For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

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## Table of Contents

## Transcription

### Gamma Distribution (with Exponential & Chi-square)

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
- Gamma Function
- Formula for the Gamma Distribution
- Key Properties of the Gamma Distribution
- Exponential Distribution
- Chi-square Distribution
- Chi-square Distribution: Overview
- Chi-square Distribution: Mean
- Chi-square Distribution: Variance
- Chi-square Distribution: Standard Deviation
- Example I: Graphing Gamma Distribution
- Example I: Graphing Gamma Distribution
- Example I: Describe the Effects of Changing α and β on the Shape of the Graph
- Example II: Exponential Distribution
- Example III: Earthquake
- Example III: Estimate Using Markov's Inequality
- Example III: Estimate Using Tchebysheff's Inequality
- Example III: Summary
- Example IV: Finding Exact Probability of Earthquakes
- Example V: Prove and Interpret Why the Exponential Distribution is Called 'Memoryless'

- Intro 0:00
- Gamma Function 0:49
- The Gamma Function
- Properties of the Gamma Function
- Formula for the Gamma Distribution 3:50
- Fixed Parameters
- Density Function for Gamma Distribution
- Key Properties of the Gamma Distribution 7:13
- Mean
- Variance
- Standard Deviation
- Exponential Distribution 8:03
- Definition of Exponential Distribution
- Density
- Mean
- Variance
- Standard Deviation
- Chi-square Distribution 14:34
- Chi-square Distribution: Overview
- Chi-square Distribution: Mean
- Chi-square Distribution: Variance
- Chi-square Distribution: Standard Deviation
- Example I: Graphing Gamma Distribution 17:30
- Example I: Graphing Gamma Distribution
- Example I: Describe the Effects of Changing α and β on the Shape of the Graph
- Example II: Exponential Distribution 27:11
- Example II: Using the Exponential Distribution
- Example II: Summary
- Example III: Earthquake 37:05
- Example III: Estimate Using Markov's Inequality
- Example III: Estimate Using Tchebysheff's Inequality
- Example III: Summary
- Example IV: Finding Exact Probability of Earthquakes 46:45
- Example IV: Finding Exact Probability of Earthquakes
- Example IV: Summary
- Example V: Prove and Interpret Why the Exponential Distribution is Called 'Memoryless' 52:51
- Example V: Prove
- Example V: Interpretation
- Example V: Summary

### Introduction to Probability Online Course

### Transcription: Gamma Distribution (with Exponential & Chi-square)

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

*We have been working through the continuous distributions.*0007

*We have done the uniform distribution and the normal distribution.*0010

*Today, we are going to talk about the Gamma distribution which is actually a whole family of distribution.*0014

*I will describe that as we get into it.*0019

*There are a couple of very important special cases of the Gamma distribution,*0022

*which are the exponential distribution and the Chi square distribution.*0027

*I will talk about those as we get into it.*0031

*First, I’m going to talk about the Gamma distribution, in general.*0034

*And then, I will talk about the specific special cases of the exponential distribution and the Chi square distribution.*0038

*We will also do plenty of problems on it, I hope you will be an expert by the time we get through this.*0045

*Before I talk about the Gamma distribution, I have to tell you about the Gamma function.*0051

*It is a very confusing thing because there is a Gamma distribution which is a probability distribution and*0056

*there is also something called the Gamma function which is part of the Gamma distribution.*0062

*It gets a little bit confusing, I will try to keep it straight for you.*0067

*The Gamma function is defined by this integral formula right here.*0071

*The input to the Gamma function is a number α.*0077

*You think of α as being a number like 3 or 2 ½, something like that.*0082

*You plug in a positive number into that capital Greek letter Γ right there.*0089

*That is Γ right there.*0095

*You take a value of Α and you plug it in right here.*0098

*And then, you work out this integral and since it is a different integral, it comes out to be another number.*0103

*This whole thing works out to be another number.*0111

*I want to emphasize here, the Gamma function is just a function.*0120

*You plug in a number and it spits a number back out to you.*0124

*It has some very nice properties which is, the first of which is that Γ of N + 1 = N × Γ of N.*0128

*That is very similar to the factorial function but it is slightly different.*0138

*Basically, what happens is the Gamma function behaves similarly to the factorial function, but it shifted over by 1.*0143

*In fact, when N is a whole number, Γ of N is equal to N -1!.*0152

*For example, Γ of 4 is equal to 4 -1! which is 3! which is 1 × 2 × 3 which is 6.*0159

*One way to think about the Gamma function is that, it is a close cousin of the factorial function.*0171

*The difference is that, the factorial function you can only plug in whole numbers.*0178

*There is no such thing as 3 ½!.*0182

*For the Gamma function, you can plug in numbers that are not whole numbers.*0186

*You could find Γ of 3 ½.*0191

*It will be a little tricky to do the integral but it is possible.*0193

*People do calculate Γ of numbers that are not whole numbers.*0197

*I want to remind you again, this is the Gamma function.*0203

*We have not talked about the Gamma distribution yet.*0206

*This is just a function that it takes in the number as input and it spits out an answer as output.*0209

*It takes a number and it spits out an answer.*0218

*Next, we are going to move on and talk about the Gamma distribution.*0222

*The Gamma function is just one ingredient in it, be careful not to mix the two up.*0226

*I'm going to start out with the formula for the Gamma distribution.*0233

*You start win two parameters, I’m going to call them α and β, they are both positive numbers.*0238

*They can be anything you want, as long as they are positive.*0244

*We have this quite complicated density function for the Gamma distribution.*0248

*Let me look at each ingredient of this.*0253

*It is a mixed polynomial parts, there is this polynomial term here, Y ⁺α – 1.*0257

*There is this exponential term, E ⁻Y/β.*0266

*There is this denominator, β ⁺Α/Γ of Α.*0276

*Let me remind you that Γ of Α is the Gamma function.*0281

*That we learn about on the previous slide, that is the Gamma function.*0288

*Remember that Α and β are both constants, which means Γ of Α is a constant.*0293

*This entire denominator here is nothing but one being constant.*0301

*You should think of it as being kind of the less important part of the Gamma distribution.*0307

