For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

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### Density & Cumulative Distribution Functions

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

- Intro
- Density Functions
- Cumulative Distribution Functions
- Properties of the CDF (Density & Cumulative Distribution Functions)
- Example I: Density & Cumulative Distribution Functions, Part A
- Example I: Density & Cumulative Distribution Functions, Part B
- Example II: Density & Cumulative Distribution Functions, Part A
- Example II: Density & Cumulative Distribution Functions, Part B
- Example III: Density & Cumulative Distribution Functions, Part A
- Example III: Density & Cumulative Distribution Functions, Part B
- Example IV: Density & Cumulative Distribution Functions
- Example V: Density & Cumulative Distribution Functions, Part A
- Example V: Density & Cumulative Distribution Functions, Part B

- Intro 0:00
- Density Functions 0:43
- Density Functions
- Density Function to Calculate Probabilities
- Cumulative Distribution Functions 4:28
- Cumulative Distribution Functions
- Using F to Calculate Probabilities
- Properties of the CDF (Density & Cumulative Distribution Functions) 7:27
- F(-∞) = 0
- F(∞) = 1
- F is Increasing
- F'(y) = f(y)
- Example I: Density & Cumulative Distribution Functions, Part A 9:43
- Example I: Density & Cumulative Distribution Functions, Part B 14:16
- Example II: Density & Cumulative Distribution Functions, Part A 21:41
- Example II: Density & Cumulative Distribution Functions, Part B 26:16
- Example III: Density & Cumulative Distribution Functions, Part A 32:17
- Example III: Density & Cumulative Distribution Functions, Part B 37:08
- Example IV: Density & Cumulative Distribution Functions 43:34
- Example V: Density & Cumulative Distribution Functions, Part A 51:53
- Example V: Density & Cumulative Distribution Functions, Part B 54:19

### Introduction to Probability Online Course

### Transcription: Density & Cumulative Distribution Functions

*Hello, and welcome to the probability lectures here on www.educator.com.*0000

*We are starting a chapter now on continuous distributions.*0004

*My name is Will Murray and I’m your guide today.*0009

*First section here is on density and cumulative distribution functions.*0012

*Anytime you have a continuous probability distribution, you will have a density function and a cumulative distribution function.*0017

*I’m going to explain what those are and make sure that you do not get those 2 mixed up.*0026

*We use F for both of them, that is always a little confusing but I will try to highlight the differences there,*0032

*and then we will do some example problems to practice them.*0039

*Let us jump in with density functions.*0043

*Y is a continuous random variable.*0046

*What that means is it contained values over a whole continuum of possibilities.*0049

*Instead of having discrete probability were Y would be a whole number or a certain number of possible values,*0056

*the values now for Y can be a whole range of things.*0066

*Things like the normal distribution would be a typical example of a continuous random variable.*0070

*The values of Y can be anything.*0077

*It can be any real number at this point.*0081

*It has a density function that, we are going to use f for density functions.*0085

*Be very careful here to distinguish between the density function and what we are going to learn next,*0089

*which is the cumulative distribution function.*0095

*We will use f for the density function and we will use F for the cumulative distribution function.*0097

*If you have sloppy handwriting, now is the time to be very careful to be clear about the difference between*0106

*f of Y which is the density function, and F of Y which is the cumulative distribution function.*0113

*We will learn about that in the next slide.*0121

*Be very careful about the difference between those two.*0123

*The density function, the properties it has to satisfy, it is always positive.*0127

*Essentially that means you cannot have negative probabilities, the probability is always positive.*0132

*The total amount of area under this graph from -infinity to infinity is always exactly equal to 1,*0140

*that is because it is a probability function.*0151

*The total probability of something happening has to be equal to 1.*0155

*Something has to happen and it happens with probability 1.*0159

*The way you use the density function to calculate probabilities is, you always talk in terms of ranges.*0163

*You never talk about the probability of Y being equal to a specific value.*0170

*Back when we are talking about discreet distributions, like the Poisson distribution,*0176

*the binomial distribution, geometric distribution, we would say what is the probability that Y is equal to 3?*0180

*What is the probability that there will be exactly 3 forest fires next year in California?*0187

*What is the probability that the coin will come up heads up 3 ×?*0191

*We would ask what is the probability that Y is equal to a particular value?*0196

*Now, we ask about ranges.*0201

*We will never ask the probability that Y is equal to 3, that would just be 0 because we have so many possible values.*0203

*Instead, we will ask about what is the probability that Y is between one number and another?*0212

*For example, we will find a probability that Y is between A and B.*0220

*The way you find that probability is, you calculate the area under the density function.*0225

*In order to calculate that area, what you do is you take an integral.*0232

*The probability that Y is between constant values A and B is the integral of the density function F of Y DY from Y = a to Y = b.*0240

*That is how you calculate probabilities from now on.*0255

*Remember, that is the density function, that is not to be confused with*0258

*the cumulative distribution function which is the next thing we are going to learn about.*0262

*The cumulative distribution function that is closely related to the density function but it is not the same thing.*0270

