 10 so say it again right right so so the question was in problem number five in this homework which is refers to problem number ten in chapter four part B says estimate the maximum number of new cases in any one week so the new cases infected right so it's basically looking at the infected population over the span of that week hold on in any one week right because because this is a weekly thing rate it's an updated weekly is that okay okay so be the maximum of all the iterations I mean all the values in of new cases so right so the difference Delta I whatever right well I mean at each iteration you know how you can count how many new cases there are you know is the number of infected I don't remember the exact so I is the number of infected sees the number of immune right yeah by the way those that model is kind of a again a fame not a famous but I mean it's original models for this kind of infectious disease population so right how do you account for the new cases count the number of infected that week subtract the number of infected the previous week a s times I that's the rate okay because then it subtracted the the number of of immune cases that the change in the number of immune humanized people so so I guess it makes sense to to look to look at that rate a times s times I right so s times I the product of s times I let's see any other questions yeah it doesn't help that actually skip to signing this problem back in chapter 4 I should have signed the problem chapter 4 so then it would have been much easier I guess but all right so let me say one thing so what so there isn't really anything new in chapter 6 that that we need to cover for you to work on the other four problems I believe one one of them involves so let's see it's 2021 25 and 26 so number 21 actually is asked for an implementing of the Runge Kata method and notice that there isn't much well so this is just an exercise in modifying that code for the you know instead of one line you got to put four or five lines but also what I'd like to point is you remember how an Oilers method we look at the kind of how things change if the step size age changes right if it gets smaller or if it gets larger right so one way it's kind of you're approximating the truth the true dynamics of a continuous system if you make age smaller if you make age larger of course you're going away from the continuous system and in turn you get this chaotic sometimes this chaotic behavior right period doubling and all this so I guess it's a it's the same question that would would or should or it could be posed for the other methods like Runge Kata method right so if H is instead of making made smaller and smaller to get a better approximation for the continuous dynamical system what happens as you make H larger I don't know if there's any particular question for you to to do that but if you're curious I think you could take any of the any of the other problems and instead of implementing Oilers method implement the Runge Kata and increase the age value of course you're going to see different things than you see in the India when you when you use Oilers method but you will still see chaotic behavior no matter what so so try I mean again you can try them even now and let me know by Wednesday if there are any issues with those problems okay also so why is I like to or even today I think if time permits I'd like to start talking about chapter 7 which which takes us in the realm of probability models so and basically stay with it sort of the rest of the semester except possibly move to I mean I certainly will talk about Markov chains which is a type of desk this could dynamical system that has probability kind of well it's probabilistic model but so so the nature of chapter 7 is that it's just sort of a review or sort of a rephrasing of things that you should have seen in other courses and how they kind of fit with what we've been talking about so before we start with chapter 7 I still like to talk about some of this chaotic behavior that is seen with we've seen certainly we've seen it in discrete dynamical systems and also I mentioned in continuous dynamical systems there are you know very relatively simple you know physical systems where this kind of sense sensitive dependence on initial condition is is observed with this I want to emphasize the fact that we don't have in all these models there's nothing probabilistic so there isn't I mean this model sort of ignore everything anything that's random okay so even in this Lawrence model for you know let's say whether prediction or whether so it's basically a three variables that are being observed in let's say the portion of the atmosphere and these variables represent let's say one of them says represents the rate at which convection rolls rotate x y represents the temperature difference between ascending a descending air currents and z represents deviation for linearity of the vertical temperature profile so so again it's not it's not exactly what I was saying last time that you talk about pressure temperature and humidity okay but the striking feature of this of this model is that and how easy it is to write it down and it basically I'm gonna use x y and z since I think last time I'm sorry like that so so it is it consists of terms that are simply either linear or quadratic right it's like predator prey type terms right and there's absolutely nothing nothing that's so this is a deterministic model right so no random effects are observed or you know even if there are in the model I mean they are ignored and we're just simply saying that at time zero the x of zero y of zero z of zero are known or are given and then this this dynamical system is deterministically projecting what the variables x y and z will do right still what I said last time is that it's done deterministically but it is very sensitive to the values of this thing so for practical purposes even if this are I mean this cannot be known as exact values right so so there's some you wouldn't call it random or it's not you know this model is not supposed to to take into account any random fluctuations of that initial condition it's just simply to solve these differential equations and tell you the behavior of the system and let's see so last time we had this and it's not p-plane of course right okay so I think it should be the same system well some of the parameters are renamed right like the like this is instead of a should be sigma instead of b should be r instead of r should be b so these are kind of flipped around okay so maybe maybe I'll change them just so we can a is sigma looks like sigma so I'm gonna leave it like that okay so and we remember that's sort of this dynamics okay so what do you do with every system well with every dynamical system what have we been doing with every dynamical system like this the first thing you do you do a steady state analysis and of course you study the equilibrium the stability of the equilibrium right so so this is simply a system of algebraic equations okay many of them I mean some of them are even linear excuse me zero which you can actually do it explicitly by hand okay although it's you know you run into that risk that it's it's not going to give you