*This is just a big constant and the point of that constant is to make the area under the Γ curve come out to be 1.*0315

*It is just a constant to make the area equal to 1, when you graph the Γ density function.*0327

*When you graph to get the Γ density function, we will graph some of these later but it might look something like this.*0336

*You remember, in order to be a density function, the area under the graph always has to be equal to 1.*0343

*If it is not equal 1 then you want to take your function and we will multiply or divide by whatever it takes,*0351

*in order to make that area equal to 1.*0357

*In this case, let me start out with, for the Γ density function is this polynomial and this exponential term.*0362

*Those are the more important terms in the definition of the density function.*0370

*And then, we just divide by these constants, in order to make the area come out to be 1.*0377

*It is defined from 0 to infinity, it is only defined on positive numbers but the numbers can get as big as you want.*0384

*It is always defined on the right hand side, it goes on forever but only in one direction.*0394

*It cuts off in the other direction.*0399

*Let me only remind you again, not to mix up the Gamma function and the Gamma distribution.*0402

*The Gamma function here, is this part right here, that is the Gamma function that we define on the previous slide.*0409

*And then this whole thing is the density function for the Gamma distribution, which one ingredient of it is the Gamma function.*0417

*It is rather confusing but we will practice with it, hopefully, you will learn to keep it straight.*0427

*A key properties of the Gamma distribution, the mean is given in terms of Α and β.*0435

*The mean and expected values are the same thing, they come out to be α × β.*0441

*The variance turns out to be α β².*0447

*The standard deviation is always the square root of variance, that is just the definition.*0451

*The standard deviation is just the square root of α × β².*0457

*Now you know the key values associated with the Gamma distribution.*0463

*I want to move on and talk about the special families of the Gamma distribution.*0472

*The special cases which are really the most important × when we use the Gamma distribution.*0477

*The first one of those is the exponential distribution.*0484

*Let me go ahead and describe the physical situation in which you would invoke the exponential distribution.*0488

*Essentially, if you are waiting for a random event to occur and this event occurs every so often,*0494

*and there is no correlation between previous instances and future instances of the event.*0501

*The waiting time is a random variable which is an exponential distribution.*0508

*Typical exponential distribution is waiting for just something happen out of the blue,*0516

*maybe you are waiting for an earthquake.*0521

*You really have no idea how long it is going to take for an earthquake to strike.*0525

*We know that we do have earthquakes on average, every many years*0529

*but there is not a lot of correlation between one earthquake and the next.*0534

*The waiting time for an earthquake would be an exponential distribution.*0539

*Maybe you are working in a call center and you are waiting for the next call to come in.*0544

*That again, is an exponentially distributed.*0550

*The time for the next call to come in is exponentially distributed.*0553

*You could be sitting by the side of a country road and waiting for a car to come along.*0559

*You really have no idea how long it will be, before a car comes along.*0564

*The waiting time is exponentially distributed random variable.*0568

*This sounds a lot like something we have talked about before, back when we are talking about discreet distributions.*0575

*I did a video on the Poisson distribution.*0583

*You might think, did we use all the same examples to describe the Poisson distribution?*0586

*The answer is yes, but the thing we are keeping track of is different.*0593

*In the Poisson distribution, we were keeping track of how many times that random event occurs.*0597

*If you are working in a call center, how many calls do you get in an hour?*0605

*With the exponential distribution, we are keeping track of how long it takes for the next call to come in.*0610

*We are keeping track of the length of time until the next call comes in.*0617

*That does not have to be a whole number, it can be any number.*0622

*It could be half a minute, it could be 3/10 of a minute, and it could be 15.2 minutes until the next call comes in.*0625

*With the Poisson distribution, we are keeping track of how many calls come in, in the next hour.*0633

*That does have to be a discreet number.*0641

*We can get 4 calls or we can get 17 calls, but we could not get 4.5 calls, because calls only occur in whole numbers.*0644

*That is the difference between the exponential and the Poisson distribution.*0654

*They both describe the same physical situations but they are keeping track of different quantities.*0658

*The Poisson distribution is keeping track of how many times something occurs.*0664

*The exponential distribution is keeping track of the waiting time until the next occurrence.*0669

*That was all just describing the physical situation for the exponential distribution.*0676

*I have not even started telling you about the math.*0681

*Here is the way the math works.*0683

*The exponential distribution, you start out with the Gamma distribution.*0686

*Let me write down the formula for the Gamma distribution, just to remind you of the density function.*0690

*F of Y was equal to Y ⁺α – 1 × E ⁻Y/β.*0695

*Those are the important terms, the exponential term, and the polynomial term.*0704

*We also had these constants just to keep the area right, β ⁺Α and Γ of Α.*0708

*The exponential distribution is one of the Gamma distributions.*0719

*It is the Gamma distribution where you take α equal to 1.*0725

*The β you still leave it arbitrary, there is still going to be a β in there but the α are all going to turn into 1.*0731

*What that means is that, we have Y ⁺α -0, that turns into Y =0 which just turns into 1, that term drops out.*0739

*The β ⁺Α, since α is 1 just turns into β.*0751

*Γ of α, that is the Gamma function.*0755

*It is Γ of 1 which is equal to 0!.*0760

*The Gamma function on whole numbers is the same as the factorial function,*0765

*except it shifted over by 1 and 0 factorial is just 1.*0771

*This whole function simplifies down into E ⁻Y/β × 1/β.*0778

*We still have, that should be Y is greater than or equal to 0 and less than infinity.*0785

*That is our density function for the exponential distribution.*0794

*It is a special case of the Γ family but it is probably the one used most often in a probability situation.*0799

*Once we know that it is a special case of the Γ family, we can immediately say what it is mean, and variance,*0808

*and standard deviation are because we just look up the mean, and variance, and standard deviation from the Gamma distribution.*0815

*It was Α β before, plug in α = 1 and you get the mean is just β.*0822

*The variance was Α β² before, plug in Α = 1 and you get the variance to be β².*0829

*The standard deviation was the square root of Α β², plug in Α = 1, you get the square root of β² which is just β.*0836

*Those are all very easy, as long as you remember the corresponding quantities for the Gamma distribution*0848