*The cumulative distribution function we use F of Y.*0277

*Remember, the density function was f.*0281

*The cumulative distribution function is the probability that Y is less than or equal to a particular cutoff value of Y.*0283

*Let me show you how you find that.*0295

*If this is the density function that I'm graphing right now, this is f of Y.*0297

*You have a particular cutoff value of Y and you want to find the probability of being less than that value.*0303

*The way you do it is you calculate the area to the left of that cutoff.*0310

*You can do that as an integral.*0317

*We already used y as the cutoff, I cannot use my Y as my variable of integration.*0319

*I cannot use of Y, I’m using T here, F of T DT.*0325

*By the way, that is a very common mistake that I see students make in doing probability problems is,*0330

*they will mix up their variables.*0336

*They will have a Y in here and then they will also have that Y over there in the limit.*0338

*That is very bad practice, you can get yourself in lots of problems that way.*0342

*Do not do that, use T when you are calculating the cumulative distribution function.*0346

*Use T as your variable of integration, and then use Y as your limit.*0352

*We can also use F, the cumulative distribution function to calculate probabilities.*0359

*The probability that you are within one range between A and B.*0365

*Let me graphically illustrate this.*0373

*The probability that you are between A and B.*0378

*That is the area in between A and B.*0386

*One way of calculating that is to calculate all the area less than B, and then to subtract off all the area that is less than A.*0389

*To subtract off all this area less than A.*0402

*What you are left with is exactly the area that you want, which is the area between A and B.*0406

*The way you calculate the area that is less than B is to use the cumulative distribution function F of B.*0414

*The area less than A is F of A.*0421

*Once you worked out what F is, you do not have to do any more integration.*0424

*You just plug in your limits B and A.*0429

*Essentially, this is the fundamental theorem of calculus coming through in a probability setting.*0432

*It means once you have done this integral, you just plug in the 2 limits F of B and F of A.*0437

*We are going to study some properties of the cumulative distribution function,*0446

*that is what CDF stand for cumulative distribution function.*0451

*What today's functions look like, remember that if the density function often looks like something like this.*0455

*This is the density function, f of Y.*0465

*The cumulative distribution function represents the area underneath or to the left of any given cutoff.*0468

*The cumulative distribution function therefore, as you start in the left hand side of the universe, at –infinity,*0478

*there is no area to the left.*0488

*It always has to start at 0, let me draw that in black here.*0490

*It always has to start at 0, that is why F of -infinity is always 0.*0495

*What I’m doing here is I’m going to graph F of Y.*0504

*As you start to increase Y here, you get more and more area until you get to the right hand edge of the universe at infinity.*0511

*You got all the area under the density function.*0523

*We said that total area is 1.*0526

*It always has to increase and it always has to end up at 1.*0529

*The cumulative distribution function always has the same general shape.*0539

*It always starts at 0, at –infinity, and it always increases and it finishes up at 1, at infinity.*0544

*That is why we have these properties F of infinity is 1, F is always increasing.*0556

*Its derivative, this is the fundamental theorem of calculus, get you back to the density function which is f of Y.*0562

*We will be using this property in particular, as we solve some problems.*0571

*Let us jump in and solve some problems with density functions and cumulative distribution functions.*0579

*The first one we are given that Y has density function f of Y, be careful, that is the distinction between f and F.*0589

*C × Y, when Y is between 0 and 2.*0598

*C × 4 – Y, when Y is between 2 and 4, and 0 elsewhere.*0602

*I will draw a little graph of that.*0607

*We do not know what C is.*0611

*In fact, the first part of the problem here is to figure out what C should be.*0612

*When Y goes from 0 to 2, we do know that it is something increasing.*0619

*It is linear and it is increasing.*0623

*4 - Y it will be decreasing again, as we go down to 4.*0626

*It is something like that but we want to find the exact value of C.*0630

*What we are going to use for that is, we are going to use the property of density functions which is that the integral of F of Y DY,*0635

*the total area always has to be equal to 1.*0645

*That was the first property of density function, I guess it was the second property of density functions.*0650

*It always has to be equal 1.*0655

*That represents the fact that this does represent probability,*0657

*the total probability of something happening always has to be 1.*0661

*Let us find that integral based on the information we are given.*0666

*I'm only going really look at the range between 0 and 4.*0670

*The integral from 0 to 4 of f of Y DY, I’m solving part A here.*0676

*It is supposed to be equal to 1.*0683

*Whatever that turns out to be, I'm going to set it equal to 1.*0686

*We have got 2 different parts of our density function.*0691

*I’m going to split up into integral from 0 to 2.*0695

*We also have an integral from 2 to 4.*0700

*Each one of those is multiplied by C, I will go ahead and factor the C out, put a C out here.*0704

*Then, I will have Y DY here on this range and 4 - Y on this range.*0712

*I'm just going to do the calculus there.*0720

*This is C × Y²/2 evaluated from 0 to 2 + 4 – Y, that is 4Y - Y²/2, evaluate that from 2 to 4,*0722

