*0020

*The way you calculate the expected value of a random variable is you find all the possible values of the random variable,*0022

*you multiply each one by the probability that that value will come up and then you add those up.*0032

*The notation that we use for that is actually a little confusing because *0044

*they are 2 different notations that are used interchangeably there.*0048

*E of Y means expected value and μ is the Greek letter μ and that stands for mean.*0052

*Those mean exactly the same thing, we use those interchangeably.*0061

*Remember from now, if we say the word mean, if we say the phrase expected value, those are exactly the same.*0066

*They are both defined the same way, it means you look at all the possible values and *0074

*the probability of each one, multiply those together and add them up.*0079

*That is the definition, that is not the hugely illuminating.*0083

*The intuition behind expected value is that you want a think of it *0087

*as the average value of that random variable over the long run, if you repeat an experiment many times.*0092

*If you do the experiment many times, on average the, random variable Y should be that value, the value of the mean.*0101

*It does not mean it will ever be exactly equal to that value but in the long run, it will average out to that value.*0110

*Another way to think about this is if you think of Y as being a payoff for a fair game.*0120

*We talked about this a little bit in the previous lecture, when we first learn about random variables.*0126

*We talked about how you some× want to think of the random variable as being a payoff on a game.*0131

*We play a game, we do some kind of experiment, and then based on that experiment*0136

*I pay you a certain amount of money and that amount of money is the value of Y.*0141

*If you think of it that way and then if you want to make this game fair then the expected value is the amount that you should pay me, in order to play this game once.*0146

*If you play this game many times, on the average, the amount you pay me to play and*0157

*the amount I pay you will balance out and will be even.*0164

*It is the amount that a game should cost, in order to make it a fair game.*0169

*That is the intuitive idea of expected value or mean.*0174

*We got several different notions we need to explore with this.*0180

*The first one is indicator of random variables.*0183

*Indicator random variables are very simple random variables.*0187

*What an indicator random variable does is it just looks as particular event and *0191

*it takes on the value 1 if that event is true and the value of 0 if that event is false.*0197

*It is an indicator for whether that event happened.*0205

*A very common example of an indicator random variable would be if you are flipping a coin a number of times*0208

*and then you would set up an indicator random variable to indicate whether or not a particular flip was heads.*0215

*That is a kind of an example of an indicator random variable.*0222

*We will see in some of the problems later on in this lecture how you use indicator variables.*0226

*We will use it for that example of flipping a coin many times.*0231

*The important thing about an indicator random variable is the expected value of that *0234

*indicator random variable is just the probability that that event is true.*0240

*It is very easy to calculate the expected value of an indicator random variable.*0246

*You just calculate the probability of that event being true.*0251

*For example, if you are flipping a coin many × and you have an indicator random variable for the first flip being heads, *0255

*if it is a fair coin, there are is 50-50 chance that that first flip is going to be a head.*0262

*The expected value of that indicator random variable is just ½.*0267

*Let me show you a little more about how we are going to use these ideas.*0273

*A very key notion with expected values that is very useful in probability is this notion of linearity of expectation*0278

*The idea there is that you can combine random variables and you can break them apart.*0288

*The expected value is preserved across those kinds of combinations.*0293

*If you have 2 random variables Y1 and Y2, or if you have constants A and B,*0298

*the expected value of this combination just breaks apart linearly into A × the expected value of Y1 + B × the expected value of Y2.*0307

*A very key formula there, we usually use it to break apart complicated random variables into simple random variables.*0319

*The most common use of that is to break apart a variable into indicator variables *0328

*because it is very easy to calculate the expected value of indicator variables.*0336

*This is all a bit theoretical right now.*0341

*We are going to get some nice examples in just a couple of moments.*0343

*You will get to see how a complicated variable splits up into indicator variables and *0346

*it makes the expected value much easier to calculate.*0352

*There is one more notion that we need to explore before we jump into the examples*0358

*which is the expected value of a function of a random variable.*0363

*Instead of just looking around the variable by itself, we will look at some function G of Y.*0368

*Probably the most common example for G of Y will be Y², that is very common example there.*0373

