For more information, please see full course syllabus of Physical Chemistry

For more information, please see full course syllabus of Physical Chemistry

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## Download Lecture Slides

## Table of Contents

## Transcription

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### Probability & 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
- Probability & Statistics
- Example I: Calculate the Mean of x
- Example II: Calculate the Second Moment of the Data in Example I
- Define the Second Central Moment or Variance
- Continuous Distributions
- Probability Density
- Example III
- Part A - Show that P(x) is Normalized
- Part B - Calculate the Average Position of the Particle Along the Interval
- Important Things to Remember

- Intro 0:00
- Probability & Statistics 1:51
- Normalization Condition
- Define the Mean or Average of x
- Example I: Calculate the Mean of x 14:57
- Example II: Calculate the Second Moment of the Data in Example I 22:39
- Define the Second Central Moment or Variance 25:26
- Define the Second Central Moment or Variance
- 1st Term
- 2nd Term
- 3rd Term
- Continuous Distributions 35:47
- Continuous Distributions
- Probability Density 39:30
- Probability Density
- Normalization Condition
- Example III 50:13
- Part A - Show that P(x) is Normalized
- Part B - Calculate the Average Position of the Particle Along the Interval
- Important Things to Remember 58:24

### Physical Chemistry Online Course

### Transcription: Probability & Statistics

*Hello and welcome back to www.educator.com and welcome back to Physical Chemistry.*0000

*Today, I thought we would do a little discussion on probability and statistics.*0004

*Probability and statistics, probability plays a huge role in quantum mechanics because quantum mechanical systems are probabilistic.*0011

*They are not deterministic like classical mechanics, like the physics that you learn in freshman and sophomore year.*0021

*Deterministic meaning we can come up with an equation that is going to tell us exactly where something is going to be.*0027

*How fast it is moving, what its momentum is, what its angular momentum is, things like that.*0034

*We know what it is going to be, it is determine, it is predetermined.*0040

*Probabilistic quantum mechanics, elementary particles, high speeds,*0044

*we can say with certainty that something is here or there or it is moving this faster, that fast.*0049

*We speak probabilistically there is a 60% chance that it is here.*0058

*There is a 70% chance that it is moving this fast in this direction.*0064

*I thought I would just do a direction and talk a little bit about this.*0067

*A lot of what we are going to be presenting here, I would not worry too much about it.*0073

*If you cannot actually wrap your mind around some of this material, a lot of the probability and*0079

*statistical aspects of quantum mechanics, the understanding will emerge over time.*0085

*We will do a lot of computational problems.*0090

*We will just get a feel for it.*0093

*Again, quantum mechanics is very unusual.*0095

*It takes some time to get accustomed to it.*0099

*Do not feel bad if some of the stuff does not quite fit well with you.*0102

*But I do want to introduce it and see what we can do.*0106

*If we were to conduct an experiment that has impossible outcomes like flipping a coin, rolling the dice, rolling a pair of dice, whatever it is.*0114

*Let me work in blue here.*0127

*If we conduct an experiment with impossible outcomes, in the case of flipping a coin you have two outcomes, either heads or tails.*0131

*We actually repeat the experiment over and over and over again.*0156

*Just keep flipping, keep flipping and keeps flipping.*0159

*Repeat the experiment over and over then the probability of each outcome is the probability of event I,*0167

*the limit as N goes to infinity of N sub I/ N.*0206

*Where I = 1, 2, 3 all the way to N.*0216

*In the case of flipping a coin, you have got N is equal to 2.*0225

*You have 2 possible outcomes.*0229

*In the case of rolling a dice, you have 6 possible outcome so N =6.*0231

*Over N sub I is the number of times that outcome I occurs.*0238

*In other words, if you are to do 10 flips of a coin and 6 × it comes up heads and 4 × it comes up tails,*0260

*the N sub I for heads is going to be 6 and the N sub I for tails is going to be 4.*0266

*N is the total tries, you flip it 10 ×.*0273

*In this case, N is equal to 10 for the 10 flips.*0277

*N is the total tries.*0282

*Do not worry about this, this is a sort of definition.*0286

*In flipping a coin, you know the probability of getting a head is going to be 50% or 0.5.*0289