*that is supposed to be equal to 1.*0740

*This is all C, always multiply by C.*0742

*Y²/2 evaluated at 2 is 4/2 is just 2 evaluated at 0, nothing happens.*0746

*If I plug in 4 into Y for the second part, I will get 16 – 4² that is 8.*0755

*Plug in 2, I get -8, - and - is +, 2²/2 is + 4/2.*0763

*That is still supposed to equal 1.*0776

*I get C × 16 - 8 – 8, those cancel each other out.*0778

*C × 4 = 1.*0783

*C = ¼.*0786

*Let us see, that tells me the answer to part A there.*0794

*The key part there is that the total integral was supposed to come out to be 1, that is the property of a density function.*0799

*The way I worked that out was, I set the total of the integral is equal to 1.*0808

*I factored the C out of everything because that was a common factor on both parts there.*0812

*Then I worked out this integral on their respective ranges.*0823

*I had to split it up into 2 parts because the function was sort of defined piece wise.*0826

*Two different definitions on 2 different ranges.*0832

*We worked out the integral I got 2 × 4 is still equal to 1, it has to be equal to ¼.*0836

*I got my part C, I have found my value of C for part A.*0842

*Now, I have to find the cumulative distribution function of Y and that is going to be a little more work.*0848

*Let me go ahead and do that on the next slide.*0853

*With density function again, and the cumulative distribution function of Y, remember that F of Y,*0860

*the way we figure that out is, it is the integral from -infinity to Y of F of T DT.*0869

*That was our original definition of F, you can go back and you can find that.*0880

*Now, in this case, there is nothing going on between -infinity and 0.*0885

*I’m just going to start this integral at 0, the integral from 0 to Y.*0894

*I have 2 different ranges of Y here.*0900

*I’m going to split up 2 cases here.*0903

*The first case I’m going to do is Y is between 0 and 2.*0905

*What do I get there?*0913

*Then, on the previous slide, we figured out that C is ¼.*0914

*That is by the work we did on the previous slide.*0923

*If you do not remember that, just go back and check it, you will see where we got the C is ¼.*0925

*This is ¼, replacing my Y with T, I did the same mistake that I said that probability students often make, T DT.*0929

*This is T²/2T²/2 × 4 is 2²/8.*0942

*From 0 to Y which is just Y²/8, that is what my range of Y between 0 and 2.*0949

*For the range between 2 and 4, it is more complicated and this is very easy for students to mix up.*0961

*I hope you will follow me carefully here.*0967

*F of Y, it is still the integral from 0 to Y of F of T DT.*0970

*We have to split up into 2 parts because there is 2 different parts of this range.*0977

*It is the integral from 0 to 2 of ¼ of T DT + the integral from 2 to Y of the different definition which is C, C is × 4 – T DT.*0981

*I ‘m reading that off from this part of the definition here.*1005

*It is a little more complicated but if we be careful with that, we can keep it straight.*1009

*The integral of 1/4 of T DT, I can factor out a ¼ out of everything there.*1013

*I think that will make my life a little simpler here.*1021

*I factor the 1/4 out everything.*1023

*The integral of T DT is T²/2, we will be dividing that from 0 to 2 +, now that 1/4 is gone, it is 4T - T²/2,*1026

*evaluate that from T = 2 to T = Y.*1040

*This is 1/4 × 2²/2 is just 2, + I will plug in Y for T .*1048

*4Y - Y²/2, I will in 2 for T so -8.*1057

*And - is +, + 2²/2 is just 2, this is ¼.*1064

*Now what do I have here, I have 4Y - Y²/2 + 2 + 2 - 8 that is -4.*1074

*If I distribute that 4, I get Y - Y²/8 -1.*1087

*It is kind of a messy formula there but that is what we are stuck with.*1098

*Now, I found two different function for F of Y, depending on the different ranges we are in.*1101

*I’m going to summarize that.*1108

*My F of Y is equal to Y²/8 for 0 is less than Y, less than or equal to 2.*1110

*It is equal to Y - Y²/8 – 1 for 2 less than Y less than or equal to 4, that is my F of Y.*1125

*I should also mention what it does on the outsides of those ranges.*1143

*It will be 0/Y is less than 0 because, remember, the cumulative distribution function all starts out a 0*1146

*and it always goes up to 1 for bigger values of Y.*1154

*For Y greater than 4, it will be 1.*1160

*It is worthwhile somtimes to check that the values match on the endpoints.*1165

*If you plug in 0 to Y²/8, you do indeed get 0.*1170

*If you plug in 4 to Y = 4, we will get to the second part of the function 4 - 4²/8 -1 will give us 4 -16/8 is 2 -1, gives us 1.*1175

*That matches up of what we said Y should be, when we get bigger than 4.*1195

*That checks my work there.*1201

*Let me box this whole solution here because this is all part of our solution.*1203

*Finally, we have found a cumulative distribution function Y there.*1214

*Let me remind you of the steps there.*1218

*We used the definition F of Y is the integral from -infinity to Y of F of T DT.*1220

*Now, that is pretty simple when Y is between 0 and 2.*1229

*You just take the first part of the definition and you work out the integral, and you get Y²/8.*1235