anything but it's actually written down in the book of the steps there are so so there are actually three equilibrium yeah thank you this is what I have here too yeah thank you okay so again this should not be an exercise in solving a system by hand because you know it's a nonlinear system so but you can you can see I mean x and y seems to be well and also have a class here right okay so in fact x looks like it has to be the same as y and then you have these two equations which you can solve for x y and z so the first the one equilibrium is obvious is 0 0 0 and there are these two other equilibria which I think differ just by well slightly so this is b r minus squared of b r minus 1 squared of b r minus 1 so that's x equals y and that's z right and there is a minus which would just then have a minus here okay so in the x yz put in the excuse me x yz in the space you have this this is one equilibrium and then there is there are two other equilibria 1 it's let's see one it should be kind of in the first octant let's see r I believe it's chosen to be 28 so that's that is certainly positive right well excuse me here than 1 okay so it would be something like heat like this and then sort of the symmetric with respect to the z-axis or let's yeah mm-hmm okay so kind of it's if this is the projection of x y plane and these two things are kind of mirror images okay with respect to the z-axis because you have three equilibria now there's no p-plane there's no phase phase plane here right it's phase space and then it's it's difficult to to say anything about the solutions so what you do is you the next thing to do is to study stability right so the stability I mean simply amounts to to do what linearizing around each equilibrium and so remember for continuous time dynamical system that's amounts to computing the Jacobian matrix right in this case is minus sigma sigma 0 r minus z minus why is it minus 1 there oh yeah because it's minus x minus y okay and then it's minus x what's the last one yx and minus b okay so this is just taking partial derivatives okay so it's a three by three matrix and you know you can certainly do this symbolically right or whichever way you want if it gets too complicated for for x y and z equals zero this is relatively easy right but for the others it's gonna be kind of hairy to find the eigenvalues right but you need the eigenvalues of this so at zero zero zero let's see the matrix a has three eigenvalues which are lambda 1 lambda 2 that's minus 11 minus square root of by the way this this values up here the the reason for this number is because of the choice of sigma r and b okay so I'm not actually writing well guess I guess we should we should try to see what the second values look in terms of sigma r and b but if you don't have any any kind of choice of sigma let's see I'm sorry so this is this is kind of a sensitivity with respect to one of them the r right so for the sigma and b there there are values that are being kind of fixed right so the point is that you want to you want to kind of choose one of the parameters and make a discussion based on the possible values for that parameter so you have to you have to choose the other ones right otherwise it would be kind of difficult to to make any you know conclusions if you had sigma and b and r in this in this in this eigenvalues okay so so what do we look when we compute the eigenvalues for this Jacobin matrix we look whether they are positive or real part is positive or negative right so let's see I think but certainly this if r is a positive number this is real right and this this is negative obviously this is negative but the problem is a lambda 2 could be positive right so if r is large enough right in fact r equals 1 looks like makes this lambda 2 0 so so the real part of lambda 2 well there's not even real part because this this are real so if r is positive or greater than 1 let me say greater than 1 then lambda 2 is as positive so this means that e 0 is unstable okay this conclusion is actually tells you very little about the whole system right it just says that if I start somewhere close to the origin with my initial conditions I'm not gonna approach that the zero solution right so we can start with something very close even well I don't know I'm just I'm just picking some value but you can see that it it it would be very hard to actually find initial conditions for which everything goes to zero right on the contrary almost everything you put for initial conditions will actually looks like eventually is going to do the same thing that you saw earlier in some some form or another right so and as I said last time that there is the better way of seeing this right in which I mean simply viewing the what is this simply viewing that the trace of this solution not necessarily the temporal dependence of the of the components but just the trace of the solution shows that every looks like every every initial condition well that one can try would actually eventually have this behavior now this is for this particular values for the for the parameters right but if you change the parameter so far is let's try one I don't know what one will do but you see you see the the change that it's a drastic change in the parameter right that's for sure but that kind of suppresses everything that was and does it look like it goes to zero maybe not but if I choose anything or that's like below 1 0.5 then this clearly should it shows that it actually approaches the origin right again this this and I'm choosing values that are very close to zero but if I choose the original values that are far away from zero for that change of the parameter I forgot which value with which initial condition was here I don't know you still see what that the solution goes to zero so so yes we knew that for R less than one all those dragon values are negative so this is asymptotically stable but that information alone doesn't tell you what happens with solutions that are start far away from it right so there's there's a lot you know a lot more that needs to be studied you know to draw those conclusions and again here we're just kind of you know experiment I'd say right you see even here I went really far away and I'm still getting to do that so it has to do with the dynamical system in three or more dimensions which is very peculiar feature I mean very very complex can be very complex okay but so the surprise was that there is a range of parameters that causes this kind of behavior to exist okay okay so again the study what do you think the other two equilibrium might do just from this picture will it be stable you you kind of see the other two equilibria are somewhere like where these two lobes are right and of course the origin and I rotated but it's it's somewhere here right so this certainly is is unstable what do you think is happening with these two they'll also be unstable right but again that alone doesn't even doesn't explain this so this this steady state and stability of the steady state is kind of the an epsilon step towards understanding the this kind of longtime behavior okay please there is there is no limits that's a good point we'll have to kind of compute those eigenvalues excuse me we have to compute