 Good morning. Okay, I have some announcements. I have some announcements and the first announcement is that there will be an IBK colloquium an IBK Colloquium is Is basically a lecture given by an invited guest who has something interesting to say And this particular colloquium will be on May 8th, which is a Tuesday and It will take place at 1700 hours. So five o'clock in the afternoon in this room and It will be a presentation. It will be a lecture by Peter Buckland, who is one of the directors of the Canadian very famous bridge engineering company called buckland and sailor it's It's a company which is owned by the Consulting company called covey Which is the Danish company? well The lecture will be About the lion gate bridge and Some recent Advances in suspension bridge engineering. So I can promise you this this presentation Will be extremely interesting. They are among the world top in in in bridge engineering considering especially Cable-supported bridges. Okay, so I can warmly Recommend this lecture to you. I have another announcement and namely that there is a list outside the door and for those of you who are Repetitions, so those of you who attended this lecture last year and Wanted to follow it again this year those of you Who are doing that and who would like to take these? Small tests during this semester here. You have to sign up on this list So that we know that you will be attending these small test exams, okay for all of those Who are in this lecture for the first time this year? You don't need to sign because we know that you will have to take those tests So it only concerns The other people, okay any questions in this regard and for these small tests Also in order to avoid any confusion you are allowed to bring any material you would like Okay, the only thing you cannot bring our computers who can communicate with other computers or With the outside or with mobile phones, etc Please don't do that and and and please don't bring Computers which have strange devices which could be antennas and things like that because it will make it very difficult for us to Conduct the exam even though that they are not working or you have You have switched the antenna off or things like that. Please don't do these things. It will only make things difficult, okay? So you know that this small exam is coming up. Are there any any questions in this regard? Then it's probably a good time to raise the questions now just for clarification You have seen the tests from from previous years The tests are going to be in the same style so basically like a multiple choice Sheet of paper and a lot of exercises. They're really not too complicated, but of course It all depends a little on how well prepared you are Okay Now the last announcement before we continue And I start talking a lot about the material for today Is that we have planned a small Classroom assessment exercise So that means that together with the usual color cards use you picked up entering the classroom there was also one sheet of paper and in this sheet of paper that there are two parts you can fill out what you find was Say dubious or difficult to understand in this lecture here and You can fill out what you find was nice or easily understandable. So basically some negative Evaluations and also some please also some positive evaluations and then please fill out this piece of paper during the during the break and When you come back from the break, please put this filled out piece of paper on the table outside in the in the Entrance of this lecture room, okay then we will we will study the answers in in great detail and we will try to make an statistical evaluation of the results and I will try to present. I don't know what will come out of it But I will try to present the results of your answers in the next lecture, okay? Maybe also if we have time I will try to pick out at random a couple of the answers in the second lecture today That's basically That was my opening announcement. Are you ready for the lecture? Okay that Let's do it so this this picture here illustrates Really to a very large degree how we apply probability and statistics in engineering this picture illustrates the on the On the right hand side we have the real world over here and What we want to do is we want to make decisions We want to optimize decisions whereby we can Manage the risks which we are dealing with in engineering So we want to optimize engineering decisions but we have of course the problem is that we have to do it in the real world and In between us and the real world when we are sitting at the desktop when we are trying to Comprehend what could be the optimal decisions we need to operate with models and What you see here in between Here are The way we usually try to attack the problem of modeling the real world and the details of the Figure in the middle are What is being treated in the lectures we have on the seven semester and risk and safety in civil engineering? We cannot go into the details here, but what we really are trying to do generally when we are managing risk is that Especially considering risks due to natural hazards. We are trying to model the exposures So exposures could be considering natural hazards. It could be Earthquakes it could be typhoons. It could be tsunami events It could also when we are looking at normal engineering problems as they also occur in our daily lives exposures could be the processes leading to severe corrosion of infrastructure like Bridges We need to be able to describe these events and we do it in a probabilistic way Because that makes it possible for us to assess the risks, but that's only the first step We also need to assess the the damages which are induced by the exposures So the damages to the infrastructure the immediate loss of lives when we are dealing with earthquake events or tsunamis We need to have models Taking the exposures to the damages Now the the final step is That in some cases the damages the immediate damages they can they can lead to follow-up events Like what we saw when the World Trade Center collapsed of course The exposure was the event of a terrorist attack the immediate damages were the collapse of the of the of the building and the loss of lives of all people inside the buildings and the tremendous monetary losses associated with the contents of the buildings But the indirect consequences of that event where you can consider to be the damages to the surrounding buildings not only the two trade towers but also the the other parts of the World Trade Center the other many buildings which were completely destroyed and also the infrastructure in the area and Very importantly also the damages to the economics so there were very large economical consequences due to loss of business interruption of normal business possibilities and Also you saw the effect on the stock exchange market So these in a societal context are what we call indirect consequences and we need of course In order to be able to describe these there are lots and lots of uncertainties and to evaluate the risks we need to Also to be able to treat and describe these uncertainties and model them Okay, so that's tremendously important So we are really dealing with the problem of trying to establish models for these Uncertainties in order to support the