*The expected value of functional random variables is defined very similarly to how the expected value of the original variable was.*0379

*Let me just remind you how we define the expected value of the original variable.*0389

*E of Y was the sum overall real number, of all possible real numbers of P of Y × Y.*0394

*That was the expected value of Y.*0406

*That Y on the right there came from that Y in there.*0409

*That is what we are finding the expected value.*0414

*Here we are finding the expected value of a function G of Y.*0417

*We replace that Y on the right with G of Y.*0421

*We replace that Y on the right, instead of just having P of Y × Y.*0428

*We did P of Y × G of Y and then we calculate that sum.*0433

*Often, it will be Y², we will see some examples of that.*0438

*I think this is a bit abstract but after we do some examples then it will make a lot more sense.*0441

*Let us work through some of these examples together.*0448

*First example here, this is actually the same experiment that we had back in the previous lecture *0451

*but we are going to calculate the expected value now.*0457

*The example is that you are going to draw a card from a standard 52 card deck.*0463

*If it is ace through 9, I pay you that amount.*0468

*If you draw an ace, I will pay you $1.00.*0471

*If you draw a 2, I will pay $2.00.*0473

*If you draw a 9, I will pay you $9.00.*0475

*If it is a 10 though, you have to pay me $10.00.*0478

*If it is any face card, a jack, queen, or king, then you have to pay me $10.00 for that privilege.*0481

*I want to figure out what is the expected value for you for this random variable.*0489

*Let us calculate that out together.*0495

*Remember, the expected value of a random variable just by definition is the sum of all possible values of Y , of P of Y,*0497

*the probability of that particular value × the value itself Y.*0509

*Let us think about the different values that this game could take for you. *0514

*It could end up winning $1.00.*0518

*If you win $1.00 from this game that is if you get an ace.*0520

*There are 4 ways you can get an ace out of 52 cards.*0527

*You have a 1/13, that is 4/52, 1/13 chance of winning exactly $1.00.*0530

*By the way, we calculated this probability distribution as an example in the previous lecture.*0537

*If this is completely mysterious, you may want to go back and *0542

*watch that example in the previous lecture then this one will make a little more sense.*0545

*What else could you make, you could make $2.00 out of this experiment.*0554

*There is a 1/13 chance that you are going to make $2.00 because there are 4 cards out of the possible 52 that will give you $2.00.*0558

*1/13 chance you are going to make $3.00 because there are 4 cards, there are 4 3’s in the deck out of 52 cards.*0569

*All the way on up to you can make $9.00 and there is a 1/13 chance that you can make $9.00.*0578

*You could also end up losing money on this experiment.*0588

*If you get a 10, jack, queen, or king, you pay me $10.00.*0591

*For you, that is a -$10.00 difference.*0595

*The odds of getting that were 16 out of 52, we calculated that last time.*0603

*There are 16 cards out of 52 possible that count as 10, jacks, queens, or kings.*0608

*16 out of 52 reduce to 4 out of 13.*0617

*That is the probability that you will end up paying me $10.00.*0621

*Let us calculate this together.*0625

*You see you got a 1/13 everywhere, if we factor out that 1/13 and then I will just have 1 + 2 + 3*0629

*up to 9 -4 × 10 because I factor out the denominator of 13, so 4 × 10 there.*0638

*1 + 2 + 3 + 4, 5, 6, 7, 8, 9, that turned up to be 45 - 4 × 10 is 40 which is 5/13.*0651

*What that means is that is the expected value of a random variable, it is also the mean.*0669

*Remember, the mean and expected value are exactly the same thing, so use those interchangeably.*0676

*Μ and E of Y, you want to get use to using those interchangeably.*0681

*What that means, I guess 5/13 is a little bit less than ½.*0686

*In the long run, if we play this game many times, on the average you are going to make about 5/13 for a dollar per game.*0692

*Obviously, in each game, you will never make exactly 5/13 because we are only trading back and forth a whole dollar here.*0703

*But it means that in the long run, every time we play 13 games, you should expect to make about $5.00 on average.*0712

*You, if we are going to make this a fair game, you should pay me for the privilege of paying.*0726