*Getting a tails is going to be 0.5.*0296

*If you are to roll a dice, you have 6 numbers, 6 faces on the cubic dice.*0298

*You have 6 numbers 123456.*0303

*The probability of getting a 1 is going to 1/6.*0305

*The probability of getting a 2 is going to be 1/ 6.*0308

*Do not worry about this definition, this is just a formal definition.*0313

*All this says that if I keep flipping it over and over again, in other words if I do the 10 and I get 6 × heads, 4 × tails.*0317

*That is going to be 6/10 and 4/ 10.*0327

*Will that is not quite 0.5, it is 0.6 and 0.4.*0329

*If I do it 20 ×, if I do it 30 ×, if I do it 50 ×, if I do a 100 ×, a 1000 ×, as N goes to infinity it is going to end up pretty evenly.*0336

*Half of the time it is going to be heads, half of the time it is going to be tails.*0346

*That is why this limit is there.*0349

*This is just a formal definition, do not panic about this.*0351

*It is not that important.*0354

*N is the total number of tries.*0356

*This is just for our formality so that we actually see it but that is all it says.*0359

*It is that the probability of an event is the number of times that a particular outcome occurs ÷*0364

*the total number of times and then you just keep trying.*0371

*You keep trying and keep trying, eventually you are going to get the probability of the event.*0374

*Clearly, N is the number of times that a certain outcome occurs and N is the total number of times.*0386

*We know that this N sub I is definitely less than or equal to N and is greater than or equal to 0.*0391

*In the case of flipping a coin, 6 × heads, for this particular outcome 6 is going to be less than*0400

*or equal to 10 and it is greater than or equal to 0 because you are going to have that outcome.*0406

*If we divide by N we end up with the following.*0411

*0 less than or equal to the probability N sub I/ N is the probability, less than or equal to 1.*0420

*The probability is always expressed in terms of a percentage or a fraction.*0427

*I think it is best to express it as a fraction.*0432

*Of course, all those fractions for all of the different possibilities they add up to 1.*0435

*That is actually the real important thing here.*0440

*And we are going to formalize this in a minute.*0442

*If the probability, if P sub I = 1 that means that outcome is 100% certain.*0449

*That means it happens all the time.*0464

*There is no uncertainty there.*0468

*If the probability is 0, this means the outcome is impossible.*0472

*It is never going to happen.*0476

*Of course, most probability are going to be somewhere between 0 and 1.*0482

*We also know that the sum of the N sub I is actually equal to N.*0488

*In other words, if I have 2 outcomes and the 10 × that we actually flip a coin, N is the total number of times.*0497

*The N sub 1 which is heads, that is a fix.*0508

*The N sub 2 is going to be the tails so 6 and 4 they add up to 10.*0511

*That is all this is saying.*0515

*Again, we are just formalizing everything.*0516

*I = 1 to N = N.*0519

*The number of times, if you add up all the different outcomes, how many times each one happens,*0522

*you are going to get the total number of tries which is the experiment.*0527

*Once again, let us go ahead and divide by N.*0530

*If I divide by N, I get 1/ N × the sum as I goes from 1 to N of N sub I = 1.*0533

*I’m going to pull this back inside so I get the sum N sub I/ N, goes from 1 to N = 1.*0551

*This N sub I is just N sub I/ N, that is just the probability.*0563

*This just says that when I add up all of the probabilities, for all of the outcomes, I get 1.*0575

*This is very important, this is called the normalization condition.*0584

*All of the probabilities have to add up to 1.*0600

*In the case of a flipping a coin, 0.5 + 0.5 = 1.*0602

*In the case of the dice, 6 faced dice, 123456, 1/6 1/6 1/6 + 1/6 1/6 1/6 all add up to 1.*0606

*All the probabilities have to add up to 1.*0617

*Profoundly important.*0619

*This idea of the normalization condition is going to be very important in quantum mechanics later*0621

*when we introduce the wave function because we are going to normalize this wave function.*0626

*Because we are going to interpret the wave function as a representation of the probability that is something is here or there.*0632

*Or the probability of the energy being this or that, things like that.*0640

*Again, this is the important relation, the sum of the probabilities = 1.*0652

*If you do not take away anything from this lesson normalization condition.*0656

*Suppose that some number X is associated with probability P sub I.*0667

*Let us say if I had a probability of 0.2 and let us say that 0.2, there are some number 10 associated with that.*0698