decision-making Now if you look at one of the classical examples here from the domain of Switzerland Rockfall is one of the Events which are always returning every year. We have a couple of severe and many not so severe rockfall events and One of the one of the problems is engineers We have to deal with is of course the safety on the roadway network so here you see a car which is Well, it has been involved in a in a rockfall event And and what's one of the options is what we can do of course We can install all sorts of protection structures in order to reduce the risk of rockfall One of the ways we can reduce the risk is first of all We can try to secure the the rock so that the rocks are not falling the next step We can take is that we can install nets to catch the stones when they are falling and at the last So the last resort more or less is to build structures Like galleries which you have seen many many times. Maybe you're not always realizing what they are for But these structures are to protect you when you're driving on the roadways from falling rocks But when we are designing These engineering measures like the rockfall nets or the rockfall barriers the the galleries Then we need to have an idea on what we're dealing with We need to know the parameters which are important in order to design the structures And just to give you a feeling for that There are two phenomena when we are dealing with rockfall which are very important the first The first phenomena is the main is is the Is the release of the rock Up on the cliff So This phenomena is very uncertain Because you're basically you're you're you're you're standing there with this with this cliff which Is a very old thing and it's in many cases very inhomogeneous and the knowledge you have it is Is very sparse You can you can make tests you can drill holes in in the cliff but In reality There is a limit to the information you can you can get out of the cliff and it takes a lot of expertise Geologists expertise in order to evaluate evaluate come up with some models on How much volume would this cliff release Per year and what are the sizes of the rocks falling out of this cliff? So that's extremely important and we need to be able to assess that and And this here is is one of the typical representations So an exceedance probability curve for releases of different volumes from A certain location up on the cliff Now the next step is to model Okay, so we have a certain volume. It has been released now. How will that fall down? And in the end Of course, you might realize that the falling process is also Subject to a significant amount of uncertainty The rock is falling down. It's bouncing on the ground. It's jumping up again Maybe it's splitting into two or splitting into three. It's continuing to fall and in the end There will be a spatial distribution of the falling rocks at the location where we have the possibility to build some sort of protection Now we know we need to know this spatial distribution of falling rocks at the location and based on that We are able to derive for instance the joint probability density function of the energy and the volume at any given location And this now forms the basis for design of the structures which we can install to protect you Mind my handwriting Okay, sorry about that So this is one of the examples where You can very clearly see Where do these probability distributions come into the picture? Why are they important? And what I also wanted to say is that the engineering problems Which we're dealing with they are basically they are always Different, it's not like we are repeating everything like thousands of times any new engineering problem We have to we have to think we have to start with basics In order to be able to see what is needed in order to develop the models Which are required to support the decision-making and it's not like that there's one unique solution You can just look up into a book In order to solve these challenging problems You need to try to think what would be the nicest way of representing my model the nicest way say Providing the possibility for work with With as small as possible uncertainties utilizing the informations we have in terms of data To the maximum extent and of course any engineering model Should be verified So we cannot just think out engineering models and and postulate them without having some sort of a fixed point whereby we can evaluate the goodness of our engineering models So the validation of engineering models is important as mentioned the engineering problems Also those involving uncertainties are very often specific or even unique and being able to solve such problems requires basic tools Physical mathematical natural sciences human sciences and engineering of course We are trying to combine all these disciplines into the decision-making process Understanding these different disciplines also gives you an idea on What would be the appropriate way of modeling the problem in order to provide the answers which are needed in the decision context It's not only mathematics. It's not only physics. It's not only engineering It's trying to get an understanding from a holistic perspective and because Basically all problems When they are real problems, of course in some cases you can look up into a cookbook When you really don't need to think when it's something which really has been solved like a thousand of times You don't need to think too much but Of course Some of you will be doing this all of us are basically doing that sort of exercises but the good engineers Who are really in need outside ETH? So the engineering practice needs really the good engineers and the good engineers are those who are able to innovate solve problems Which have not already been solved 1000 times before So we need we need to be able to train innovation We need we need to be interested. We need to really want to try to find good ways of solving problems And that requires training. It's not something which comes for free You really need to train the ability to be innovative. It's not something you're born with That's a misunderstanding You you can be born open-minded That's clear. You can train being open-minded. You can try to remember yourself Okay, maybe we should try to look at problems from many different angles Instead of just looking at problems from the same angle every time And it takes training Training is important because it provides experience By training we start to realize patterns and it's basically the same thing as when you're playing chess or any other game Which is say more complex The more you train the more immediately You you are able to overview situation and this is because you recognize patterns So you recognize situations you have seen before You recognize The possible potential of tools which you have in your toolbox We talked about the toolbox the last time the toolbox is very important. You need you need to understand the tools, but training practice provides you provides you Really with the ability to try to see In what situation would I get the best benefit out of This combination of tools so you will be able to select to select the tools Appropriately to your problem Sometimes it's even possible To reformulate your problem so that they fit to your toolbox And this is also of great value