*because the exponential distribution is a special case of the Gamma distribution.*0856

*We are going to be seeing a lot of the exponential distribution.*0864

*Let me mention another very common sub family of the Gamma distribution of which is chi square distribution.*0868

*Chi is a Greek letter, it looks like a X.*0878

*Sometimes people will use the Greek letter Chi for Chi square distribution.*0882

*Sometimes people spell it out as chi.*0889

*It is used most often in statistics, and. Because of that, I'm not going to be doing a lot of work*0893

*with the chi square distribution in these videos.*0899

*But it is something that is occasionally studied in probability, I wanted to mention it to you.*0903

*The important thing I want to emphasize is, it is a special case of the Gamma distribution.*0909

*You have another new Greek letter that you have to learn.*0916

*It is the Greek letter ν, that is pronounced ν, that is the Greek letter ν.*0920

*It looks like a V but it is pronounced ν.*0927

*Do not call it V because people will think you are an idiot if you call it V.*0930

*It is the Greek letter ν.*0934

*For each whole number ν, we have a Chi square distribution with new degrees of freedom.*0936

*Some people will talk about Chi square with 3° of freedom, or Chi square was 17° of freedom.*0943

*What you do is, you build a Gamma distribution.*0950

*Remember, the Gamma distribution has an Α and β.*0955

*The Α is going to be ν/2 and the β is going to bev 2.*0958

*Then, you can build the formula for the Chi square distribution, out of the formula for the Gamma distribution.*0965

*I did not bother to do that because we are not going to use the Chi square distribution so much.*0973

*We will be spending more time with the exponential distribution.*0977

*I do want to calculate the mean and variance of the Chi square distribution.*0981

*Remember, the mean of the Gamma distribution was α × β.*0988

*In this case, if you multiply R Α × R β, you get ν.*0992

*The variance was α × β².*0998

*Α × β², in this case is ν/2 × 2².*1002

*If you multiply that through then you get two ν.*1009

*That is the variance of the Chi square distribution.*1013

*The standard deviation of the Chi square distribution, like all standard deviations is just the square root of the variance.*1016

*You just take the square root of what we have above.*1023

*Those are kind of the basic facts about the Chi square distribution.*1026

*I’m not going to spend more time on the Chi square distribution because*1030

*it is not common in probability classes, as the exponential distribution.*1033

*That is where we are going to spend most of our time.*1037

*Now, you have the basic facts on Gamma distributions and their special cases, the exponential and Chi square distributions.*1043

*Let us jump into some problems.*1050

*In our first example, we are just going to be drawing some graphs.*1052

*We want to kind of understand how the density function for the Gamma distribution plays out.*1055

*Let me remind you what that density function was.*1061

*It is F of Y is equal to Y ⁺α – 1 × E ⁻Y/β.*1063

*Those are the two important terms.*1075

*If you forget everything else, you want to remember those two terms.*1077

*The less important terms but they are still there are, β ⁺Α Γ of Α.*1080

*Now, the reason I was a little bit snooty about the importance of those terms, is that there just constants.*1090

*They are just thrown in there to make the total area equal to 1.*1097

*They do not really change the shape of the graph that much.*1100

*The important terms are the ones in the numerator.*1104

*We are going to draw some combinations of α and β here.*1108

*I will draw them in different colors.*1114

*In the first one here, I will draw the first one I n blue.*1118

*This is Α is equal to ½ and β is equal to 5.*1123

*That means I'm graphing Y⁻¹/2 × E ⁻Y/5.*1132

*There is also a constant which I'm not even going to bother to write.*1144

*I want to figure out what that does.*1150

*The important thing here is to look at, remember all these are defined from 0 to infinity.*1156

*When Y is equal to 0, I got this Y⁻¹/2 here.*1163

*That means I'm dividing by Y because it is like 1/Y ^½.*1169

*That means, when you try to plug in Y = 0, the thing blows up to infinity.*1174

*I'm going to show something that blows up to infinity here at Y = 0.*1180

*As Y increases, here is Y = 0, and here is Y going out to infinity.*1189

*As Y increases, what happens is the exponential term kicks in pretty fast.*1199

*Since it is negative, E ⁻Y/5, it pulls it down pretty quickly to 0.*1205

*It kind of goes down fairly quickly and is asymptotic to 0.*1211

*That is the Α = ½, β = 5, part of this graph.*1220

*Let me do the next one in red.*1231

*This is α is equal 1 and β is still 5.*1234

*I got Y⁰ × E ⁻Y/5, Y is 0 is just 1.*1244

*Again, I'm not writing the constants because they do not change the shape in any fundamental way.*1253

*They just shift the graph up and down, and pull it back to get an area 1.*1258

*This one, if I plug in Y = 0, the Y⁰ drops out, this is E ⁺Y/5 divided by some constant.*1267

*I’m not going to worry about the constant.*1280

*If I plug in Y = 0, that gives me just E⁰ which is 1.*1282

*Now, I do not want two say that it is actually equal 1 because that value of the constant might affect it.*1288

*But, I do know that it is going to be some finite number here.*1295

*Let me draw that, I cannot make it entirely underneath the blue curve.*1302

*Because then, it would have smaller area and all of these things have area equal to 1 here.*1309

*I'm going to have a crossing above the blue curve at some point.*1319

*There it is, that is α = 1 and β = 5 there.*1324

*I did actually graph these using a computer and I checked that the intercept there turns out to be about 0.2.*1334

*Even if you plug 0 in here, you get 1 because that constant pulls it down a little bit.*1346

*It turns out to be about 0.2.*1353

*That is what that graph looks like.*1356

*Let me do the last one in green.*1360

*My α is 3 and my β is 2.*1363

*What I'm graphing here is Y ⁺α-1 that is Y² × E ⁻Y/β.*1367

*Again, there is some constant there but I'm not going to worry about the constant.*1376

*The key thing here is that, when you plug in Y = 0, Y² is going to be 0.*1380

*This one starts at the origin.*1387

*My green graph is going to start right here at the origin and it has to go up because it is a probability density function.*1390