*In other words, the number 10 is going to show up 20% of the time.*0706

*Let us say the number 20 is going to show up 30% of the time, that is 0.3.*0710

*And the number 30 is going to show up 50% of the time, that is 0.5.*0717

*That is what we are saying, that given a particular probability of a certain event,*0722

*that there is some number associated with that particular event, we are going to define the following.*0726

*We define the mean or average, we will use both words more often than not.*0734

*We will probably talk more about the average than we will the mean.*0747

*In quantum mechanics we can use the word average more than mean.*0752

*We define the mean or average of all the X is this way.*0756

*The average value of X is equal to the sum of the each X sub I × the probability that X is actually shows up as I goes from 1 to N.*0772

*What we are saying is, it will make more sense if we do an example in just a minute here.*0785

*X is the particular number and X sub I is a particular number and P sub I is the probability that, that number occurs.*0795

*In the case of rolling a dice, 1, 1/6, 2, 1/6, 3, 1/6, 4, 1/6, 5, 1/6, 6, 1/6.*0806

*In this particular case, all of the probabilities happen to be the same.*0818

*There is no guarantee that the probabilities will actually be the same.*0822

*You might have a number in some situations that might end up showing up 20% of the time,*0825

*and another number shows up 70% of the time, and a third number shows up 10% of the time.*0830

*The probabilities add to 1, you have 3 numbers represented.*0836

*The average of those numbers is not going to be right down in the middle.*0839

*You do not just add them and divide by 3 because one of the numbers is going to end up showing up more often,*0844

*the average is going to be weighted more towards that number.*0851

*And that is what this definition of average actually takes into account.*0855

*It takes into account the weight that a particular number actually occurs.*0858

*Again, angular brackets this is the symbol for the average quantity.*0865

*Later on, we are going to see something like this.*0869

*E is going to be the average energy of that particular particle.*0872

*Let us go ahead and do an example.*0878

*I think it will start to make sense.*0879

*Very important definition, this is the definition.*0881

*The definitions are profoundly important.*0884

*If you ever lose your way in something, go back to the definitions and start again.*0887

*That is why definitions are very important.*0891

*Let us see if we can do an example here.*0896

*Given the following discreet probability distribution, calculate the mean of X.*0900

*When we talk about what discreet probability distribution means.*0905

*A distribution is exactly what you think it is.*0909

*In a dartboard, if I start throwing darts at it for an hour, there is going to be distribution of holes on that dartboard.*0916

*That is what a distribution is.*0925

*It is the attempts and how they have distributed themselves over a particular interval or over an area, or over a volume.*0926

*Whatever it is that a particular situation calls for.*0934

*Discreet probability distribution, discreet means there are specific numbers 2, 5, 7, 12.*0937

*When we talk about a continuous distribution, we talk about the entire number line.*0945

*Every single number, every single fraction, every single decimal, is represented.*0950

*It is continuous.*0954

*That is what continuous means vs. discrete 0, 1, 5, 10, 15 nothing in between.*0956

*That is all discrete probability distribution means.*0962

*We want to calculate the mean of X.*0964

*In this case, we have 2, 5, 7, and 12, we have 4 numbers.*0967

*Here is the X value and here is the probability of that number showing up.*0976

*2 is going to show up 20% of the time.*0980

*This is the probability distribution.*0983

*5 shows up 50% of the time, 7 shows up 35% of the time, and 12 shows up 30% of the time.*0987

*Notice that the sum of the probabilities is equal to 1.*0995

*We want to find the average.*0999

*You have learned that the average is just add 2, 5, 7, and 12, and divide by 4*1003

*It is not going to happen here.*1007

*Once I talk about the relation of this definition that we gave to the definition of average that you learn ever since you are a kid in school,*1008

*but notice the probabilities are not the same.*1016

*Each number has a different weight.*1019

*This 7 shows up more than this 5 does.*1022

*The average, you should represent the fact that average is going to be a little bit closer to 7.*1025

*The 7 has more weight.*1030

*Let us just go ahead and use the definition, use the math.*1032

*You do not necessarily have to understand everything that is going on conceptually.*1036