to be able to do that So much for that Now coming back to the tools The last time was really an exercise of presenting some tools some Some basic understanding of of characteristics of Random variables, okay And what we looked at Were the expectation operator We looked at the variance operator and the corresponding characteristics We also looked and introduced the jointly distributed random variables Like we have in this case For discreetly distributed random variables Finally, we went over to Have a look at functions of random variables and we looked at one particular function namely the sum of two random variables And we looked at how could the probability density function for such a sum be established Knowing the joint probability density function of the two random variables We had a look at more general constellations like non-linear a non-linear function of one random variable And we were able Based On such functions to derive the probability density function for such random variables for the case Where we are dealing with continuous Distributed random variables and the very final thing was the case Where we were looking at functions of random variables y y are functions of random variables contained in the vector x to the functions g i and we assume that the The inverse relation also could be established. So the x's could be related to the to the y's to another set of function and also in this case And I realized I was very fast in the end of the last lecture But we will come back to this with a small example today We even for this relatively complex situation We were able to establish the joint probability density function of the random variable y's expressed In terms of the probability density function of the random variable x So these are tools And I want you to get some training With the tools because it provides you to pattern recognition It provides you with a very valuable toolbox to solve problems which in general may be specific in specific situations Now today We are again looking at the topic of random variables And as a part of that I want to introduce a central limit theorem I want to introduce a normal distribution And I want you to see also and get acquainted with the log normal distributions These are two very important probability distribution or cumulative distribution functions And then we will we will start looking at random processes random processes are opposite to random variables They describe phenomena which has a time dependency So the random variables we have we have talked about up until now Represent uncertainties Which do not change in time but now because many uncertainties actually have a significant Time dependency we also need to talk about that sort of uncertainties And we call those stochastic processes Random sequences binomial distributions and geometric distributions are what we will look at today and this is relatively simple Now the first insight here concerns the central limit theorem And the central limit theorem says that the the the cumulative distribution function for a sum of a number of random Influences or random variables approaches the normal distribution It's also sometimes called the Gaussian distribution as the number Becomes large so if you have a number Of random variables x1 x2 up to xn and you sum them up together Then the result of that sum namely y will become normal distributed as in Becomes large There's a couple of of conditions For this to be to be valid. So first of all n needs to be relatively large Secondly None of the individual terms in the sum should dominate the sum so it shouldn't be like x1 is Contributes with 90 percent to y and all the the other n minus 1 random variables contribute only with 10 percent in that case it does not hold But if these contributions are more or less of equal equal weight in the sum then it holds For the same reason I'm sure you're discussing the central limit theorem For the same reason the individual components in the sum They cannot be too strongly dependent If there was a strong dependency between the components of the sum it would simply have the effect That it in a sense it would reduce n So if you have a strong dependency between the random variables in effect That would correspond to a situation where you had independent random variables, but Where you did not have as many Okay, and then we come back to the first Requirement that n should be large Now there's another thing coming out of the central limit theorem Say the law of large numbers and that is that if n is really really big Then the uncertainty associated with y Goes to zero So y would become for a really large n it would become A deterministic variable There would be no uncertainty. So if you're summing up very very large numbers of of terms Each associated with uncertainty then in the end The uncertainty associated with the sum would reduce to zero That's a very nice property to keep in mind It gives good support now the normal Dent the normal probability density function is is given as it's written here You will see that many many times It's true an exponential function and you see the parameters Of the probability density function is the standard deviation And the mean value here you have the standard deviation again It's it's very straightforward to calculate and to program Now the cumulative distribution function Again the parameters sigma you see here and here Is simply defined through the integral of the density function Which is the normal definition of course for a cumulative distribution function But in this in this particular case this integral Is not very easy to solve Okay We can do it numerically Of course, we can do that but to find an analytical solution This is not easy, okay, and In general What you will you will be able to find is in software tools such as excel or any other spreadsheet There are some functions already incorporated which you can call if you need to calculate the cumulative distribution function of a normal distributed random variable And in those solutions they They have made some nice evaluation of this integral based on good approximations Here you see the shape of the density function And for this case the expected value Is zero And here you see the corresponding Probability the cumulative distribution function We already talked about that Oops Now as an illustration you have also in the lecture notes you have this small example Which concerns the repetition of measurements So if you're measuring some distance and You You're you're repeating measurements in order to measure the Entire distance then if And and every time you make one measurement There is an error associated with a measurement in the order of plus minus 0.5 millimeter And we assume that this measurement uncertainty Is uniformly distributed Now you you may imagine that we can measure distances at different lengths and depending on the lengths We need to measure in total we need to repeat these measurements and in the cases Where we have only one measurement two measurements four measurements and eight measurements What we can do is that we can we can we can establish The histograms corresponding to the distribution of the measurements So of course if we only make one measurement Then the measurement uncertainty is uniformly distributed as I said in