*Because on the average, I’m going to end up paying you a little bit per game.*0732

*You should pay 5/13 for a dollar to make it a fair game.*0736

*If I'm going to open a casino then I will probably round that up to 50¢.*0749

*I will make you pay 50¢ to play this game because I want to make sure that I will make a profit in the long run.*0756

*If I’m in a casino, I have to charge a little bit more than 5/13 for a dollar,*0763

*in order to guarantee that I will make a profit in the run long and cover my other expenses.*0767

*Maybe I will charge you 50¢, maybe I will charge you an even dollar to play this game but *0773

*I have to charge you more than 5/13 for the dollar or else I’m going to lose in the long run and*0778

*you will take money away from me by playing this game multiple times.*0782

*Just to remind you where everything came from here.*0787

*We are calculating the expected value of a random variable, that means we look at all the possible values and the probability of each one.*0789

*Those possible values, you can make $1.00, $2.00, $3.00, up to $9.00, or you could end up having to pay me $10.00.*0798

*And then, we calculated the probabilities of each one.*0806

*We did this example in the previous lecture but those probabilities just depend on how many cards give you those particular payoffs.*0809

*We figure out those probabilities and then this just was a fairly easy set of fractions to simplify down to 5/13 for the dollar.*0818

*That is the expected value which is the same as the mean of this game.*0829

*That means that if we are going to make it a fair game, a game that in the long run *0833

*there is no winning or losing between us, you should pay me 5/13 for the dollar.*0838

*If I want to make a profit off this, if I'm running a casino, of course, I want to make a profit.*0843

*I’m going to charge you a little bit more than 5/13 for the dollar, maybe 50¢, maybe a whole dollar to play this game.*0848

*Example 2 here, it is the same example as above, at least in the beginning.*0857

*You are going to draw a card from a standard 52 card deck.*0861

*If it is an ace through 9, I pay you that amount.*0865

*If it is a 10 or a face card, you pay me $10.00.*0867

*That part is the same, let Y be the amount that I pay you.*0870

*What is the expected value for Y²?*0874

*That is the new part, we are looking at Y² instead of Y.*0877

*The difference between this example and the previous one.*0882

*If this were a casino game and the casino promise to pay you Y², how much Y would the casino charge you to play?*0885

*What we are really doing here is we are looking at the expected value of a function of a random variable.*0894

*We are trying to find the expected value of Y².*0900

*Remember, the expected value of a function of a random variable, *0906

*let me show you up here the formula that we learned earlier.*0910

*It is the sum over all the possible values in the variable of the probability of each one × that function applied to the value.*0915

*The difference there is instead of having Y by itself, you put G of in here.*0928

*I will add up all the probabilities and all the Y values, except instead of putting in the Y values, I’m going to put in Y² each time now.*0933

*Our probabilities are 1/13, the Y is going to be 1.*0943

*Let me actually write the sum generically first, just to make it a little more clear here.*0953

*We are going to find the sum of Y/Y × P of Y × GL of Y, that is the sum on Y of P of y.*0958

*G of Y, that is this function right here, it is Y².*0971

*Our probabilities were 1/13 and our first value was 1, but we are going to square it, 1/13.*0976

*The next value was 2² 1/13, 3², up to 1/13 × 9².*0984

*Finally, the last value we could get was our probability 4/13 and it was -10 then we have to square that.*0997

*Now, we have to do some arithmetic to simplify this.*1007

*Again, I still see that I have a 1/13 everywhere.*1010

*1/13 × 1 + 2² is 4, 3² is 9, and so on, up to 9² is 81.*1014

*We have this last value of 4 × -10².*1026

*Squaring it makes it positive, + 4 × 100.*1030

*It is just some arithmetic to simplify that.*1038

*There is a formula, by the way, for adding up the sum of cubes.*1040

*It is a clever formula, I did not bring that in here because it is not really relevant to what I'm trying to teach you today.*1044

*I will just to add up those numbers quickly.*1052

*1 + 4 + 9 up to 81 + 400 turns out to be 685.*1055

*Our real answer there is 685/13, that is our exact answer.*1064

*If we are in a casino, you do not to deal in 13 for the dollar.*1075

*That is a little bit around $53.00, I round it out there to $53.00.*1079

*That means if you play this game, on the average, on the long run, you can make about $53.00.*1091