*It is going to be a lot of the case with quantum mechanics and in other things too.*1040

*But if you at least trust the math and let the definition of the formulas work for you,*1044

*then you will at least get accustomed to just you doing it mechanically.*1050

*There is no problem with that.*1053

*There is no sin in that, in just doing things mechanically, that is how we get a sense of what is going on.*1054

*We have this average value definition which is the sum as I goes from 1 to N of the X × the P sub I for that particular X.*1060

*Well, that is going to equal I = 1 to 4 because we have 4 values of the X sub I and the P sub I.*1073

*Let us just go ahead and do it.*1084

*It is going to be the X value × this probability.*1086

*2 × 0.2 + 5 × 0.5 + 7 × 0.35 + 12 × 0.30.*1089

*I will go ahead and write it out 0.4 + 0.7 + 2.45 + 3.6 the answer ends up being 7.2.*1112

*There we go, that is the average given the probability that these numbers show up.*1128

*Notice ,7 has the highest probability followed by 12 and 2.*1131

*Certainly, it is going to be that is what this takes into account.*1137

*That is what the definition takes into account.*1142

*This is how you find an average.*1145

*If you are wondering about why this definition of average looks different than the definition that you have been using ever since you are a kid.*1156

*In other words, take all the numbers, add them together, and divide by the number that there are.*1164

*In this case, 2 + 5 + 7 + 12 ÷ 4.*1169

*Here is what is going on.*1173

*The one that you learned in school is this one.*1175

*You take the sum of all of the X sub I, you add them all together, and you divide by N, the number that there are.*1177

*This is N up here.*1186

*This is the one that you have learned in school.*1190

*This definition is actually this definition that you learned in school and is actually the same as the definition I just gave you.*1192

*The only difference is this definition presumes that the probability of each number showing up is equal.*1200

*In other words, if I have 15 numbers, the probability of each one is just 1/15.*1210

*If I have 36 numbers, the probability is 1/ 36 for each number.*1215

*And I will show you.*1220

*If I have the numbers 1, 3, 5, 7, 9, 11, and 13, and if I said take the average of these numbers, just take the average.*1223

*Notice, I have not set anything about the probability of each one of these numbers.*1243

*The presumption is, the natural assumption that we have to make is that the probability is going to be the same for each.*1247

*The average is going to be this.*1254

*The average is going to be based on what you have learned from school.*1255

*It is going to be 1 + 3 + 5 + 7 + 9 + 11 + 13 all over 7 because there are 7 numbers.*1265

*Let us pull out the 7 as 1/7.*1280

*It is going to be 1/7 × 1 + 3 + 5 + 7 + 9 + 11 + 13.*1284

*I will distribute so I get 1 × 1/7 + 3 × 1/7 + 5 × 1/7 + so on and so forth until + 13 × 1/7.*1296

*That is equal to the sum of each X sub I, the 1, 3, 5, 7, 9, 11, and 13 × the probability.*1311

*The probability is 1/7.*1320

*Notice the probability is the same.*1323

*That is what they do not tell you in the definition of average that they teach you in school.*1326

*The presumption is that the probability is the same but it turns out to be the same thing.*1332

*It is just that the probability is 1/7 but it is the same definition.*1336

*In this case, Y = 1/7.*1341

*It is still just a particular number × this probability added together, that is all.*1344

*I hope that makes sense.*1352

*Let us go ahead and calculate the second moment of the data in example 1.*1358

*We are going to define something else called the second moment.*1365

*Here is our definition for that.*1369

*Define the second moment.*1373

*It is symbolized this way and that is exactly what you think it is.*1381

*Here, this time we are going to add from 1 to N.*1385

*The X sub I, the squares of the X sub I × their probabilities.*1389

*It is called the second moment.*1393

*Do not worry about what it means, just take it as it is a mathematical definition, plug the numbers in.*1399

*In this particular case, we get the second moment is equal to 2² × 0.2 + 5² × 0.15 + 7 squares × 0.35 + 12² × 0.30.*1407