the interval plus minus half a millimeter But if we repeat it Then the sum The sum of these two corresponding Measurement uncertainties You see it's already significantly different to a uniformly distribution. It's it's very different And if you make yet another two more two more measurements, so n is equal to four You see that now it really It already now has a very nice curved shape for n equal to eight You can you can start to recognize The shape the shape of the normal density function this thing here So very very fast Meaning for even for relatively small n You see that this normal this uniformly distributed random variable adding up On other uniformly distributed random variables Very very fastly converges towards a normal distributed random variable Now of course also using the result from the last lecture. We can also derive The shape so the function of the normal distribution By summing up Random variables during the convolution integral Then we have a new random variable then we can sum that up with a with a new uniformly distributed random variable If you like or any other distribution we can continue that And This process will also lead Successively To the development of the normal distribution density function so The normal distribution function is very frequently applied in engineering modeling When we can assume that the random phenomena is composed of a sum Of other random influences So this is already one physical argument Which we can we we can keep in in our back head as one of the tool in the in the toolboxes And we can recognize ah this problem is dominated It's by influences which are summed up influences which are uncertain Like for instance the traffic volume on on some major Highway here in switzerland. It's a contribution of inflowing traffic from many directions And in such cases it might be an idea to use a normal distribution To model the traffic flow Again, there can be other arguments Which Could lead to other choices for instance looking at the traffic flow If you're looking at a freeway and then you want to have the flow on the freeway then it's clear That the traffic flow cannot be negative Okay, but the normal distribution is defined in the interval from minus infinity to infinity So it would allow also for negative values And for that reason, well, it might not be the most appropriate Random variable to represent that uncertainty now That Has to be considered in every particular case. What makes sense? To use as a model now if if you If you have a linear combination Say a linear sum of Normal distributed random variables then of course also Due to the central limit theorem It it comes out automatically that the sum of normal distributed random variables is also normal distributed So the normal distribution is also said to be closed in regard to linear operations So a linear operation on a normal distributed random variable will also be a normal distributed random variable any linear operation The normal distribution can also result from other considerations uh, for instance Concerning the distribution of energy in an isolated system that relates to so the distribution of energy and the concept of entropy from the second law of thermodynamics, which says that that the the energy Distributed in a system will over time Tend to be distributed such as to maximize the entropy of the system That means it will reduce the energy available to produce work in a system That's the that's the second law of thermal thermodynamics And it's well, of course, there are some interesting features with this concept You see here in in the first illustrations up in the right corner You see two systems one system where we have a separation between high energy and low energy Particles then we and when we take up this the the separating wall in this container Then over time the low energy and the high energy particles they will they will mix together So it's the principle of mixing trying to Reduce say to maximize the potential energy in the system reduce the the energy available available for for work Another aspect which Which is related to this Is the aspect of time so if you have here on the left hand side You see some particles For instance gas molecules And these are the locations of these gas molecules And now you look you look at the at the same container But at a different time and you you see a different ordering of these particles in space Of course, you would assume that the picture on the right hand side Would be prior to the picture on the left hand side because on the left hand side You have this order on the right hand side you have order And this is implicitly the way we understand time Time goes from order to disorder and The concept here is that it is not necessarily so It is not necessarily so Based on on on on normal physics. We could also go the other way around so there's nothing in physics Which for instance could not unster a cup of coffee When we are mixing sugar and milk into a cup of coffee We are stirring it when you're mixing it But there's there's there's nothing in physics which say we cannot unster a cup of coffee So taking a cup of coffee already mixed with sugar and milk and then we would stir it and then We would basically get we would get some sugar and we would get some milk clearly separated from the coffee in physics. It's possible, but It's extremely unlikely So the probability that it happens is simply so small That we we we very rarely would observe this event So the concept of probability came in very very early in thermodynamics in order to to be able to analyze the energy in in isolated systems And one of the one of the guys Coming up with this concept was an austrian called bolsman And then later another physicist called maxville operated with this in the area of thermodynamics and the concept evolved into information Siri and There a very famous guy called shannon Showed that the the information entropy was actually a more General concept than than entropy in in thermodynamics It's very interesting and it's really relates to the probabilistic description of of information or say energies If you look at at At the accumulation of random movements like also arrows Can be considered as a special case of In in in a system like this i will try to To make this work somehow Should not not like that. Let's see what we can do Arrow Sometimes it works. Sometimes it doesn't work It's very probabilistic Well, I think I will do it this say the safe way And we'll just block this discard And then you can see here now you can see what is really going on Up in the in in the top up where the cursor is A small particle is released and then it hits one of these small Pins on its on its way falling down and every time it hits a pin It has an equal probability of going to the right or to the left And then you can see how the distance from the center Is distributed by by counting up Here in the bottom the number of balls falling down at the different locations And as you might already guess from the curve which is suggested There where the balls are piling up in a kind of histogram In the end it will become a normal distributed random Verbal so the distribution of the distance And this concept Was also very intensively studied In the in the early literature in in europe Concerning studies