*Of course, you will not make exactly $53.00, you will make one of these values.*1097

*You either make $1.00 or $4.00, or $9.00.*1101

*You might make $81.00, you might make $100.*1106

*You are going to make one of those values but in the long run, on the average, you are going to make about $53.00.*1110

*If I'm the casino running this game, I had a 100,000 people play it on a given night then*1116

*I'm going to expect to pay out about $53.00 per customer on average.*1125

*That means, I will payout about $5,300,000 to people playing this game on any given night.*1131

*That is a lot of money.*1139

*If I'm the casino then I want to make sure I get a profit on this game.*1143

*A casino would have to charge a little more than $53.00, in order to make sure that *1148

*they make a profit in the long run on this game.*1158

*They might charge, if we are going to run this casino, my charge maybe $60.00 to play.*1163

*That means on every customer that plays this game, on average, *1173

*we expect to make $7.00 off this customer and in the long run we will make a nice, healthy profit.*1178

*In the short run, we might to lose money because certain customers are going to win $80.00 or $81.00.*1183

*Certain customers are going to win $100 from their $60.00 play.*1191

*In the long run, we will make about $7.00 from each game.*1195

*We will be happy as a casino even if we lose a few games in the meantime.*1199

*Let me recap what we are doing here.*1205

*We are really finding the expected value of the function of a random variable here.*1207

*I’m using this formula that was given on one of the early slides of this lecture, the third or fourth slide.*1212

*I think it is the 4th slide of this lecture, the expected value of a function of a random variable.*1219

*What you do is you look at all the possible values in the random variable, *1225

*the probability of each one, and you multiply it by that function of that value.*1229

*In this case, the function is Y².*1235

*Instead of having P of Y × Y, the way we had in the previous example, we have P of Y × Y square.*1237

*We do the same calculations before except the new element here is I’m squaring each of the Y before I run through the calculation.*1245

*The most notable change there is that makes that -10, makes it positive.*1255

*We now get 4 × 100, instead of 4 × -10.*1260

*The arithmetic there comes out to be about $53.00 which means if the casinos can offer you Y², then on average, *1265

*you are going to make about $53.00 per game.*1274

*If the casino wants to make sure that they make a profit, they will charge a little more than $50.00,*1277

*maybe $60.00 would be enough to guarantee them that they are going to make about $7.00 per game.*1283

*You are going to make more money on some games and lose some money on some games but make about $7.00 per game.*1291

*It is a nice, healthy profit for the casino.*1297

*The next example here, I want to flip a coin 3× and we are going to calculate the expected number of heads.*1303

*The second part of this problem, we are going to flip a coin 100× and calculate the expected number of heads.*1312

*We are going to use two different strategies to think about this problem and *1319

*I'm sort of trying to illustrate the principle of linearity of expectations.*1322

*We are getting to that in a few minutes but let me start out just by calculating the answer to the first problem directly.*1327

*We are going to set up our random variable first.*1336

*Let Y, we are going to define Y to be rows columns equals and it is defined to be the total number of heads.*1339

*That is the random variable that I'm going to keep track of here, the total number of heads.*1351

*I want to think about all the possible outcomes of this experiment and then what the value of Y would be for each one.*1362

*Let us think about all the possible things that can happen, when you flip a coin 3×.*1371

*You could get head-head-head, you could get head-head- tail, could get head-tail-head.*1377

*I’m sort of running them through in a binary fashion here, head-tail-tail.*1385

*I’m imagining myself counting in binary, tail-head-head, tail-head-tail, tail-tail-head, and tail-tail-tail.*1392

*There should be 8 outcomes total.*1405

*Let me make sure I have got them all, 1, 2, 3, 4, 5, 6, 7, 8 outcomes total.*1407

*Let us think about what the value of Y would be for each one of those.*1414

*The total number of heads, in the first one we got 3 heads, then 2 heads, 2 heads, 1 head, 2 heads, 1 head, 1 head, and 0 heads.*1418