*When we actually do the math here.*1444

*I have written 164.9.*1448

*We are not going to assign any meaning to it now.*1454

*It will be coming up later when we talk about quantum mechanics.*1456

*Important thing to note here.*1463

*Notice that the average value was 7.2 from example 1.*1468

*We are calculating the second moment of the data from example 1.*1473

*The second moment ended up being 64.9.*1476

*Notice, the average value², in other words 7.2² does not equal the average value of X².*1488

*The square of the average value of X does not equal the average value of the square.*1504

*The mean² is not equal the second moment.*1509

*Be very careful with that.*1515

*Be very vigilant about what the exponents on the outside or on the inside.*1517

*They are 2 different things.*1521

*Let us go ahead and define the third entity.*1528

*This is called the second central moment or the variance.*1533

*Define the second central moment or variance.*1539

*It is symbolized this way, as a σ².*1555

*The variance of a set of data.*1560

*The symbols right here.*1582

*As those moves us like that so that is equal to the sum = 1 to N of our individual X sub I - the average value of X² × the probability.*1585

*I find the mean of the set of values.*1605

*I take the difference of each individual value from that mean.*1612

*In other words, the difference from the distance of a particular number from the mean and I square it.*1616

*I multiply that number by the probability of the number and I add that up.*1625

*Do not worry about where these come from.*1629

*I know in statistics there is a real sense of some of these things just sort of dropping out of the sky.*1631

*To be completely honest with you, sometimes I would say that is exactly where they come from.*1638

*It just dropped out of the sky.*1641

*I wonder about some of these definitions myself.*1643

*Do not try to assign any meaning just now.*1645

*Just deal with the mathematics.*1648

*We also have the symbol.*1658

*If we take this value which is the variance and if we actually take the square root of it, we get that and this is called the standard deviation.*1661

*This is the one that you are actually more familiar with.*1678

*When we talk about standard deviation, the standard deviation of a set of data is going to be the square root of the variance.*1682

*This is the variance and you end up getting this number.*1692

*You add all these together and then you take the square root of it.*1696

*You do not add the square root of it.*1698

*You do this number then you take the square root and you get this.*1701

*These are just symbols, they do not have mathematical value.*1704

*We do the square and here without the two, simply to differentiate symbolically.*1708

*Show there is some relationship between them, that is all this is.*1714

*Now either of these variance or the standard deviation, does not matter which one you use.*1718

*Either of these is a numerical measure of the overall deviation of the points X sub I from the mean X.*1725

*It just gives you a measure of the example that we just did, we found an average value of 7.20 for the numbers 2, 5, 7, and 12 given the particular probability distribution.*1762

*The average was 7.2 and it gives you a measure of how far each individual point, the 2, 5, 7, and 12 are from the 7.2.*1778

*2 from 7.2, 5 from 7.2, 7 from 7.2, and 12 from 7.2, that is all it is.*1787

*It gives you some numerical measure.*1794

*It tells you how far apart is, how close it is to the mean.*1796

*We say that is a measure of the spread of the data.*1800

*How spread out are they or how close are they, actually to get to the mean value of all that data.*1805

*That is all that we are talking about here.*1811

*Let us go ahead and do some math with this.*1815

*Σ² of X is equal to the sum of the X sub I - the average value of X² × P sub I.*1819

*Let us go ahead and multiply this out.*1834

*This is just something² and we have a binomial so we can multiply this out.*1836

*This is going to be X sub I² -2 × X sub I × X + X² and P sub I.*1840

*This is a linear so I can go ahead and separate these into 3 sums.*1860

*This is actually equal to the sum of the X sub I² × π.*1864

*This π distribute over each.*1872

*This is going to be - 2 × the sum of X sub I × X × π + the sum of the average value of X², above the average value, × π.*1877

*Note also, this σ², the variance is a sum of positive terms.*1904

*These are positive, the probability is positive.*1916

*What we want you to notice is that the variance is actually greater than or equal to 0.*1920

*It is going to be very important.*1925

*Now we have first term, second term, third term.*1928

*First term, the sum of the X sub I P sub I² that is just equal to, by definition, the second moment.*1940

*That is the first term.*1957

*Our second term is -2 × the sum of the X sub I × the average value of X.*1960

*This average value of X is just a number. Because it is a number, I can pull it out of the summation symbol.*1976