related to human sciences The statistics When you really go back You will you will realize that the statistics were mainly developed in the in the social and human sciences and then later the concepts were Were also utilized in the area of natural sciences So that's that's also quite interesting I I told you previously that the normal distribution is also called a gauss distribution because of the famous German mathematician called gauss But There were a number there actually there were a large number of of of very good mathematicians and statisticians Involved in the development of this distribution Before gauss even came into the picture and I think it's fair to say that demover Was was one of them Also a swiss guy called banuli Who actually started by studying law school and then Because he was interested in trying to evaluate the chances of having a fair outcome in a in a in a court trial He started to get interest in probabilities and statistics and So he he developed into becoming really a mathematician of very significant importance The banuli family Included nine brothers And they were all excellent scientists So there are some very famous and strong swiss traditions Okay, uh, let's let's stop this thing and let's uh, respect the let's respect the break But please feel out Please feel out this this this small note with the two type of questions. Okay Thank you very much ladies and and gentlemen that I hope that That you tried to fill out these things what I did notice is that Very many of you did not take one of these and I wonder if you could just pass them through again So those of you who did not fill it out. Maybe you can fill it out during the lecture. Can we do it like that? Thank you Now due to a minor technical problem We missed a couple of small exercises during the first lecture, but let's come back to that Very fast So here you are facing a joint probability density function of discreetly distributed random variables x and y and Let me see This is the start of the exercise We are looking at a village The traffic passing through the center is being counted by a device an automatic device You have seen these wires lying across the road sometimes they're used to count the number of vehicles driving On the road Now of course there is uncertainty associated with that type of traffic counting because sometimes Due to the dynamics of the vehicles Maybe one of the wheels is actually not really touching very Hardly the ground when it's passing one of these wires. So in there will be some miscounting also other effects Will contribute to the uncertainty Now what you can do is that you can go out and then you can you can you can place Some poor engineer next to the device and you can ask the engineer to count the The number of cars and note up the number of cars per time unit and then you can compare afterwards The visually counted vehicles with the Automatically counted vehicles and this is what you're looking at right now It's the joint probability density function of visually And automatically counted cars per time unit and we are looking at a time unit I think about one minute one minute intervals and you see that the number of cars coming on this road Is distributed between zero and four over one minute intervals now the question is What is the probability that the imprecise measurement Will show the actual number given that the actual number is three So the actual number is what is visually observed And this is what we have down here on the x-axis Those are the numbers which are correct These are the numbers which are associated with uncertainty Okay Now given that the actual number so the correct number is three What is the probability? That the automatic device will actually also give three And you have these choices Red means zero green means 0.07 And yellow means 0.7 Okay, thank you. I see a lot of green. I see A few yellow I see a few yellow now Let's try to analyze this situation We want the probability That y Is equal to three given that x is equal to three. This is what we were looking for The probability that the measurement Would come out with three given that the true number is equal to three This is a conditional probability and what you have here What you have here in the histogram Or in in the joint probability density function Is the joint probabilities of the different possible events And we want to calculate This probability and then we remember how we calculate conditional probabilities We calculate the intersection of the two events and then we divide it By the probability of the conditioning event and the conditioning event here Is the event that x is equal to three and in order to get the probability of x is equal to three We need to sum up Or the the different possibilities of the joint outcome of y So this is why we sum up for x equal to three We marginalize By summing up Over these probabilities and that comes out to 0.1 And in the end we have 0.7 Okay, now I've tried to go back to where we were For some strange reason the the small exercises I had prepared for you in this presentation. They were hidden It's possible to hide parts of your presentation But of course that can be a nice thing and it can be a very bad thing if you don't realize whether they're hidden or Not hidden I wonder how that could happen Either I wonder how that could happen Oh my god Anything can happen Okay Now as we So I just want to say One more thing about this entropy thing So the thing is with the with the principle of maximum entropy that the the maximum the the distribution of the energy in the particles of such an isolated systems will according to the principle of maximum entropy it will be Distributed according to a normal distribution This distribution is what is called a maximum entropy distribution And it's a very particular Kind of distribution it's it's it's a distribution which is a maximum entropy distribution for the situation where the The characteristics of the particles of the system Are unbounded In both directions So there's no bound downwards, there's no bound upwards So they can go from minus infinity to infinity in that case a maximum entropy distribution is a normal distribution Now let's look at another type of of Random phenomena if we are looking at the random variable y which is a product Not now a sum of many individual components, but a product of many random components x's Then what we can do And we many times encounter such products and we would like to be able to say something about the probability distribution of y What we can do is that we can take the logarithm on on both sides And this is what I've done here And then you see that on the right hand side By taking the logarithm we can we can rewrite the product as a sum And of course whether we are talking about a A sum of logarithms of random variables or whether we are talking about a sum of random variables then the central limit theorem holds so that means That the logarithm of y becomes normal distributed and This is what leads us to the to the log normal distribution. We say That when the logarithm of a random variable is normal distributed then the random variable itself is log normal distributed and the log normal distributed Random variable has a probability density function and a commutative distribution function which