*That is where we got this part of the answer.*1241

*But when Y is between 2 and 4, it is much more complicated because you have*1243

*to take into account both parts of this definition.*1248

*You have to use both parts of this definition and split up the integral into two parts,*1252

*and evaluate both of those using T as your variable.*1258

*You do one from 0 to 2, one from 2 to Y, and then simplify that down into a much more complicated function.*1263

*That is how we got this more complicated function on the range between 2 and 4.*1271

*I want you to hang onto the answers that we got here for example 1,*1277

*because we are going to use the same density function and therefore, the same cumulative distribution function for example 2.*1281

*I want to make sure that you understand these answers for example 1.*1293

*Make sure you understand this example very well, before you move on to example 2.*1296

*In example 2 here, we are taking the same density function from example 1.*1304

*Remember, we figured out that the constant had to be 1/4 there.*1309

*I went ahead and write that in on example 2.*1312

*It is ¼ Y for the range between 0 and 2, and 1/4 of 4 – Y from the range between 2 and 4.*1315

*You got a density function that looks kind of like this.*1325

*What we want to find here is the probability that Y is between 1 and 3,*1331

*and the probability that Y is less than or equal to 2, given that Y is greater than or equal to 1.*1335

*Those are some conditional probability involved in there.*1342

*The useful thing to use at this point is not the density function that is given,*1345

*but the cumulative distribution function that we worked out in example 1.*1351

*If you have not just watched the video for example 1, you should go back and work out example 1*1356

*because we are going to use that answer from example 1, to calculate the answers for example 2.2*1361

*Let me remind you right now what the answer was from example 1.*1368

*F of Y, the cumulative distribution function was, I broke it down into two important parts there.*1372

*It was Y²/8, when Y is between 0 and 2.*1380

*It was more complicated Y - Y²/8 -1, when 2 is less than Y less than 4.*1387

*That was the cumulative distribution function, we did figure that out in example 1.*1400

* Quite a lot of integration we went into that, we are not going to redo it.*1404

*If you do not know where that is coming from, it is worth going back and working through example 1 because it will make sense.*1408

*For part A here, to find the probability that Y is between 1 and 3.*1416

*What we can do know is, we can use the cumulative distribution function F of 3 – F of 1.*1421

*We can also use an integral of the density function but then, we just end up redoing the work we did from example 1.*1430

*It is much easier to use F, if you already down that work.*1437

*F of 3, now, 3 is between 2 and 4.*1443

*Let me use this second version of the formula.*1447

*It is 3 -, 3²/8 is 9/8 – 1.*1453

*-F of 1, I have to use the first part of the formula because 1 is in the range between 0 and 2.*1460

*Y square/8 is 1/8 and I will just simplify those fractions.*1468

*3 -1 is 2 – 9/8 - 1/8 is 10/8.*1474

*10/8 is 2 - 5/4 and 2 is 8/4, that is just ¾.*1480

*That is my probability that Y is between 1 and 3.*1487

*3/4 probability that Y is between 1 and 3.*1492

*Let me show you that on the graph because I think it will makes sense there.*1495

*There is 1, 2, 3, 4, being between 1 and 3.*1499

*If you figure out how much of that area is between 1 and 3, if you do a little triangle geometry there,*1505

*you will figure out that, that ¾ of the total area is between 1 and 3.*1512

*We also checked that using our arithmetic here, using our integration.*1520

*I’m going to jump over onto the next line to do part B.*1526

*It will be a little more complicated, I need more space.*1529

*Just remind you how we did part A here.*1532

*I have recalled from example 1, the cumulative distribution function, the F of Y.*1535

*That is what we worked out in example 1.*1541

*And then, I just had to plug in F of 3 - F of 1.*1543

*The wrinkle in that was those are two different ranges so I have to use two different formulas.*1549

*One for F of 1, there is the F of 1 using that formula right there.*1553

*There is F of 3, using that formula right there.*1559

*Once I take those two values and plugged them in, I got some easy fractions to simplify.*1567

*It ended up with ¾ there.*1573

*We are still working on example 2.*1578

*We still have to do the second part of the problem here.*1579

*It is going to be really helpful to use the cumulative distribution function that we figured out in part 1.*1583

*Let me remind you what our cumulative distribution function was.*1592

*We figure this out in example 1.*1597

*There was two parts to this function, Y²/8 for Y between 0 and 2.*1599

*Y -1²/8 -1 for Y between 2 and 4.*1611

*We got to find conditional probability, it has been a long time since we did condition probability.*1621

*If you look back into some of the early videos in the series, you will find one that covers condition probability.*1626

*I will remind you of the formula that we have for condition probability.*1633

*The probability of A given B is the probability of A intersect B divided by the probability of B.*1636

*Remember, intersection is like saying N, you want both of those things to be true.*1651

*That formula is way back in the early videos for this course.*1657

*You can find it, just scroll back to the videos here on www.educator.com.*1660

*Our probability of Y being less than 2 given that it is greater than 1 is the probability of Y*1666