*This sum of the X sub I π, this is just the definition.*2000

*It = -2, that is just the definition of average value so it becomes -2.*2004

*That is that one.*2017

*Let us do the third term.*2023

*I was able to go from here to here because this is just the number.*2026

*It is an average, that is a number, it has nothing.*2034

*I’m not adding so this stays the same.*2035

*It is these that change, this is the one that is indexed.*2038

*This I here, I goes from 1 to N.*2041

*This is the number that changes so this just pulls out of the constant.*2045

*The third term, we have the sum of the X² P sub I.*2048

*Once again, I can pull this out, this is just a number.*2063

*This is the average value of X² × the sum of the P sub I.*2066

*Normalization condition, the sum of the probabilities is always equal to 1.*2073

*Therefore, this is just equal to that².*2078

*That is our third term.*2084

*Our σ², our variance is equal to the second moment -2 × the square of the average value + the square of the average value.*2089

*Therefore, our variance is equal to our second moment - the square of our average value.*2107

*Now this is of course, σ² is greater than or equal to 0,*2120

*Which means that the second moment is going to be greater than or equal to the square of the average value.*2125

*You do not particularly have to know this derivation.*2136

*We are just throwing some things out there so that you see them and start to become comfortable with them, familiar with them.*2138

*Let us talk about some continuous distributions.*2148

*I will go back to blue.*2151

*Now, we discuss continuous distributions.*2156

*We said discreet earlier.*2170

*A discreet, we have a number line, this is 0, and number here, as few or as many as you like.*2173

*Only specific values are possible, that is what discreet means.*2185

*Only specific values are possible.*2190

*Now the difference of continuous distribution, not all values on a real number line.*2199

*All values along the real line.*2212

*They all have a shot, they all have a chance.*2220

*That is the only difference.*2227

*Recall from calculus, if you do no recall, this is what you actually learned when doing the integration.*2232

*When we go from discreet to continuous, we go from the summation symbol to the integral symbol.*2254

*That is what the integral symbol is.*2267

*It is a limit of discreet sum.*2270

*That is what we are doing, we are taking the limit.*2272

*We are just taking the integration smaller, smaller.*2274

*We are taking the limit of an actual summation which is a discreet.*2278

*We are adding a finite number of areas.*2281

*They are discreet numbers that we are adding.*2285

*When we passed the limit, we all of a sudden get this new object.*2287

*We get the integral of a function.*2291

*When we go from discreet to continuous, we go from summation to integration.*2293

*Let me see, do I just introduce them or do I want to actually talk about probability density.*2301

*I will talk probability density.*2315

*All of the definitions that we just gave, the average value, the second moment, and variance, notice we gave them in terms of sums.*2318

*Those were for discreet distributions of specific number of numbers.*2325

*10 numbers, 20 numbers, 50 numbers, 1000 numbers.*2330

*If now all the numbers on a real line are possible and infinite in any direction,*2333

*we are going to use the same formulas but now we are going to define them with integrals instead of sums.*2338

*Other than that, they actually stay the same.*2343

*We will get to those in just a moment.*2349

*But before I do, I want to talk about something called probability density*2350

*which is going to play a huge role in quantum mechanics and the wave function.*2354

*As it is going to turn out, we interpret the square of the wave function as a probability density.*2361

*And let us talk about what that means.*2368

*I want to introduce it here.*2373

*Probability density, first and physical analogy.*2376

*You already know what density is.*2387

*I’m going to use the physical analogy of a mass density.*2391

*Now, I'm going to use this equation right here DM = D of X × DX.*2395

*What this says is that a differential linear amount of mass, let us just call it an amount of mass K is equal to the linear mass density.*2408

*In other words, the mass of something per the unit length × the differential length element.*2437

*I will just say × a length, × a differential length.*2449

*Basically, what is happening here is this.*2456

*We are saying gram is equal to g/ cm.*2458

*Let us take cm as our unit of length × some cm.*2466

*A density is just that.*2473

*A density is just the amount of something per unit something else.*2474

*It does not matter what that is.*2480

*In chemistry, we normally do g/ cc or g/ ml.*2482

*We can do g/ L, we could do mcg/ ml.*2488

*It does not matter, it is the amount of something per something else.*2495

*When we multiply that by the another unit of the denominator, we actually get what we want.*2501