I've shown here You see it it looks somewhat similar to the normal probability density function Here we are not working with distribution Parameters directly related to the moments So we don't see the standard deviation We don't see the mean value of the random variable here in the definition of the Density function and the commutative distribution function but Here on this side you see how these parameters are related to the mean value and the standard deviation Okay So that means if you're starting out with knowledge on the first moments Mu and sigma Then you can calculate the parameters which are required to evaluate the density function and the commutative distribution function And this is often the case But of course if you know the parameters so the The lambda and the zeta from the very beginning then it's straightforward to calculate the density function and the commutative distribution function if We are dealing with a random variable Given as a product of log normal distributed random variables Y i to the power of a i and this is also a situation which we also encounter that we can write our random functions in this way Then it's very very nice because the product turns out to be a log normal distributed random variables A variable for which the parameters are easily achieved to the exponents And the parameters of the individual log normal random variables entering into the product So that's that's a very convenient result Which is nice to have in the back hit Is one of the tools so products of log normal distributed random variables Turn out to be log normal distributed And we have a nice way of calculating the parameters of these density functions and Commutative distribution functions try to remember that don't try to remember the expressions. It's not so important You can always find it in the book Yeah log normal distributions are very often applied to model uncertain phenomena, which cannot have negative values So the log normal density function or distributed random variable is only defined to have outcomes In the positive domain. There are special variants of the log normal distributed Random variable which can shift the distribution to the right or to the left and but It cannot have negative values, okay And therefore it's it's often used for also for modeling fatigue lives steel and concrete resistances daily river flows and phenomena like that And whenever as I said a random variable Is the result of a product of several random variables As an example, I just wanted to show you That we here we have the expressions for the mean value and the standard deviation related to the parameters Of the log normal density function Now what we often have is information about the mean value and the standard deviation and in this example here relating to the concrete Compressive strengths of of cubes from the laboratory We have evaluated that the standard deviation is 4.05. Let's imagine that And the mean value is equal to 32.677 Now based on this information What we know Is the coefficient of variation which I've calculated here And this information is very useful when we are dealing with log normal distributed random variables because Then it becomes easy to calculate their parameters So now we can take the ratio of this To this and you see that this term here Disappears when you take this ratio and in the end you only have this term Which is equal to the coefficient of variation And then you can calculate theta when you have theta you can calculate lambda Okay, this is the way we normally do this And then it becomes easy To calculate for instance the density function. We can also calculate the probability distribution function Here's the density function Using these data and now if we are worried about say the probability that we will have That we will have a value smaller than or equal to 25 mega Pascal Then we calculate this using the standard normal Community distribution function, which I will show you in just a second And then evaluated with this argument corresponding to what I showed you on the previous overhead To this argument here So we can now calculate probabilities of events which are of special interest to us for some reason I will just jump back a few overheads because Because of this example I showed you we missed one particular aspect of the lecture And that is that we are often working On standardized random variables and the standardized random variable is a random variable From which we have subtracted the mean value and we have normalized by the standard deviation This is what we call a standardized random variable Okay And what happens is that if we start here With a random variable, which is not standardized then the standardization is shifting the distribution Such that it has mean value zero Because we are subtracting the mean value And we are normalizing such that the standard deviation of the Standardized random variable becomes equal to one So a standardized random variable has mean value zero Standard deviation equal to one And when we are dealing with a standardized normal Distributed random variable then we call it a standard normal distributed random variable And then of course the density function reduces to this Because now the the standard deviation is equal to one and the mean value is equal to zero So the Normal density function looks like this and then we call it by this symbol here. This is a standard standard Normal density function. Yes, sorry How do you see that? Yes No But the shape you see here The this here corresponds to zero And another interesting phenomena Which if you are really interested you can try to investigate by function analysis Is that the location where you have turning gradient? so On on this curve of the density function In in the first in the first part here, you see the gradient will be rising The gradient is always rising Up to a certain value and then it starts decreasing again the location here Where you have turning gradient Corresponds to the location For the standard deviation equal to one So you can try to prove that mathematically, okay Now we call this a standard normal density function and correspondingly the cumulative distribution function For a standardized normal distributed random variable. We denote by this symbol here These are the standard Ways of writing The standard normal density function and cumulative probability Or distribution function Okay, now we can proceed There are many many many different Types of probability distribution functions To model random phenomena And here I just have taken some of those from from the lecture notes You see here the uniform distribution You already saw that the normal is there this the shifted lock normal is there The shifted exponential the gamma and then there's one I did not include which is the beta And there are many many more I mean these are only a selection of those Which we often work with And to be able to choose the right one to work with in a given type of problem Requires as I indicated from the very beginning Mathematical insight physical insight insight into the natural sciences engineering, etc, etc And it's something Which takes time to learn it's we cannot take all of that in this course Already today you learned two very important cases the normal distribution and the log normal