*less than or equal to 2 and is greater than or equal to 1 divided by the probability that Y is greater than or equal to 1.*1675

*I’m just filling in my formula for conditional probability.*1685

*Now, another way of saying that it is less than 2 and greater than 1 is to say 1 is less than or equal Y less than or equal to 2 divided by,*1688

*Now, the probability that Y is greater than or equal to 1, that is hard to compute directly but it is easy to compute*1700

*if write is a 1- the probability that Y is less than or equal to 1.*1707

*That is the easy way to calculate it because that sets it up to be something that we can answer*1712

*using the cumulative distribution function.*1721

*Let me write that on a new line here.*1724

*The probability that Y is less than or equal to 1 is just F of 1.*1727

*The probability that Y is between 1 and 2 is F of 2 – F of 1.*1734

*Now, I can just input all these values into my cumulative distribution function, F.*1742

*F of 2, it looks like everything here is on the first range, the Y²/8 which is nice, because that is the easier formula.*1749

*F of 2 is 2²/8 that is 4/8.*1757

*F of 1 is 1²/8, 1/8, 1 – F of 1 is 1 – 1/8.*1763

*This is 3/8 /7/8, and if you do the flip on the fractions, the 8’s will cancel.*1770

*I will just multiply top and bottom by 8, I will get 3/7.*1780

*That is my probability that Y is less than or equal to 2 given that Y is greater than or equal to 1.*1784

*You can also check that geometrically, if you like.*1793

*The graph we have on f of Y looked like an elongated triangle, f of Y.*1797

*There is 2, there is 1, and there is 3.*1805

*The probability that Y is greater than or equal to 1 is all that range there.*1811

*The probability that Y is less than or equal and 2, let me draw that in another color.*1820

*Less than or equal to 2 is that range right there.*1827

*If you break this up into little triangles, you can see that there is 1, 2, 3, 4, 5, 6, 7 triangles total.*1833

*3 of these 7 little triangles are in that region.*1844

*It does check graphically that we get this 3/7 answer.*1853

*But I probably, do not want to rely on that, I do like using the formulas.*1858

*Just to remind you how we did use the formulas there.*1862

*This formula for the cumulative distribution function, this F came from example 1, we work that out in example 1.*1865

*You can go back and check it, rewatch the video from example 1.*1871

*This break down here, we are using conditional probability.*1878

*This came from a very old video but it is on the series for conditional probability.*1882

*I'm using that formula for conditional probability here.*1888

*The probability that Y is less than or equal to 2 and greater than or equal 1, that just means it is between 1 and 2.*1893

*The probability that Y is greater than or equal 1 is hard to evaluate directly.*1899

*I flipped it around and that is because the probability that Y is less than or equal to 1*1904

*is something we can answer easily using our cumulative distribution function, our F.*1910

*This is F of 2 - F of 1, 1 – F of 1.*1918

*I just dropped those values of Y into this F formula, since they are all in the first range there,*1923

*and simplify the fractions down to 3/7.*1930

*In examples 3, we are given a new density function for Y, F of Y, f of Y, f is the density function, is some C.*1939

*It does not tell us what C is, I guess we have to figure that out, on the range between 0 and 1.*1951

*2C is on the range between 1 and 2, and everywhere else it is going to be 0.*1956

*To the first task here is to find what c should be.*1961

*The second task is to find a cumulative distribution function of Y.*1966

*Let me go ahead and graph this.*1971

*We got between 0 and 1, the value is c.*1975

*Between 1 and 2, it jumps up to 2c.*1984

*That is our density function right there, it is 0 everywhere else.*1994

*We have a density function that looks like that, it is a step function there.*2000

*You can answer this question pretty easily graphically, if you know what you are doing.*2005

*Let me go ahead and show you the arithmetic here.*2010

*Just to make sure that it makes sense using either method.*2013

*The key point here is that the total area under the density function is always equal to 1.*2018

*In this case, the area is the integral from, in this case 0 to 2 of F of Y DY, should be equal to 1.*2024

*I want to figure out what the value of c should be, to make that equal to 1.*2039

*Let us evaluate that integral, that is the integral from, since we got the function defined differently*2044

*on the 2 different ranges, I will write it as the interval from 0 to 1.*2050

*We also have an integral from 1 to 2.*2056

*It looks like they are both going to be multiplied by c.*2061

*I will go ahead and factor that c out.*2063

*That will leave me with 1 on the first range, 1 DY.*2067

*2 on the second arrange, 2 DY there.*2071

*That is supposed to come out to be equal to 1.*2079

*Now, that is c × the integral of I DY is just Y from 0 to 1 + 2Y from 1 to 2 is equal to 1.*2084

*That is C × evaluation of Y from 0 to 1 is just 1.*2101

*2Y is just 2 × 2 -2 × 1, that is + 2 is equal to 1.*2108

*It looks like c is going to be 1/3, in order to make this be a valid density functions.*2116

*That tells us the answer to part A.*2125

*That is really not surprising if you go back and look at your graph here.*2127

*If you go back here, the area of that first block is definitely c, because height × width is I 2C.*2133