*It is just a question of units.*2509

*In this particular case, the grams of something = the grams per length,*2511

*whatever the density happens to be in that particular case × the particular length.*2518

*Let us write this out in words.*2525

*In general, this is very important.*2531

*A quantity of something of a given unit, does not matter what the unit is,*2544

*of a given unit equals the density of that unit per another unit × this other unit.*2549

*And by unit, I'm talking about grams, centimeters, liters, things like that.*2585

*The gram of something is equal to the density in g/ cc × a cc.*2592

*Another example, the number of particles of something, in other words, a quantity of something of a given unit.*2601

*In this case, the number of particles is equal to a number of particles per in square × that particular unit in square.*2609

*You know this already, this is just units of cancellation.*2623

*2 mi/ hr × 3 hr is 6 mi.*2626

*That is all that is going on here.*2631

*In this case, mi/ hr you might call it a length density per time.*2633

*That is all we are doing. That is all that is happening here.*2639

*Let us use this analogy to talk about something called a probability density.*2644

*The probability that some particle is between X and X + δ X is in some interval.*2653

*Δ X is going to equal the probability as a function of X × the differential X element.*2665

*This is the same exact thing that we did before.*2678

*The analogy is DM = D of X × DX.*2680

*Except now, we are talking about the probability density.*2686

*The probability at this particular point in space.*2690

*The probability is now a function of X at this point in space.*2695

*This point along the real number line, the probability of me finding something is this.*2700

*There is some probability per some length element.*2704

*This is a probability density, that is what is going on here.*2709

*The probability that some particle is between A and B, I integrate this thing.*2718

*In other words, if I were to integrate this, I would end up with my total mass.*2740

*It is just the integral of the density.*2744

*I’m just adding up all of the mass elements.*2749

*Here, if I want the probability in some small length, it is this thing.*2753

*It is the probability density × the particular length element.*2760

*This is what is important right here.*2764

*This is the probability density.*2766

*If I multiply by some length element, this whole thing is actually the probability.*2770

*The whole thing is probability, this particular thing is probability density.*2780

*If I want the total mass, I just integrate it.*2786

*If I want the total probability that something is between this point and this point, I just integrate this thing.*2789

*In other words, I add up all of the probabilities.*2795

*Let us go ahead and do our integral definitions.*2799

*Again, do not worry if this does not really make sense, if you cannot wrap your mind around it.*2802

*We will be discussing it more and more especially when we talk about quantum mechanics and the wave function.*2805

*But I do want to introduce it to you.*2809

*Let me actually stay with red.*2812

*The normalization condition we said that the sum of the probabilities has to equal 0.*2818

*Will now that we are talking about continuous distributions, the sum becomes an integral.*2830

*The integral for - infinity to infinity of this PX DX which is the probability has to = 1.*2836

*Profoundly important equation.*2845

*This is the normalization condition.*2847

*The integral over the particular region in space that we happen to be dealing with,*2850

*it is the most general form, from -infinity to infinity of the probability density × a length element = 1.*2853

*This is the probability.*2861

*Let us give our definitions.*2865

*Let me write that again.*2872

*Normalization condition, we have the integral from -infinity to infinity of the probability is equal to 0.*2874

*This is just the continuous version of the sum of the probabilities = 1 not 0.*2896

*The average value of a continuous distribution = the integral from -infinity to infinity of X ×*2902

*the sum we have X × the probability of X.*2914

*Here the probability of X is this, the second moment = the integral from -infinity to infinity of X² × the probability which is PX DX.*2917

*We have a final one which is the σ² of X which we call the second central moment or the variance.*2938

*It is the integral from - infinity to infinity of X sub I.*2947

*That is fine, I will just do this X – I cannot do X sub I here, this is continuous distribution.*2959

*× PX DX.*2974

*The PX of X DX is the probability.*2982

*P of X alone is the probability density.*2993

*Let us do a final example.*3014

*Let us go back to blue here.*3017

*If a particle is constrained to move in 1 dimension only, the interval from 0 to A.*3020

*0 to A in a real line, here is 0 and here is A.*3031

*It is constrained here, back and forth.*3037

*It turns out that the probability that the particle will be found between X and DX is given by this function right here.*3040