When we have a sum of many influencing random phenomena, it turns out to be a good choice to use a normal When it's a product of many uncertain phenomena, then a good choice would be the log normal Okay, and I also told you That the normal distribution corresponds to the maximum entropy Distribution for isolated system And for this reason if we don't know anything about the phenomena If we have no prior knowledge Then it's very often argued that a good choice For a distribution function, which is not bounded downwards and which is not bounded upwards would be the normal distribution Because that would correspond to a maximum entropy situation the way nature behaves So if we don't know anything only that it's not bounded downwards or upwards Then a normal distribution function could be a good choice From that argument alone and the same applies for the north for the uniform distribution and the exponential distribution If we have with a bounded phenomena Upwards and downwards If we are dealing with a problem like that and we don't know anything then the uniform distribution is a maximum entropy distribution If we are dealing with a phenomena, which is not bounded upwards, but only bounded downwards Then the exponential distribution is a maximum entropy distribution and in in such situations If we don't know anything it would be An argument for choosing sorts A distribution function Now Small exercise Expectation operator, can you write the expectation of one divided by x As one divided by the expected value of x Yes, no, don't know Come on We need more cards This is a very very important thing and it's a very easy question I see I see A green fraction over there. I see a little yellow reddish over here with some greens also in between I think the majority is green And that is a very good choice And we try to remember that the instance inequality says that we cannot do that Okay, there's there's no equality between these two And whether we are getting a higher value than the true value or lower value depends on whether the function g of x Is convex or concave Actually, I think I'm going to skip that little exercise. I will Distribute you the solution. Okay, but I just skipped one in order to save some time Random quantities In many cases have a time dependency Which means That they can take on new values at new times or What we call new trials And for instance throwing dices you can you can you can consider that to be a time dependent phenomenon So we have new realizations every time we toss the dice Now if new realizations occur at discrete times And they have discrete values. So discrete realizations Then we call the random quantity a random sequence And a random sequence Could be events of failure So failure can be described as no failure or failure. So it's two discrete states And they may occur at discrete Times also it could also be traffic conditions If we are looking at random say time varying phenomena where new realizations occur Continuously in time. So at any given point in time they can take place And they take continuous values when they Come up with new realizations Then we call the random quantity a random process or stochastic process And examples of these are wind velocities Their variations over time Wave heights Snow levels Etc Well a sequence of experiments with only two possible outcomes Which are also mutually exclusive. So like failure or no failure Or getting a six by throwing a dice or not getting a six these are two discrete outcomes of of tossing a dice Of course You could also split it up into many other events But it's a discretization of the sample space of tossing dices getting a six or not getting a six this type of Phenomena is called a banuli trial I already already introduced you to banuli And typically the outcomes of banuli trials are Are Denoted successes or failures now If the probability of getting a success in one trial is constant over time and equal to p Then the probability density Of y namely the y successes in n trials This probability density function is given by this expression here Where This Thing here is what we also know as the binomial operator Which gives the possible number of different combinations of y into n Yeah And then of course what we can do is that we can we can integrate up this thing This thing here the density function we can integrate this up and then we get the probability or say the cumulative distribution function so now we have to sum up or All the possible y's Which we get here And this is what we call the binomial Community distribution function. This is not correct Community distribution function. Okay And how does it look? Well here I probably I plotted the probability density functions for two different choices of n And the probability of success So n is that in both figures equal to 5 and the probability of success is equal to 0.15 and 0.5 And you you see how We end up with Fundamentally different shapes of the density function But it's just to give you an impression that we are dealing with the density function here And we can calculate the densities of the different possible outcomes Now I want to come back to one of the issues we looked at at the last time when we were looking at tools Look at this small exercise We remember that we can establish the probability density function of a function of a random variable y is a function of a random variable x to this expression here where this gradient is is the first order partial derivative of the x With respect to y So let's see how easily that really works if we have a very simple functional relationship between y and x as an example y is equal to x squared Then we can make the relation x to the square root of y Then look in at this expression and Realizing that we need to take dx to dy We differentiate this thing here one time in regard to y And then we become Then we become this thing here Realizing that the square root of y Of y Is equal to y To the power of one half, okay Then it's very easy to differentiate this thing And then we can very easily write up The probability density function of y Expressed in terms of the probability density function of x So you see All you need to do is to get a little training In in doing these things And then you will be able to solve Many different cases Now a random sequence coming back The expected value and the variance of a binomial distributed random variable Is simply given by The expected value is equal to n times p n is the number of trials p is the probability of success in every trial So the expected value of the number of successes is equal to n times p And the variance of the number of successes Is equal to n times p multiplied by one minus p and you can You can easily when you look at the probability density function you can you can try to Convince yourself that this is correct Now the probability density function for the number of Independent trials before the first success is in engineering Interesting because it it could give us for instance the time until We will have failure the time until we will have an earthquake The time until we will have a flood And this is geometrically distributed In terms of the p of n We call this the geometric probability density function And the corresponding probability distribution function is established from this