*Those where the values of Y for each one there.*1429

*Let me calculate the expectation of that random variable, the mean of that random variable.*1434

*I’m just going to use the basic formula which was the sum/all possible values of Y of P of Y × Y.*1441

*I’m just using the basic definition of expectation of the mean of the random variable.*1451

*Remember, expected value and mean are the same thing.*1457

*Those are just absolutely synonymous expressions.*1460

*You use them interchangeably that you do not get any different information from one as from the other.*1463

*The probability of each one of these outcomes is 1/8.*1470

*It is 1/8 × 3 + 1/8 × 2 +, there is going to a 1/8 on all of these.*1477

*Let me just factor out the 1/8 and add up all the values of Y there.*1487

*It is a 3 + 2 + 2 + 1 + 2 + 1 + 1 + 0.*1493

*Let us see, 3 + 2 is 5 + 2 is 7 + 1 is 8 + 2 is 10 + 1 is 11 + 1 is 12, this is 12/8.*1504

*Of course, that simplifies down to 3/2, that is the expected number of heads.*1517

*It is a little curious there, if you flip a coin 3 ×, there is no way you can get 3/2 heads.*1524

*You are either going to get 0 heads, or 1 head, or 2 heads, or 3 heads.*1529

*It is not saying we expect to get 3/2 of a head because we cannot get 3/2 of a head.*1534

*What it is really saying is if you do this experiment many times, on the average, you will get 3/2 heads per experiment.*1540

*If you do this experiment, for example 8 ×, you would probably expect to see about 12 heads in total because 12/8 is 3/2.*1551

*That is what this number is saying.*1561

*That was a bit of computation in order to find that.*1564

*Let me show you another way to calculate this and I think it is a better way.*1568

*We are going to use linearity of expectation and we are going to use indicator variables.*1572

*Here is a better way to calculate this expected value.*1579

*We are going to define the indicator variables.*1589

*Remember, indicator variables are variables that just take the value of 1 or 0*1600

*depending on whether some event is true or false.*1605

*I have talked about indicator variables on one of the early slides back in the beginning of the same lecture.*1607

*You can just scroll back and check that out, if you do not remember what the indicator variable is.*1613

*In this case, I would define y1 to be the indicator variable for the first flip being heads.*1619

*Y1is going to be 1 if the first flip is a head and it is going to be 0 if the first flip is a tail.*1626

*Y2 is very much the same thing.*1642

*It is going to be 1 if the second flip is a head and 0 if it is a tail.*1646

*Y3 is the same thing for the third flip.*1658

*If the third flip is a head and 0 if it is a tail.*1663

*The point of that is that y1is just an indicator variable for the first flip being a head.*1669

*Y2 is an indicator variable for the second flip being a head.*1674

*Y3 is an indicator for the third flip being a head.*1678

*Let us think about y1by itself, the expected value of Y1.*1684

*The expected value of y1is just the probability that the first flip is a head.*1694

*The first flip is a head, that was what we learn about indicator variables in an early slide for this lecture.*1703

*It is very easy to calculate the expected value of an indicator variable.*1715

*It is just the probability that that event is true.*1721

*What is the probability that the first flip is a head, that of course is ½.*1724

*And then, we can find the expected value of Y2 the same way is also ½ and the expected value of Y3 is also ½.*1730

*Here is the beauty of this system.*1742

*Y is our total number of heads.*1744

*Our total number of heads is going to be the number of heads you get on the first flip +*1752

*the number of heads you get on the second flip + the number of heads you get on the third flip.*1757

*Y breaks down into a sum of these 3 indicator variables.*1763

*What we can use now is the beautiful linearity of expectation.*1769

*Linearity of expectation says the expected value of Y is equal to the expected value of Y1+, let me put them all together here.*1784

*Y is the same as Y1+ Y2 + Y3, the total number of heads is equal to the number heads you get on each flip.*1797

*The expected value of Y1, now we are using linearity + expected value of Y2 + the expected value of Y3.*1805

*But we figure all those out that is just ½ + ½ + ½ which is 3/2.*1816

*That is the same answer we got before but the advantage of that is,*1824

*we did not have to scroll down through all these 8 different possible outcomes.*1828