*We will see this again, do not worry about that.*3049

*Later, when we talk about this thing called a particle in a box, we will see this again.*3051

*That turns out to be very basic and elementary.*3055

*The probability that the particle will be found between X and DX is given by this right here.*3062

*Notice PX, that is the probability density.*3070

*This whole thing is the probability.*3073

*This is the probability density and it is going to be equal to 1, 2, 3, and so on.*3075

*We want you to show that PX is actually normalized.*3079

*We want to calculate the average position of the particle along the interval.*3084

*They want us to show that PX is actually normalized.*3096

*The normalization condition is from -infinity to infinity of this PX DX = 1.*3100

*We need to show the PX is normalized.*3118

*We need to show the following.*3123

*We need to demonstrate that this integral, in this particular case are space that we are dealing with.*3125

*It is a -infinity to infinity, it is just to 0 to A of this function 2/ A × the sin² of N π X/ A DX.*3132

*We need to actually show that it is equal to 1.*3149

*Let us go ahead and do this integral and see if it is actually equal to 1.*3153

*This integral, it is going to equal, I’m going to pull this 2/ A out.*3159

*0 to A of sin² × N π X/ A DX.*3169

*I can use math software or I can go ahead and look this up in a table of integrals.*3183

*I get the following.*3188

*I get 2/ A × this integral is X/ 2 - the sin of 2 N π X/ A ÷ 4 N π/ A.*3189

*We are taking this from 0 to A.*3211

*When I do this, it is going to be 2/ A.*3216

*I will put in A, put 0 in here, I'm going to end up getting A/ 2 -0.*3222

*When I put A in here and here, I get A/ 0 -, now I put 0 in here.*3235

*0 -0, so I end up with 2/ A × A/ 2 which = 1.*3240

*Yes, this is already normalized.*3253

*Calculate the average position of the particle along the interval.*3258

*Let us just use our definition of average.*3264

*Nice and straightforward.*3267

*Part B, the average position of a particle or the average of whatever is going to be equal to 0 to A of X × the P of X DX.*3271

*It is going to equal the integral from 0 to A of X × 2/ A × sin² of N π X/ A DX.*3289

*Again, I can use math software or I can look this up in a table of integrals.*3308

*I get the following 2/ A ×, this is going to be X² / 4 - X × the sin of 2 N π X/ A / 4 N π/ A -,*3313

*this is going to be the cos of 2 N π X/ A ÷ 8 × N² π²/ A².*3334

*I'm going to do this from 0 to A.*3352

*When I put this in, I get the following.*3355

*I get A²/ 4 -0 -1/ 8 N² π²/ A² -0 -0 -1/8 N² π² / A².*3359

*- this, - and - that becomes +, these cancels.*3390

*And I'm left with 2/ A × A²/ 4 which is equal to A/ 2.*3396

*And the average position.*3411

*It says is the following.*3412

*This says that the particles spends half of its time to the left of A/ 2 and half of its time, the other half of its time to the right.*3415

*The average position between 0 and A is A/ 2.*3453

*That is what this is saying.*3460

*On average, some× it is going to be here, sometime just going to be there.*3462

*Overall, the average is going to be right down the middle because it is going to spend*3467

*an equal amount of time to the left and to the right.*3470

*From your perspective, to the right.*3474

*It says that the particle spent half of its time to the left of A/ 2 and half of its time to the right of A/ 2.*3480

*This average is 2A/ 2.*3488

*This confirmed this averages to A/ 2.*3492

*We will definitely see this again when we talk about the particle in a box.*3499

*The things that I would like you to actually take away from this lesson.*3505

*The important things to remember.*3509

*The P of X is a probability density.*3526

*P of X DX is the probability of finding something between the X and DX.*3540

*Of course, the normalization condition -infinity to infinity of this probability = 1,*3569

*which is completely analogous to this one for the discrete probability.*3582

*The sum of the probabilities = 1, this is the continuous version of it.*3588

*Thank you so much for joining us here at www.educator.com.*3594

*We will see you next time, bye.*3597

1 answer

Last reply by: Professor Hovasapian

Mon Jun 27, 2016 7:55 PM

Post by Joseph Cress on June 16, 2016

Can I ask a specific question within the subject matter being taught,but the question is from a specific text question.