*But it also has to level off at some 0.20.*1398

*It is going to go down and just like the others, it would asymptotic to 0.*1402

*That is the graph corresponding to 3, 2.*1408

*I got my three different graphs here.*1414

*What can I learn from this and what can I learn by looking at the equation.*1416

*The difference here is that, the value of α really seems to change the shape of the graph.*1423

*Because if Α is less than 1, it is seems to blow up.*1429

*Let me write this down as I'm saying it.*1436

*Α controls the shape of the graph in some fundamental way, the shape especially at Y = 0.*1439

*If Α is less than 1, we got that that 1/2 to 5 graph, it blows up to infinity at 0.*1463

*Α = 1 goes to a finite number, goes to a positive number, positive at 0.*1479

*Α bigger than o1, it looks like it is going to pull it down to 0.*1497

*It goes to 0 at 0, let me put a little color coding on here.*1503

*The blue one was the α less than 1.*1510

*The read one was the α equal to 1.*1512

*The green one was the α bigger than 0, I meant α bigger than 1.*1518

*Now, we know the Α really seems to control the shape, especially around Y = 0.*1533

*Different values of α will make it go up to infinity, or go a finite number, or be pulled down to 0.*1540

*Β is not as important, what β does is, I can see the effect of β right here.*1548

*Β is just a scaling factor on the exponential term. A larger value of β will point out to the right.*1558

*It does not change the shape in such a fundamental way.*1570

*It stretches the graph out to the right.*1573

*If you are going to stretch the graph out to the right, if you going to stretch it out to the right,*1585

*the total area always has to be 1.*1591

*It would have to smash the graph down a bit, to maintain area 1.*1594

*In answer to the question here, where we have our nice graphs, but Α seems to affect the shape when Y = 0.*1611

*Β stretches it out and stretches it down but it does not fundamentally change the shape of the graph.*1621

*In example 2, we have an actuarial calculation.*1633

*We are going to park a car on the streets of Long Beach California.*1637

*We are going to see how long it takes for that car to get stolen.*1642

*This is the kind of thing that actuaries working for insurance companies, calculate all the time.*1646

*They want to know how long it takes for a car to be stolen because if it is stolen, then they will have to pay you.*1651

*If you bought car insurance, they will have to pay you to replace the car.*1660

*They need to know how often they will have to make those kinds of payouts.*1665

*This is the kind of thing that the exponential distribution works for.*1672

*Because again, you are waiting for some random event to happen and there is really no telling when it is going to happen.*1678

*It might happen tomorrow, it might happen a month from, it might not happen for the next 100 years.*1683

*If you get lucky, you can leave your car out, in 400 years no one will steal it.*1689

*But if you are very lucky, it might get stolen tomorrow and you might buy a new car the next day,*1693

*and that new car might be stolen after that.*1699

*This is an exponential distribution.*1703

*We have calculated that your car is stolen once every 12 years, you have a bad day and your car will be stolen.*1706

*Although that is just an average, it might happen twice in a year, if you are in a really bad year.*1713

*It might not happen for 30 years, if you are very lucky.*1718

*We want to figure out the chance that our car will last 24 years without being stolen.*1721

*Can we go for the next 24 years without being stolen.*1728

*I want to make a little computation using the exponential distribution, kind of in general.*1732

*And then, I will apply it to this particular example.*1738

*If I make a general computation, I think it can be useful for several different problems.*1741

*I’m going to say the answer, the general answer first.*1746

*The exponential distribution, the density function remember is, F of Y is 1/β × E ⁻Y/β.*1751

*That is on the range Y goes from 0 to infinity.*1761

*In order to calculate probabilities with the exponential distribution, we have to integrate that.*1773

*I would like to calculate the probability that Y will be a bigger than any particular value C.*1780

*I'm going to say that and use it for several other problems.*1791

*You are really want to make sure that you understand this computation.*1793

*By the way, β there is the mean of the distribution.*1797

*That was something we figured out on a previous side, you can go back and look that up.*1801

*The probability that Y is greater than or equal the C, we will use an integral for that.*1807

*It is the integral from C to infinity because it could be any value from C to infinity of the density function.*1811

*1/β × E ⁻Y/β × DY.*1820

*This is going to work a lot better, if I make a little substitution.*1828

*I'm will make u substitution, I will define u, := means defined to be.*1832

*I'm defining my variable here right now, to be Y/β.*1842

*Any time I make any kind of substitution in an integral, I also have to find DU.*1847

*DU would just be 1/β DY.*1854

*That is quite convenient because I have a 1/β DY in the integral.*1858

*My integral will convert into the integral of E ⁻u DU.*1864

*I’m not going to put bounds on it because I'm going to go ahead and integrate it, and then I will convert it back into terms of Y.*1872

*We would not actually finish the integrals in terms of u.*1879

*The integral of E ⁺u is just E ⁻u × – 1.*1882

*You can work that out doing another little substitution, if you like.*1890

*Or you can use of the opposite of the chain role which of course is substitution.*1895

*That is the same as E ^-, u is Y/β.*1903

*We are supposed to evaluate that from Y =C to, we will take the limit as Y goes to infinity.*1911

*That is –E ⁻infinity/β – E ⁻C/β.*1925

*A lot of negatives in here, but fortunately, E ⁻infinity that is 1/E ⁺infinity.*1938

*This is just 0, that term goes away.*1946

*These negatives cancel and we get positive E ⁻C/β.*1949

*I want to hang onto this result, the probability of Y being greater than or equal to C is equal to E ⁻C/β.*1958

*We are going to use that in several different problems here with the exponential distribution.*1967

*Make sure you understand that calculation.*1971

*Make sure you are able to repeat that.*1974

*As long as you do understand that, we would not have to go back and work it through every time, we just sight this result.*1977

*In this case, what do we want to calculate.*1983

*We want to calculate the probability that our car will last 24 years without being stolen.*1986

*The probability that the waiting time for a car to be stolen is more than 24 years.*1992

*The probability that Y will be greater than or equal to 24.*1999

*Now, using the formula that we just worked out, that is E ⁻C/β.*2005

*Β was the mean, that is the average amount of time until somebody's car is stolen.*2011