*That is a good way of calculating our answer to part A.*1836

*All those was just in the service of calculator answer to part A.*1840

*Let us think about part B now, I have not left myself much space here.*1844

*If we want to answer part B by listing all the outcomes, we would need to list 2 ⁺100 outcomes because we are flipping a coin 100 ×.*1848

*Each time we flip it, there are 2 possible things that can happen.*1866

*There are a string of 100 heads and tails for each possible outcome and there is 2 ⁺100, *1869

*2 × 2 × 2 a hundred × possible outcomes.*1876

*There is no way we can list that many outcomes.*1881

*I certainly do not have time on this video and you do not want to watch all of that.*1884

*There is really no way we can do this problem using the first method that I taught you for part A here.*1888

*We are pretty much stuck until we learn about linearity of expectation.*1893

*Instead, use linearity.*1901

*If you break it up the exact same way we broke up the first one, E of Y is equal to E of Y1 up to E of Y100.*1907

*We have broken it up into 100 little indicator variables and*1926

*the expected value of each one of those little indicator variables is just the same as the variables before, that is ½ + ½, 100 × over,*1930

*We are adding up a hundred ½ here and the expected number of heads,*1952

*the average number of heads, this is not too surprising, in a hundred flips is 50.*1958

*We are really depending heavily on that linearity of expectation, *1964

*in order to simplify a problem from having to write out 2 ⁺100 possible outcomes down into just adding up a bunch of ½.*1969

*That is really showing the power of linearity of expectation there for larger experiment.*1980

*Let me recap here.*1989

*We cannot do this, the first part of this problem 2 different ways.*1990

*First way, we just listed all the outcomes here, all the possible things that can happen when you flip a coin 3 ×.*1993

*All the possible strings that can happen, head-head-head, head-head-tail, and so on.*2002

*For each one of those, we wrote down how many different heads were listed in the string*2007

*and then we added up the probability of each one which is 1/8 × the number of heads we saw each time.*2013

*This set of numbers comes from this set right here.*2021

*If you just do the arithmetic there, it all simplifies down to 3/2.*2026

*In the long run, we expect to see 3/2 heads when you do this experiment, on average.*2030

*You will never see exactly 3/2 heads because there is no way to have half of the head.*2037

*What this means is that the long run, on average, will be that you will see 3/2 heads per experiment.*2042

*That was sort of the first way to do it but then I said there is a much better way *2052

*which is to set up these indicator variables which keep track of each flip, the first flip, the second flip, the third flip.*2056

*It is just 1 if that flip is a head and 0 otherwise.*2064

*It is the number of heads you see on the first flip.*2067

*You either see 1 or 0.*2071

*The expected value for each one of these indicator variables, the expected value of any indicator variable is the probability that*2074

*that event is true which in this case, what is the probability that the first flip is a head, its ½.*2082

*For each one of the other flips, the probability is ½.*2088

*Y is the total number of heads which breaks down into the number of heads on the first flip + *2092

*the number heads on the second flip + the number of heads in the third flip.*2098

*Here is where we invoke linearity of expectation.*2102

*The expected value of Y turns to be the expected value of Y1+ Y2 + Y3 and that breaks up.*2105

*Here is the linearity right here, that breaks up into the expected value of the individual random variables*2112

*which we already calculated to be ½ each so we get 3/2.*2121

*That may not seem much better on this small example A here but when we get to part B, we have a hundred flips.*2126

*The number of outcomes we would have to list would be 2⁺100 because it is 2 × 2 × 2 a hundred ×.*2135

*There is no way we can list that.*2142

*There is no way we can use that first method to solve that one.*2144

*But instead, if we use linearity, it breaks up very easy way into 100 little indicator variables.*2148

*Each one has expected value ½ and if you just add up a ½ a hundred ×, we get 50 which is a very easy answer.*2157

*It is not surprising that if you flip a coin 100 ×, on average, you expect to see 50 heads.*2165

*It does not mean that it is very likely that you will see exactly 50 heads.*2172

*It means that in the long run, if you do this experiment many times, you should see an average of 50 heads per experiment.*2177

