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Lecture Comments (2)

1 answer

Last reply by: Professor Hovasapian
Mon Jun 27, 2016 7:55 PM

Post by Joseph Cress on June 16 at 07:38:48 AM

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

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 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

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 41003

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 density2350

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