*We are given that, it is your car is stolen every 12 years.*2017

*If you just park your car as usual and go about your daily business, on the average,*2021

*once every 12 years, you are going to wake up and say oh my gosh, they stole my car.*2027

*The C value is 24 and the β value is 12, this is E ⁻24/12 here, which is E⁻².*2033

*I work that out on a calculator, I just threw that into a calculator and it came out to be approximately 0.135, that is 13.5%.*2063

*The probability that your car will last for 24 years without being stolen is fairly well.*2082

*Probably, within the next 24 years, it happens to most people sooner or later, you are going to lose a car.*2090

*Certainly, the actuaries working for the insurance companies want to know what that probability is,*2097

*so they know how likely is that the company will have to pay you to replace your car.*2103

*And in turn, they know how much to charge to cover that kind of insurance.*2108

*That is the answer here, the probability of lasting 24 years is 13%.*2115

*Let me recap the steps here.*2121

*In particular, I want to emphasize this initial calculation because we are going to use it over and over again.*2123

*I do not want to re do it again, I do now want to recalculate these integrals every time, because it is the same every time.*2129

*Here is the density function for the exponential distribution, 1/β × E ⁻Y/β.*2136

*If you want to calculate the probability of Y being bigger than or equal to any constant C, E of Y bigger than or equal C,*2144

*we plug in those values for this integral C ⁺infinity because the exponential distribution does go on to infinity.*2156

*Little substitution got us through that integral, plug in the values, to get this,*2164

*we took a limit but it is an easy limit because E ⁻Y as Y goes to infinity is just E ⁻infinity.*2171

*1/E ⁺infinity is 0.*2181

*Here, the negatives all canceled and we got E ⁻C/B.*2184

*Let me summarize that, that C/β because that is what I want you to remember.*2189

*I want you to be able to just recall it for future problems.*2195

*For this particular problem, our C value was 24, our β was 12 that came from the average of 12 years.*2201

*The mean of the exponential distribution is β.*2209

*I just plugged in 24 and 12, and we got E⁻².*2214

*I convert that into a percentage of 13.5%.*2220

*In examples 3, we have seismic data indicating that the time until the next major earthquake*2227

*in California is exponentially distributed.*2235

*Again, this is kind of a classic application of the exponential distribution.*2238

*You are waiting for something to happen and it happens kind of randomly.*2244

*Sometimes it happens, sometimes it does not.*2248

*In this case, it happens on average once every 10 years but*2250

*you might have two earthquakes in one decade and no earthquakes in the next decade.*2253

*We want to find the chance that there will be an earthquake in the next 30 years.*2257

*In this case, we are not calculating exactly, we are going to estimate.*2262

*We are going to use our two Russian inequalities, Markov's inequality and Chebyshev’s inequality.*2267

*Let me remind you what Markov’s inequality went.*2276

*It is in the probability that a random variable will be bigger than a particular constant,*2280

*is less than or equal to the expected value of that variable divided by the constant that you are interested in.*2286

*In this case, the constant we are interested in is 30 years because I am estimating the chance that*2295

*there will be an earthquake in the next 30 years.*2305

*My A is 30, the expected value is the mean of the variable.*2308

*In this case, we are given that there is a mean of 10 years, this is 10.*2316

*10/30 simplifies down to 1/3.*2321

*That was the probability that Y is bigger than A.*2325

*That is the probability that it will take longer than 30 years to have an earthquake.*2329

*But we want the chance that there will be an earthquake in the next 30 years,*2336

*meaning the next one comes in less than 30 years.*2340

*Our Y here is the waiting time.*2345

*We are trying to estimate the probability that there is an earthquake in the next 30 years.*2350

*That would be Y less than A, the probability that Y is less than A.*2355

*We have to flip the inequality here, if the probability that it is greater than A is less than 1/3,*2364

*this is the probability of being less than A.*2372

*Let me go ahead and fill in 30 here, is greater than or equal to 1 -1/3, 2/3.*2375

*That means there is at least a 2/3 chance that there will be an earthquake in the next 30 years.*2384

*That the waiting time for the earthquake is less than 30 years.*2390

*Within the next 30 years, we are due with a probability of at least 2/3.*2396

*Maybe, it is even higher than that, I do not know just by using Markov’s inequality.*2401

*But I can say for sure just for Markov that, it is at least 2/3 chance we are going to have an earthquake in the next 30 years.*2405

*Let me calculate the same thing using Chebyshev’s inequality.*2414

*Again, I will remind you what that is.*2417

*This is Markov’s inequality right here, Chebyshev’s tells us that the probability of Y minus μ, the mean, being bigger than K σ.*2419

*Σ is the standard deviation is less than or equal to 1/K².*2434

*By the way, I have some earlier lectures right here in the probability lecture series,*2439

*right here on www.educator.com that cover Markov’s inequality and Chebyshev’s inequality.*2445

*That is why I'm not developing them from scratch for you here.*2450

*But if you do not remember Markov’s inequality and Chebyshev’s inequality,*2453

*you just go back and watch those other lectures on those two inequalities and get all caught up.*2457

*You will be ready to go with this example.*2463

*In this case, let us figure out what some of these values are.*2467

*Our μ is our mean, in this case, it is the mean of the exponential distribution is β which is 10.*2470

*Σ is our standard deviation, σ for the exponential distribution, I said that a couple of slides ago, is also β, that is 10.*2479

*We want the probability that Y will be greater than 30.*2491

*Because, I want to calculate the probability that Y is less than 30 but I will come back to that later.*2498

*I’m going to start out with the probability that Y is greater than 30.*2505

*30 is 2 standard deviations bigger than 10 because the mean is 10, then, μ = 10.*2509

*The mean is 10 and the standard deviation is 10.*2528

*In order for it to be bigger than 30, it is got to be 2 standard deviations bigger than 10.*2531

*That means our K value is 2.*2538

*The probability that Y – μ is bigger than or equal to 2 σ is less than or equal to,*2540

*Chebyshev’s tells us it is less than or equal 1/K².*2550

*1/K² which is 1/2² which is ¼.*2554

*That is the probability that Y is bigger than 30.*2560

*The probability that Y is less than 30 is, if the probability that it is bigger than 30 is less than ¼,*2564