*This one was all about the power of linearity of expectation and indicator variables.*2188

*That is the point that I was trying to drive home with this example.*2195

*In example 4, we are going to calculate your average grade in your probability class.*2202

*In your class, you are going to take 2 midterm exams and they are 25% each of your semester grade.*2208

*The final exam counts for 30%, that is a little bigger there and the homework counts for 20%.*2215

*That is where your grades is based on.*2220

*What your scores are 60 and 80 on each of the midterms, you score an 80 on the final, and then you score 100 on the homework.*2223

*The question is, what is your semester average?*2233

*This might not seem like an expected value problem.*2235

*Remember, another word for unexpected value is mean and mean really does mean average.*2239

*I will show you how this breaks down exactly into an expected value problem.*2245

*The μ, here I’m going to use the Greek letter μ for expected value.*2251

*Let me remind you what the formula is for expected value.*2257

*It is the sum over all possible values in the variable of the probability of each value × that value.*2261

*Let us think about all the possible scores you can get.*2272

*You scored 60 on 1 midterm, you scored 80 on something else, a couple of things, and you scored 100 on something.*2276

*Those are the possible Y values that we can see and we want to figure out how much weighting,*2287

*how much probability is attached to each one of those Y values.*2291

*Let us figure out the probability of each one of those Y values.*2296

*Think of that as the weight on each of those Y values.*2299

*We are going to find the probability for each one of those or weight of each of those.*2306

*How much weight did you score 60 on?*2312

*60 was on the first midterm, that was 25%.*2314

*I will put a 25/100 here, 25% there.*2318

*Where did you score 80 on?*2325

*You scored 80 on the midterm and then 80 on the final, that is 80.*2326

*The second midterm was 25/100 + the final is 30/100.*2333

*That is how much of weighting you scored on and 100 on the homework which was 20%, so 20/100.*2346

*You scored 100% on that part of the class.*2359

*I’m really using the formula for expected value to calculate your mean for the class or your average for the class.*2363

*This is a fairly easy thing to simplify here.*2372

*In fact, I did not bother to write down the intermediate steps.*2376

*I’m just going to calculate this all together.*2381

*If you calculate this all together, it turns out to simplify down to exactly 79.*2385

*That is your mean for the class, that is your average for the semester.*2401

*If you have a particular grading scale, you would se that to determine your grade for the semester.*2408

*Let me show you how what seem like a problem of calculating your semester average in class turned out to be an expected value problem.*2418

*We are calculating your μ.*2427

*Remember, μ is always the same as E of Y, it means mean or expected value.*2428

*Those are the same concept.*2433

*The way you calculate it is you look at all the possible things you could have scored and then you multiply each one by,*2435

*we said the probability but it is really the weight attached to each one of those items.*2442

*When I have looked at all the scores, I saw 60, 80, and 100.*2448

*I put a weighting on each one, 25% for the 60, 20% for the 100, and the 80 was a little complicated *2454

*because there were 2 different items that gave me an 80.*2460

*I had to put a weighting, sort of an extra weighting 25 out of 100 for the second midterm *2463

*and then 30 out of 100 for the final exam.*2471

*When I added all those things together and I simplify the arithmetic, simplify it down to exactly 79, *2475

*that is your semester average in your probability class.*2483

*In example 5 here, we are going to roll one dice and let Y be the number showing on the dice.*2491

*We are going to calculate E not of Y but E of Y².*2498

*This is another test of finding the expected value of a function of a random variable.*2503

*Let me remind you of the formula for that.*2512

*The expected value of the function of a random variable, *2514

*you calculate it quite similarly to expected value for the random variable itself.*2518

*It is P of Y, the variable itself, you just have a Y here.*2524

*Since, there is a function of the random variable, you run it through G of Y.*2529

*In this case, our function is exactly Y² and we are going to add up all the possible values *2534

*that could be showing in the probability of each one × Y² for each one.*2543

*If we are just finding the mean or the expected value of the variable itself, *2553

*that would just be a Y but now it is Y² because of what the problem is asking us.*2557