*A of that second block is 2C, the total area is 3C which should be 1.*2141

*Definitely, I want my C to be 1/3, in order for that to be a valid density function.*2149

*I got my C to be 1/3, and I still need to find F of Y.*2158

*That will be a little more work but let me recap the work here because that throws away the slide.*2164

*I use the fact that the total area under the density function is always equal to 1.*2172

*That is true for any density function, that is a requirement.*2177

*It is essentially saying that the total probability in any experiment has to be 1.*2180

*Because it was defined on two different ranges here, I just split up that integral into two parts.*2187

*It is 0 to 1, and 1 to 2.*2192

*Those were both very easy integrals, I factored out the C already.*2195

*That gave me 3C was equal to 1, I figure out that C is equal to 1/3.*2205

*I could have figured that out from the graph because I know that the total area should be 1.*2210

*If I just look at the blocks there, I get 3C is equal to 1, C is equal to 1/3.*2217

*We are going to hang onto this and make the jump to the next slide, where we will figure out part B here.*2223

*For part B of examples 3, we figured out in part A that C was equal to 1/3.*2232

*That is already done on the previous slide.*2238

*But we have to now find F of Y, we want to find the cumulative distribution function.*2242

*Let us remember the definition of F of Y.*2252

*It is the integral from -infinity to Y of f of T DT.*2255

*In this case, there is no density below 0.*2264

*I can just cut this off at 0, this is the integral from 0 to Y of F of T DT.*2269

*I need to separate two ranges here.*2280

*I have a range from Y going from 0 to 1, and then I will have another range from Y going from 1 to 2.*2283

*I need to separate this problem into two parts.*2292

*The second part will be a little more difficult.*2295

*The first part is pretty easy because then, I'm just looking at the definition of Y between 0 and 1.*2297

*This is we get that from, 0 to Y is just 1/3, that was my value of C DT.*2305

*That is 1/3 T evaluated from 0 to Y which is just 1/3Y, that part is fairly easy.*2316

*The second part is more complicated, you want to be careful about that.*2328

*On the second range, Y goes from, I said 0 to 2, and I should have said 1 to 2.*2333

*You do not just want to use this formula from 1 to 2 because we really need to find the integral from 0 to Y of F of T DT.*2340

*That breaks that up into two parts, the integral from 0 to 1 of F of T DT and the integral from 1 to Y of F of T DT.*2350

*We need to do separate integrals for each of those.*2364

*I have to do that because I know that Y is bigger than 1.*2367

*Y is somewhere in this range between 1 and 2.*2370

*The first part is not too bad, 0 to 1 of 1/3 DT +, 1 to Y between 1 and 2, the density function is 2 × C, that is 2/3 DT.*2374

*I get 1/3 T from the 0 to 1 + 2/3 T from 1 to Y, from T = 1 to T = Y.*2395

*I get 1/3 × one – 0 so 1/3 × 1 + 2/3 Y - 2/3 × 1.*2410

*I guess that simplifies a bits into 2/3 Y + 1/3 - 2/3 is -1/3.*2421

*I got two different ranges in two different functions.*2431

*I will put those together to give my answer, my F of Y is equal to 1/3Y, when 0 is less than Y less than 1.*2436

*It is equal to 2/3 Y - 1/3, when one is less than or equal to Y less than or equal to 2.*2452

*Those are the important parts of my cumulative distribution function.*2467

*We just tacked on the parts of the N.*2471

*If Y is less than 0, it will just be 0 because the left hand side is always 0, the right hand side is always 1.*2473

*If Y is greater than 2 then we will be looking at a function of 1.*2484

*You can always plug that in and check, I will plug in Y = 2 and I get 2/3 × 2.*2491

*4/3 - 1/3, I’m plugging in Y = 2 in here and it does come out to be 1.*2498

*That reassures me that I have probably been doing my arithmetic correctly.*2503

*Let me box this up.*2511

*I got my cumulative distribution function.*2518

*We will be using this again in the next examples.*2521

*I want to make sure that you understand this very well, before you move on to example 4.*2524

*Let me remind you of how we got that.*2530

*Our cumulative distribution function, the F, we always get that by integrating the f, the density function.*2532

*You integrate it, you change the variable from Y to T, and then you integrate from 0 to Y.*2539

*That is sometimes a bit subtle, especially, when you have these functions that are defined piece wise.*2545

*Because the integral from 0 to Y, Y is between 0 and 1, no problem, you just use the definition of the f of Y between 0 and 1.*2550

*You integrate that and you get 1/3Y.*2561

*If Y is between 1 and 2, then you get a more complicated integral because*2564

*you have to break it up between 0 to 1, and 1 to Y.*2570

*And then, you have to use the different definitions, the 1/3 the 1C, the 2/3 the 2C, on the different ranges.*2574

*You plug in those two different definitions, the C and the 2 C, and then you do the integral and it simplifies fairly nicely.*2583

*You want to assemble these two answers into a definition of F of Y defined on the different ranges.*2593

*You always have the left hand then being 0 and the right hand then being 1.*2600