*this would be greater than 1 -1/4 which is ¾.*2577

*That makes our prediction of an earthquake a little more dire.*2583

*It says that the probability that there would not be an earthquake within the next 30 years is at least ¾,*2587

*at least 75% chance that we will have an earthquake in the next 30 years, according to Chebyshev’s inequality.*2594

*It gets us a more accurate prediction, by the way, that is not saying that Markov’s inequality was wrong.*2604

*Markov said it is at least 66%, Chebyshev’s says it is even bigger than that, it is at least 75%.*2609

*They are both right but Chebyshev’s gives us the stronger result.*2618

*The reason it give us a stronger result is because we had to go through a little more work to do it.*2622

*We had to use more information, the mean and the standard deviation, in order to calculate it.*2627

*Let me recap how we got these results.*2634

*First of all, if you do not remember Markov’s inequality and Chebyshev’s inequality,*2637

*I have got lectures here on www.educator.com.*2642

*Just scroll up in the probability series and you will see the lectures on Markov’s inequality and Chebyshev’s inequality.*2645

*You will see where these initial formulas are coming from.*2651

*The Markov’s inequality said that, the probability Y being bigger than the cutoff is less than the mean divided by that cutoff value.*2654

*In this case, we are interest in the probability of Y being bigger than 30,*2664

*because we want to have an earthquake in the next 30 years, whether it comes before 30 or after 30.*2668

*A is 30, our mean is, we got that here, we fill that in as 10 and we get 1/3.*2675

*Remember, that is the probability of being bigger than 30, that means we wait more than 30 years to get an earthquake.*2683

*But that is not we are interested in, we are interested in waiting less than 30 years.*2693

*We have to flip it around, from 1/3 we flip it around to 1 -1/3 is 2/3.*2697

*We have to flip the inequality, the probability is more than 2/3 that we will have an earthquake.*2705

*Here is the formula for Chebyshev’s inequality, it is based on the standard deviation which is 10 and the mean which is 10.*2713

*And then you ask yourself, how many standard deviations away from the mean am I going?*2722

*In this case, we are interested in 30.*2728

*30 is 2 standard deviations away from the mean.*2731

*That is because 30 – the mean of 10 divided by the standard deviation of 10 is 2.*2735

*That is where I get my K there.*2743

*There is K is 2, and I plug in K is equal to 2.*2746

*The probability is less than 1/K², that is 1/2² is ¼.*2751

*Again, that is the probability that Y will be greater than 30.*2757

*We have to flip it around and instead of taking ¼, we have to do 1 -1/4 for the probability of being less than 30.*2761

*We have to say it is greater than ¾.*2771

*Our probability of having an earthquake is at least 75%, that is scary if you live in Southern California.*2775

*In the next example, we are going to use the same basic setup except we are going to calculate the probability exactly,*2784

*instead of estimating it, using Markov and Chebyshev’s.*2791

*You want to make sure you understand this example and understand the same basic setup,*2795

*before you go on to the next example, example 4.*2801

*In example 4, we are using the same setup that we had from example 3.*2806

*You might want to go back and check over example 3.*2811

*Same thing, we have waiting for an earthquake to happen and we know that they happen once every 10 years on average.*2814

*That is going to be our mean and that is going to be our β, follows an exponential distribution.*2822

*We want to find the exact probability that there will be an earthquake in the next 30 years.*2827

*We want to find the probability that Y is less than or equal to 30.*2835

*Remember, Y is our waiting time for an earthquake.*2841

*How long do we have to wait until the earth starts shaking?*2849

*What is the chance that that will be less than 30 years?*2853

*Now, I think the easier way to calculate this is to do 1 - the probability that Y is greater than or equal to 30.*2857

*The reason I frame it like that, is because we have a formula that we worked out back in example 2.*2866

*Let me show you that formula from example 2.*2875

*From example 2, we work this out, the probability of Y being bigger than the value C is equal to E ⁻C/β.*2880

*We did an integral to calculate that, it cost us some work.*2895

*If you do not remember that or you work that out on your own, just go back and watch example 2 again,*2898

*and you will see where that result comes from.*2905

*In this case, our C value is 30, by the way I'm being a little cavalier here in my use of greater than or equal to vs. greater than,*2907

*because these are continuous distributions, it does not matter.*2921

*Continuous distributions, the probability of any exacta value is 0.*2926

*The probability that it is equal to 30 is 0.*2931

*What is the chance that you are going to have an earthquake exactly 30 years from now?*2934

*That is not going to happen, it will not be exactly 30 years, it will be 30.1 years or 29.8 years.*2938

*You do not have to worry about being exactly equal to 30,*2946

*which means I do not have to worry about whether I write greater than, or greater than or equal to.*2950

*Using my formula back from example 2, this is 1 – E ⁻C is 30 that is 30.*2957

*My β is 10, I got that up here.*2966

*30/10 is 1- E⁻³, I threw that into my calculator and my calculator spat out the number 0.9502.*2971

*My probability of having, these are approximations, I guess, there are some small rounding involved.*2988

*That is just about 95%, what that means is that the exact or very close to exact probability of*2995

*there being an earthquake in the next 30 years is 95%.*3004

*It is really time to run for the hills because it is very likely that there will be an earthquake in the next 30 years in California.*3009

*That is not too surprising, if we have it 10 years on average, it is not very likely that*3017

*we will survive 3 decades without having an earthquake.*3021

*It is quite likely that there will be an earthquake, sometime in the next 30 years.*3025

*Now we know that the exact probability is 95%.*3029

*Notice that, this does not contradict the answer from the previous problem.*3033

*In the previous problem, we are calculating the same thing except you are*3038

*just using rough estimations using Markov's inequality and Chebyshev’s inequality.*3042

*In example 3, if you have not just watch that, you might go back and check that so you know what I'm talking about.*3048

*In examples 3, we used Markov to estimate this probability.*3054

*I have to remember the spelling of Chebyshev’s, fortunately you can spell Chebyshev’s*3060

*almost any way you like and it will be right according to some of version the name.*3064