*Let us go ahead and think about all the possible values you can have.*2563

*The possible values when you roll a dice are 1, 2, 3, 4, 5, and 6.*2567

*If we want each one through the function that means we square each one.*2574

*Let me go ahead and square each one here.*2578

*Let me put a probability on each one.*2581

*1/6 × 1² + 1/6 × 2² + 1/6 + 1/6, and I did not give myself enough space here, + 1/6 × 6².*2583

*Of course, it is easier there to factor out the 1/6.*2605

*I will put a 1/6 on the outside here.*2608

*1² is 1 + 2² is 4, 3² is 9, 4² is 16, 5² is 25, and 6² is 36.*2611

*If you add those up, 36 + 25 is 61, 16 is 77, 9 is 86, 4 is 90, 1 is 91.*2622

*I get 91/6 as my expected value of Y² or my μ for Y².*2635

*That is the end of that problem but let me recap the steps here.*2649

*We are using the formula for the expected value of a function of a random variable.*2652

*I gave you that formula back on one of the early slides in this lecture.*2657

*You can go back and check that out.*2662

*The difference between calculating the expected value of a function and*2664

*the expected value of the original random variable means instead of Y here, you just change it to G of Y.*2668

*You are no longer talking about Y, you are talking about G of Y which means that you are adding up the probabilities ×,*2677

*in this case we have our function is Y².*2684

*That is the most common function we are going to be looking at.*2687

*In this case, we have Y² here.*2692

*We take all of our values and we square each one, 1², 2², 3².*2695

*These are all the values you can get by rolling one dice, 5², 6².*2700

*The probability of each one of those is 1/6, we will multiply by 1/6.*2704

*Now, we just simplify the arithmetic.*2709

*Square all those numbers, multiply it by 1/6, and we get 91/6.*2711

*This is a kind of an early preview of something we are going to be doing in the next lecture.*2716

*We are going to be looking at variance and standard deviation.*2721

*In order to calculate that, it is going to be very useful to calculate the expected value of Y² for our random variable.*2725

*That is why I'm giving you several examples this time of calculating the expected value of Y²*2733

*because we are going to be using that very often in the next lecture.*2738

*I hope this made sense to you, if it did not, you might want to work it through a couple more times,*2743

*before you go on to the next lecture and learn about variance and standard deviation*2748

*because you really want to understand how to find the expected value of Y² first.*2753

*That is the last example for this lecture on expected values and means.*2759

*Of course, expected values and means are the same thing.*2765

*You are all watching the probability lectures here on www.educator.com.*2768

*My name is Will Murray, thanks for watching, bye.*2772

1 answer

Last reply by: Dr. William Murray

Fri Dec 11, 2015 10:10 AM

Post by daniels calvin on December 10, 2015

When I worked it out prior to watching I modeled y as the final points you earned and your class scores out of 100 as your probability. IE if you got a 60% on exam one, you got 60% of 25 points towards the final grade. Is that reasonable?

1 answer

Last reply by: Dr. William Murray

Wed Apr 8, 2015 6:25 PM

Post by Alvi Akbar on April 7, 2015

Is there any way i Can change the video Playback speed ?

1 answer

Last reply by: Dr. William Murray

Mon Feb 16, 2015 6:27 PM

Post by Anhtuan Tran on February 15, 2015

Hi Professor Murray,

On example 2, I understand the math that you did, but it just does not intuitively make sense, especially the last part. So for example, when you draw a 10 or a face card, do you still have to pay 10 dollars? But if so, it doesn't reflect that on the calculation because when you square it, the minus sign goes away.

Next question: why are we interested in calculating the E(y^2). Does it mean that if we draw for example a 4, we win $16 instead of $4?

Thank you.

1 answer

Last reply by: Dr. William Murray

Tue Sep 2, 2014 7:58 PM

Post by Ikze Cho on September 1, 2014

Can we use other values apart from 1 and 0 for the indicator random variables?

1 answer

Last reply by: Dr. William Murray

Mon May 19, 2014 6:02 PM

Post by Ying Cao on May 14, 2014

Even it is Y^2, the casino will still pay to the player for the $10^2. I think the E(Y^2) = $22, instead of 53 in the problem at 14:14.