By summing up all the possible different outcomes So this is the geometric probability distribution You see here Now this is also interesting to analyze a little more in detail But before we do that, I want to show you the final example Relating to the tools the last time namely the thing which had to go a little faster in the end of the lecture Looking at the situation where we have a vector Of random variables which are defined through functions Of a vector of other random variables. So the y's are functions of x's and the functions Are the g i's And we assume that the Relation between the x's to the y's can also be established in terms of functions f i's Now the probability density function of the y's can be given in terms of the absolute value of the yacobi determinant Which can be calculated from the yacobi matrix given here and the probability density say the joint probability density function of the x's But look this this really I appreciate this looks really complex. Okay So therefore a small example is probably very good Look at the simple case where we have y1 is equal to x1 plus x2 And y2 is equal to x2 y Is given in terms of two very simple functions of x now Looking at this linear Set of equations we can easily come to the relation between the x's and the y's as well Now then we need to remember what is central here. We need to evaluate the acobi matrix Namely the quotients dx to dy And when we do that you need to study this side here And then you need to take the first order partial derivative of x1 To y1 and when you do that, then you'll get one here. This is just the constant it gets zero So the first Member of this acobi matrix is one Then the next term here dx1 to dy2 You see the first term is gets a zero. It's just a constant in here. We get a minus one Then in the next row we take dx2 to dy1 It's zero dx2 to dy2 is one In this way, we have the acobi matrix now. We need to take the determinant which is established By multiplying this and this subtracting this multiplied by this This is what we have here. So the acobi in this case is equal to one and So now it's all very easy the joint probability density function of the y's is given as one times the joint probability density function of the x's And now on the places of the x's we have substituted their values in terms of their y's So you see It looked very complicated, but for many functions it turns out to be very simple But you need some practice in order to do these operations Now I wanted to introduce the median Of the geometric distribution Because that provides the median you remember it's the probability that It corresponds to the probability of 0.5 That the values will be smaller than or larger than so you can say that the median Of the geometric distribution corresponds to the number of trials Which are necessary in order to have equal chances of Realizing an outcome or winning a play And if if the probability of of of getting a realization is equal to p Per time unit or per Toss Then the median is is immediately given Here though that corresponds to the probability of 0.5 and we can relate that to the probability To the cumulative probability Distribution of the number of trials until a success p is the probability of success n is the number of trials So if we want to evaluate how many number of trials are necessary in a game where the probability of winning is p In order to have a probability Of winning the game equal to 0.5 Then what we need to do is to Out of this expression To isolate in This is what we need And we can do that very easily Here looking at this expression here We need to isolate in so what we can do is that we can take the logarithm Here and we can take the logarithm over here And then we get this expression Now the logarithm of 0.5 is is approximately about Minus 0.7 So we end up with this expression here Now this is a little inconvenient because we still have p Somehow involves and I would I I would like to To isolate p a little So I can use the result that the logarithm of 1 minus p So you know that the logarithm the natural logarithm around 1 is very close to 0 And in the in the very Near range Of 1 We can get a very nice expression for the behavior of the function The logarithm of 1 minus p And there are some Some exact results in terms of Of sums Is this one here? That gives us the exact result only the problem is that it goes to infinity But if you look at this sum And you assume that we have small p so the probability of success is relatively small Then only the first few terms will contribute to the sum So we don't need to sum up to infinity In reality, it's a very good approximation for small p only to use for the first term And the first term is equal to minus p And then we get this relation here So n can be isolated as being equal to 0.7 divided by p This is the number of trials we need Now we can apply this result So for instance, if we want to have a 50 chance We had this exercise in in the first lecture If we want to have a 50 chance of getting a 6 in So Tossing dices How many torsos do we need? A 50 chance of getting a 6 In order to do that You have 0.7 multiplied by 1 divided by p which is equal to 6 4 torsos Okay, if you want to have a 50 chance of getting two 6s In in two throws What is the probability of getting a 6 in one throw? It's 1 divided by 6 The probability of of getting two 6s would be 1 divided by 36. So the probability p is equal to 1 divided by 36. So 0.7 multiplied by 1 divided By 1 the same divided by p is equal to 36 here And that gives you 25 torsos In order to have an equal chance of getting two 6s 25 torsos Now interesting also is that if we have if we Are concerned about the probability Of observing an earthquake of a certain magnitude at a certain location Then we can we can we can now deduce based on this result here the The number of years We would have to wait in order to have A 50 chance of seeing such an earthquake at a given location if we know the probability per annum that this earthquake would occur So if the annual probability of seeing this earthquake is equal to 0.001 Then 0.7 divided by p Corresponds to 0.7 multiplied by by 1000 So that would mean we would have to wait 700 years to have an equal probability So 0.5 probability of seeing this earthquake Please remember that the median is not the same as the expected value Yeah, and if you're we are dealing with the expected value of a geometrical distributed random variable then it's simply equal to 1 divided by p and Looking at the earthquake 1 divided by p Would be equal to 1 divided by 0.001 Which would correspond to 1000 years, okay, so That is also what we call the return period 1 divided by p the expected value of the time until An event will happen 1 divided by p And this is a common name not only for earthquakes, but for many other phenomena with a time variation The return period when torsing dies is the return period of getting a 6 would be 1 divided by 1 divided by 6 so it would be equal to 6 6 torsos so Thank you for your attention today. We managed to Cover the whole material today And I hope that many of you who did not fill out the sheets until the break have had a chance to fill out the sheets Here during the second lecture. I would like you to put down your answers here on the desk before you leave Thank you