*That is how I’m going to spell it.*3069

*Markov said that it was greater than or equal to 2/3.*3071

*Yes, 95% is greater than 2/3, that was not wrong.*3077

*Chebyshev’s said that the probability was greater than or equal ¾.*3082

*Yes it is, 95% is greater than ¾, that checks a little bit with our previous answers.*3087

*But of course, we get a much stronger answer from calculating it exactly and actually doing an integral.*3095

*We did a rollback in example 2 here.*3102

*Just to recap here, we want the exact probability that there will be an earthquake in the next 30 years.*3106

*That means our waiting time would be less than 30.*3112

*That is 1- the probability of it being greater than 30.*3116

*We did an exponential distribution, we figure out this nice formula back in example 2 of*3120

*the probability of a variable being bigger than the cut off.*3127

*It is E ⁻C/β.*3131

*I just plug in C = 30 and then β was the mean, that is 10.*3133

*And then, I simplified that down and I got 95%, that is my exact probability or very close to around a little bit.*3140

*But, that is basically the exact probability that there will be an earthquake in the next 30 years in California.*3148

*Of course, I can check that against the answers I got in example 3 where I just estimated using Markov and Chebyshev’s.*3154

*Those were not exact calculations, those are estimations, but it certainly agrees with those two answers.*3161

*In example 5, we have an exponential distribution and it does not tell us what the mean is this time.*3172

*I guess we just have to call it β.*3178

*D and M are some constants, we will have very little concrete in this problem.*3180

*We have to prove this strange expression, it says, I see here we have conditional probability.*3185

*This line is conditional probability, I have to remember that, the formula for conditional probability.*3190

*We have to prove that the probability of Y being bigger than D + M given that Y is bigger than D*3202

*is equal to the probability of Y being bigger than M.*3209

*Somehow, that is supposed to have something to do with the word memoryless.*3214

*The exponential distribution is known as the memoryless distribution.*3218

*We need to interpret that and justify it somehow.*3223

*The first thing I'm going to do with this problem is, remind you of a formula that we derive back in example 2.*3227

*If you have not watched example 2 in the recent past, you should go back and watch that right now*3236

*because we are going to be using the formula for the exponential distribution.*3244

*It tells us that the probability that Y is bigger than C is equal to E ⁻C/β.*3248

*That is going to be very useful, we calculated an integral back in example 2 to find that,*3258

*but we are not going to recalculate it now, I'm just going to use it.*3263

*I'm going to go ahead and start working out the left hand side of this expression.*3268

*It might get a little complicated but hopefully I can simplify it down to the right hand side.*3274

*The left hand side LHS is, remember, we have to use conditional probability here.*3278

*The probability of A given B, this is an old formula, I gave a lecture of video on it many moons ago.*3287

*If you do not remember that, you can always look up on my previous lecture on it.*3296

*It is right here on www.educator.com.*3299

*It is the probability of A and B, or A intersect B, if you want to use symbols for it, divided by the probability of B.*3301

*Let us figure out what that means in this situation.*3315

*It is the probability that Y is bigger than D + M and Y is bigger than D divided by the probability of Y being bigger than D.*3318

*Let us think about that, if Y is bigger than D + M, then Y is definitely bigger than D.*3336

*I did not say it here but I'm assuming that D and M are all positive numbers.*3342

*If Y is bigger than D + M, then it is definitely bigger than D.*3349

*I do not really need to say Y is still bigger than D.*3352

*I can just say the probability that Y is bigger than D + M.*3355

*I do not need to emphasize at that point the Y is still bigger than D because it is automatic, divided by the probability that,*3360

*I do not know why I say Y + D above should have been bigger than.*3367

*Y is still bigger than D.*3372

*We can calculate each one of those and we are going to use this result from example 2.*3376

*We use this result right here from example 2.*3381

*That is E ^-/ our C is D + M, D + M divided by β, all divided by E ⁻D/β.*3384

*We have a fraction of exponents here, we can do a little flip.*3400

*Maybe flip it up to the top there.*3406

*That is E ⁺D/β, since we pulled it out of the denominator – D + M/β.*3408

*Now, that simplifies down a little bit into, D/β canceled and we are left with E ⁻M/β.*3421

*If we use this result from example 2, that is exactly, that 2 sure looks like a1, does in it?*3431

*Let me change that into a real, honest 2.*3439

*This is the probability that Y is bigger than M.*3443

*Low and behold, we have the right hand side appearing here.*3450

*We have proved our equation that we set out to prove here.*3458

*I have not really given much of an interpretation as to what that might mean, but I certainly know it is true.*3466

*I certainly know that this equation is true.*3473

*And now, I have to think about what it really means.*3476

*Let us say that the exponential distribution, what it measures is the waiting time until some random event occurs.*3482

*An example I used earlier on was some unpredictable event,*3492

*sometimes it happens and you never really know when it is going to happen.*3497

*An example I picked was your car being stolen.*3501

*Let us have Y be the waiting time until how long do you have to wait,*3506

*just leaving your car around on the street until your car is stolen.*3519

*What does this equation mean?*3524

*Let me pick THE values for D and M.*3527

*Let us say D is one day and M is one month, D + M would be one month and one day.*3531

*That is why I picked D and M in the first place because I was thinking ahead to this.*3545

*What this is really saying, the probability, we have conditional probability here.*3550

*This means, given that Y is bigger than D.*3556

*Suppose Y is bigger than D, that means you may get through the first day without your car being stolen.*3561

*Suppose your car is not stolen today, that means you made it through the first day, thank goodness.*3570

*Y is bigger than D, you made it through at least 1 day without any theft of your car.*3589

*Given that you made it through today, what is your chance of making it through a whole month more after today?*3597

*Then your chance of surviving, surviving meaning your car is not stolen, surviving another month,*3608

*an extra month on top of today.*3629

*What we are really calculating there is, the probability that your car will now survive a day*3634

*and another whole month given that it is already survive one day.*3643

*What we are seeing here is that, it is equal to, is the same as the chance of surviving a month from today, just starting today.*3650

*It is the same as the chance of surviving a month today.*3668

*I keep trying to spell surviving, maybe I should find a different word, surviving a month today.*3675