*That is our cumulative distribution function.*2606

*You do want to remember this because we will be using it again for example 4.*2609

*Let us go ahead and take a look at that.*2613

*In example 4, we are using the same density function from example 3.*2616

*F of Y is 1/3 on the range from 0 to 1, 2/3 on the range from 1 to 2.*2621

*Here, we want to find a condition probability, the probability that Y is less than 3/2 given that it is bigger than ½.*2630

*I did do a lot of work on this density function back in examples 3.*2640

*If you did not just watch the video on example 3, you should go back and work through example 3,*2645

*in order to understand example 4.*2652

*I’m going to use the results right now.*2654

*Let me remind you what our answer was from example 3.*2657

*Let us see, where do I have space for that, I will put that over here.*2666

*In example 3, this was our answer from examples 3 and it did take us a fair amount of work to get to this point.*2668

*We found the cumulative distribution function F of Y not f, you got to be very careful to keep those clear.*2676

*I will just put the two important ranges here.*2684

*The range from 0 to 1, and then we had a different answer when Y was between 1 and 2.*2688

*Our two answers there, where Y was 3, where do I have that in my notes there.*2696

*That was 1/3 Y on the first range, and then on the second range it was 2/3 Y -1/3.*2703

*We work those out in example 3, I’m not the showing you the work right now to do those.*2713

*You have to go back and watch example 3.*2717

*For the probability of what we are asking for here, this is the conditional probability,*2720

*let me remind you of the formula for conditional probability.*2726

*That was done in one of our earlier probability videos here on www.educator.com.*2729

*Let me remind you that the probability of A given B, the conditional probability is*2735

*the probability of A intersect B divided by the probability of B.*2741

*Remember, A intersect B is the same as A and B, mathematical, when events are happening at the same time,*2750

*divided by the probability of B.*2760

*That is what we are going to apply to this conditional probability that we are asked to calculate here.*2762

*This is the probability that Y is less than or equal to 3/2 and Y is greater than or equal to ½ divided by the probability of B.*2767

*This is our event A and this is B right here.*2780

*B is the probability that Y is greater than or equal to ½.*2783

*Being less than 3/2 and bigger than ½, that just means you are in the range between ½ and 3/2.*2791

*Being greater than ½, the easier way to think about that is to write that as 1 - the probability that Y is less than ½.*2803

*The reason I put it that way is because that is making it in a form where*2817

*we can easily use our cumulative distribution function.*2823

*Remember, F of Y represents by definition, it represents the probability that Y is less than or equal to some value y.*2826

*This is equal to, in the denominator that would be 1 – F of ½.*2841

*In the numerator, we want the probability of being between ½ and 3/2, that is F of 3/2 – F of ½.*2851

*Now that I have an F function, I work that out in example 3, I can just drop in all the values.*2865

*3/2 is bigger than 1, I have to use the second range there F of 3/2.*2874

*2/3 × 3/2 - 1/3 -1/2, now that ½ is less than 1, I will use the first range - 1/3 × ½ all divided by 1 - F of ½ is just 1/3 × ½.*2881

*Now, I just have some fractions to simplify.*2901

*2/3 and 3/2 that is 1 -1/3 - 1/3 × ½ is 1/6, 1 -1/6.*2903

*1/3 + 1/6 is ½, I get 1 -1/2 is just ½ in the numerator.*2915

*1 -1/6 is 5/6 in the denominator, that is 6/5 × ½ is 3/5 as my answer.*2922

*I think this is one we can also check graphically.*2938

*Let me draw that function.*2942

*We graph this out in examples 3 but that was the density function that I'm graphing right now.*2943

*It had a big jump at 1 and it really went from 0 to 2.*2954

*The probability that you are less than 3/2, let me draw 3/2 there.*2958

*There is 3/2, given that you are bigger than ½.*2964

*There is ½ and ½, we are given that we are bigger than ½, let me go ahead and color in that area there.*2970

*There is the probability that we are bigger than ½ and being less than 3/2 would mean that we are in that area right there.*2980

*I’m going to cut off at 3/2.*2991

*If you just kind of work by blocks there, there is sort of 5 blocks in the color blue1, 2, 3, 4, 5,*2994

*and three of them are colored green.*3004

*That is where we get the 3/5 coming from, if you want to do it graphically, instead of doing the equations.*3007

*You will still get 3/5.*3014

*Let me recap how we did that.*3018

*We are using conditional probability here.*3021

*I went back to my old conditional probability formula.*3024

*The probability of A and B divided by the probability of B.*3027

*In this case, my A and B where Y being less than 3/2 and bigger than ½.*3032

*I get the probability that Y is between ½ and 3/2.*3038

*The probability of Y being bigger than ½, I flip that around and said it is 1 - the probability of Y being less than ½.*3042

*The reason I flipped it around was, I could easily convert that into a value of my cumulative distribution function, my F.*3050

*I can also find this probability of the range using F.*3060

*I recalled the F that we calculated back in examples 3.*3064

*I recalled this F here and I just dropped in a different values.*3072

*Of course, I have to use the different parts of the formula because 3/2 was in this second range and ½ was in this first range.*3078