*The chances of surviving a month today is the probability of Y being bigger than M.*3686

*What that means, let us think about that.*3692

*It means, you can think of the beginning of it today, what is my chance of surviving a month?*3696

*You can calculate that out, it is the probability of Y being bigger than M.*3702

*Then maybe, at the end of the day, you get through that day and you say what is my chances surviving another month?*3707

*It is the same is your chance as this morning of surviving a month from this morning.*3715

*In other words, if you survive today, you get a fresh start tomorrow.*3721

*You get you get a fresh start tomorrow, if you survive today.*3727

*It means you just got lucky today, you get a fresh start tomorrow.*3735

*Your probabilities of surviving another month do not change tomorrow.*3738

*It is not like the bad luck will build up.*3746

*If you survive today, it does not mean you are more likely to have your car stolen tomorrow.*3748

*It just means you got lucky and you get a fresh start tomorrow.*3754

*Maybe, you will keep getting lucky, your probabilities will keep staying the same.*3758

*That is what it means to be a memoryless distribution.*3764

*The exponential distribution is memoryless.*3768

*At the end of today, it does not remember that you survived one day.*3772

*It just restarts and it calculates a new for the next month.*3777

*The exponential distribution is memoryless.*3782

*It does not remember that you made it through today and hold it against you,*3791

*and make you more likely to have a car theft tomorrow or the next month.*3802

*It just says you got a fresh start today, I will compute the probabilities for the next month just the same as if we had started today,*3807

*this morning, and calculated the probabilities for a month.*3816

*I do not remember that you got through today, I would not hold it against you.*3820

*I will not build up the bad luck, I will just count a new starting tomorrow.*3824

*That is what it means to be memoryless.*3832

*Let me show you again how we did these calculations here.*3834

*At first, I just read this as an equation and I did not try to think what it meant.*3838

*I calculated this probability as a conditional probability and I use my own conditional probability formula.*3844

*If you do not remember the condition probability formula, I got a bunch of problems on that in an earlier video here,*3850

*near the beginning of this probability lecture series.*3856

*Just scroll up to the top and you will see conditional probability.*3859

*I got this condition probability, I say it is the probability of one event and another event divided by the second event.*3865

*But, these particular events, one subsumes the other, one absorbs the other.*3873

*Because if Y is bigger than D + M, Y is automatically bigger than D.*3879

*I do not need to write that Y is bigger than D.*3884

*I can just drop that out, it just disappears.*3886

*I can just include it in the fact that Y is bigger than D + M.*3890

*Each of these probabilities are in a format that is amendable to this formula that I use in example 2.*3895

*That I actually proved in example 2, we get an integral back in example 2.*3902

*If you do not remember example 2, just go back and look, you will see this formula, same videos, just scroll up and you will see it.*3907

*The probability that Y is bigger than C is E ⁻C/β.*3914

*I drop those values as C D + M and D in here.*3919

*I did a little algebra to simplify and I got E ⁻M/β.*3923

*That was example 2 right there to get to there.*3927

*I used example 2 backwards to go from E ⁻β back to the probability of Y being bigger than M.*3930

*I noticed, look that is the right hand side of my equation.*3940

*I'm done, I have proved that that equation is true.*3945

*If one thing to prove that the equation is true, it is another thing to interpret it and really understand what it means.*3947

*I said, let us interpret this as a waiting time until something happens.*3953

*In this case, until your car is stolen.*3959

*This is saying that, if your car, if Y is bigger than D, that means your car is not stolen today*3963

*because you are waiting more than one day for it to be stolen.*3970

*What is your chance of surviving an additional month after today?*3973

*Here is that additional month, that D + M is the additional month given that you made it through today.*3978

*What we worked out is that probability is the same as if we had calculated this morning,*3985

*if we come in this morning and calculated what is the probability of lasting one month from today.*3991

*If we come in this morning, we say, what is the probability of lasting one month?*3998

*We calculate a certain number, or if we wait until tonight and say, I made it through one day,*4002

*what is my probability of lasting another month after this?*4009

*We would have gotten the same number either way because those two numbers are equal.*4013

*If we can make it through today, on the condition that we make it through today, we will get a fresh start tomorrow.*4019

*It will still be exactly the same probability of lasting through another month.*4025

*That is why the exponential distribution is called memoryless.*4031

*It does not remember that you got lucky for one day,*4034

*it just restarts and starts calculating the same probabilities this evening that it calculated this morning.*4037

*You kind of get a fresh start, assuming you are lucky enough to survive through today.*4045

*That wraps up our examples on the Gamma distribution, and exponential, and Chi square distribution. Remember that the Gamma distribution is the overall family.*4051

*And then, two special cases within the Gamma distribution are the exponential distribution and the Chi square distribution.*4063

*Probably the most common of all of those in probability is the exponential distribution, and after that,*4071

*you will be using the Chi square distribution if you take a lot more statistics, that is where Chi square distribution comes up.*4077

*That is the end of our Gamma distribution lecture.*4086

*Next up, we have a nice lecture on the β distribution on, as we keep moving through our continuous distributions.*4089

*This is all part of a larger lecture series on probability here on www.educator.com.*4098

*I am your host Will Murray, thank you very much for joining me today, bye.*4103

1 answer

Last reply by: Dr. William Murray

Mon Nov 10, 2014 3:00 PM

Post by meleksen akin on November 10, 2014

In gamma distribution class, Example II when yÄ±u define u=y/B, you also define du=1/B dy, how you did this. Is there a rule for this?

1 answer

Last reply by: Dr. William Murray

Mon Oct 27, 2014 11:23 AM

Post by Anton Sie on October 25, 2014

Americans really rock in explaining :) You have saved my life Dr Murray! ^^

1 answer

Last reply by: Dr. William Murray

Tue Aug 5, 2014 4:03 PM

Post by Humam Altayeb on August 1, 2014

Hi, I wonder if you will be able to explain the standard gamma distribution & incomplete gamma function ? thanks

1 answer

Last reply by: Dr. William Murray

Wed May 28, 2014 5:04 PM

Post by Danushka Karunarathna on May 28, 2014

Hey, do you guys offer help with assignments?