*That is why I use the second formula for 3/2 and the first formula for ½.*3088

*Then, I just simplify down the fractions and it simplify down to 3/5.*3096

*I could have done all that just by looking at the graph and by measuring up the areas.*3100

*I get 3 blocks out of 5 blocks, that is why that checks my answer to be 3/5.*3106

*In example 5 here, we have a cumulative distribution function given to us, the F of Y.*3116

*It looks like the most important ranges here are Y being between 0 and 1, and Y being between 1 and 2.*3123

*We got two problems here, we want to find f of Y, the density function.*3130

*We also want to find the probability that Y is in a particular range.*3135

*It looks like I have really left myself no space to solve this problem.*3143

*Let me jump over to the next slide and solve the problem.*3146

*Here is my cumulative distribution function, I put off the less important parts of the definition there.*3154

*We are going to find f of Y and the probability that Y is in a certain range.*3164

*This is fairly easy relative to the earlier problems that we did.*3170

*You get a little bit of break this time.*3175

*What is this X doing here, we do not use X in probability.*3177

*F of Y is equal to, remember, it is the derivative of F of Y.*3185

*I can find this just by taking the derivative of F of Y.*3191

*The derivative of Y/4 is ¼ and the derivative of Y²/4 is 2Y/4, that is just Y/2.*3196

*That is for the particular ranges 0 less than Y less than or equal to 1, and 1 less than Y less than or equal 2.*3206

*Outside those ranges, F of Y, remember it was just the constant 0 and 1.*3216

*Its derivative will just be 0 for other values of y.*3223

*If Y is less that or equal to 0, f will be 0.*3230

*If Y is greater than 2, f is greater than 2, f will also be 0.*3234

*That is my answer for f of Y, we found the density function there by taking*3242

*the derivative of the cumulative distribution function.*3250

*Those are the answer to part A.*3257

*Part B, we want to find the probability that Y is between ½ and 3/2.*3259

*You can do that quickly just by using the cumulative distribution function F of 3/2 - F of ½.*3267

*You do not have to do any integral here because we already have the cumulative distribution function.*3277

*Let us just drop those in.*3282

*Now, 3/2 is between 1 and 2, I’m going to use the second formula there, 3/2²/4.*3283

*For ½, I use the first part of the definition that is because ½ is between 0 and 1.*3294

*½ /4, there should be a - there not =.*3301

*I just work out the fractions, that is 9/4 /4 - ½ /4.*3308

*½ /4 is 9/16 - 1/8, we can write 1/8 as 2/16.*3315

*9/16 - 2/16 is 7/16, that is my answer.*3326

*You do not have to do any integrals there, and that is because the cumulative distribution function was already given to us.*3335

*If it had been a density function, we would have been doing the integral to calculate that.*3341

*Let me recap the steps there.*3345

*We are given in this one, we are given the cumulative distribution function and we are trying to find the density functions.*3348

*We are given F, we are trying to find f.*3354

*To find f, what you will just do is, go and take the derivative.*3357

*You took the derivative of Y/4, that is where that 1/4 came from.*3360

*Took the derivative of Y²/4, that is where that Y/2 came from because it is 2Y/4.*3366

*The derivatives of the constants on either end are just 0.*3373

*We get those two definitions on those ranges.*3378

*That is how we find f of Y.*3382

*To find probabilities, if you know the cumulative distribution function, it is just a matter of plugging the endpoints into F.*3384

*The only issue there is you have to be careful which of the two definitions for F you use.*3391

*You figured that out because 3/2 is between 1 and 2.*3398

*1/2 is between 0 and 1.*3403

*That is why I used those two respected definitions for F.*3406

*Drop the numbers in, simplify it down, get a nice fraction as my answer.*3412

*That wraps up our lecture on density functions and cumulative distribution functions.*3418

*This is part of the chapter on continuous probability,*3424

*which in turn is part of the probability lecture series here on www.educator.com.*3428

*My name is Will Murray, thank you very much for joining us, bye.*3434

1 answer

Last reply by: Dr. William Murray

Thu Jun 1, 2017 5:45 PM

Post by Ernest Harris on May 30, 2017

Hi Prof. Murray,

Thank you, your lectures are helping me acquire a deeper understanding of the material.

For Example 1 part B:

To be sure, the accumulation does not double count the argument 2 because only one set of values actually includes 2 where as the other set only approaches 2. Is my thinking correct?

2 answers

Last reply by: Dr. William Murray

Sat Apr 19, 2014 5:41 PM

Post by Ali Momeni on April 18, 2014

Hey Prof. Murray,

Example 1 part B is super confusing.

How did you know to add the INTEGRAL from 0 to 2 of (1/4 t) dt to the INTEGRAL from 2 to y of 1/4(4-t)dt? and why would you do that?

Does the formula -> INTEGRAL from a to b of f(y) dy = F(b) - F(a) apply here? If yes how?

Thank you very very much for your work and the time you take to answer questions! - Ali