those steady states and start close to those to see but you see this is not a slimming cycle like in the RLC of Runderpal so in other words you cannot expect the solution to just kind of circle around this thing it will still do the hold while you know the butterfly effect so so even if you start here in fact I'm pretty sure that if you start anywhere in a you know reasonable box you know in 3d you will always go towards this right so this is like an object that every is attract everything right but this is not a close the like like there's no solution that is start somewhere does a million things of this and then goes back to the same place right so you could zoom in I don't it's not a best way to do it but you could zoom in it's like the Saturn you know ring I mean you can the closer and closer you go the more and the more things you start seeing the interesting rate is that you actually visit the same region infinitely often right if you long infinitely amount of time but but yeah and so if you start inside of that region you will not only circle this low you'll do the same the same dance around both lobes now there are two two other things that I want to I mean there are there are since the discovery of this and by the way Lawrence made this discovery what in the 60s there's a fairly recent lecture that is videotaped and I wanted to find it but I couldn't I'll put the link here in which you can actually see him talk about this these things but last time I saw he actually died in 2008 so just you know year a little bit more than a year ago but since then there's been a huge amount of examples of this kind of behavior for in continuous time dynamical system so what is this called Rosler attractor so I put a link here to a scolopedia article and remember a scolopedia again it's kind of this this are actually peer-reviewed articles and the authors are some most of them are the the ones that actually originated so so this is actually written by Otto Rosler and look at the system that amazing it actually looks simpler than the Lawrence attractor I mean Lawrence says Lauren Lawrence model it is still not linear it has Z times X but actually the nonlinearity only occurs in one of the equations previously it was occurring in two of the three equations and this is what how the the attractor looks like and again for certain values of the parameter rates I don't know if you can see that okay so the values of the parameters are written down here okay so you can kind of look through this if you want again it has looks like it has only two fixed points right and what else I think both of them you know are unstable right and this is kind of the the geometry of that it's in 3d right so it's not it's something you can actually you know build or put your hand on you can try this with OD solve you can try this if you're if you're curious but but you will see you'll get this picture okay let's see so what do I want to say about this so okay I don't want to say much because it's very similar and in a way and it's actually even simpler to study but what I want to point is is there is actually very kind of natural system that one that one can you know play with and that's double pendulum so I put a link to actually the other course where we talk about this in more detail on this double pendulum so there are a bunch of there are few animations let's see it there's actually derivation of the equations of motion of a double pendulum so what you have is you have a pendulum right nonlinear pendulum and then attached to it another nonlinear pendulum so so this one is you know obviously it has how many degrees of freedom okay maybe so what I'm referring to how many variables you have to to state to I mean to identify to determine the state of the system for right you need this position and anger momentum I don't they called so to me that's right so how many I mean do you call us degrees of freedom I don't know maybe maybe in other fields you don't but so I'd say that has four degrees of freedom so it has you have four different four state variables right and this are the equations okay now what I'm saying nonlinear pendulum I'm not saying that the pendulum is like bent right I'm just saying that the motion is occurring not just near the equilibrium right so it it actually can be as you know as well as going around the clock for each of the pendulum right so for one single pendulum you saw the nonlinear equation was what it was theta double prime that was the angle Newton's law right was you have had in the right hand side something with sign of theta right that made it nonlinear and of course if that I was small sign of that I was approximate with that so it made it linear but so here you see a bunch of signs cosines cosine square of the differences right and also it involves the first derivative and this I believe ignores friction so even in the absence of friction you have terms with a momentum square with a momentum right with a velocity square okay well okay so this this is not actually working but I want to show you those those two equations and you probably have seen this well you you may have seen it actually even in reality come on but you can if you follow and by the way this is not plotting in the in the face space right it's not a four-dimensional plot it's only a plot of the theta 1 versus theta 2 but not even theta 1 versus 2 is is how the actual configuration of the pendulum looks like right but if you were to plot theta 1 versus theta 2 then it would still look like a messy right I'm predicted unpredictable motion right deterministic but right there's no there's no random factors that is causing like this this rotation or that you know that turn or that right so this is deterministic but unpredictable or sensitive to initial condition of course if you are to move it right just a little bit things are gonna oops think I don't know that's I'm not supposed to force it but this just seems to wow okay now it's has its mind of its own so there was another animation which I found but again this this you can oh yeah so this is actually the plot of the angle 1 versus angle 2 but it's not again it's not a face space plot right because face space is four-dimensional right so this picture is is what is a projection of of a four-dimensional curve right in the two dimensions now this doesn't actually look chaotic right why not because the initial conditions were very close to the equilibria so this is not but the moment you change that and I guess you clear the graph the moment you're kind of moving the nonlinear regime right well try it I mean there's movies I mean physics people like this a lot make they make it and then then you can see that but again keep in mind what chaotic means and of course am predictability is the one thing right so in other words you would you never kind of know how many I don't know revolution either will do at any time right but also the more important thing is that you change the initial condition by a little bit and it's gonna do something totally different okay yeah yeah I don't think there is you know there's there's only limited amount of mathematical theory that you can actually apply to predict the ranges of initial conditions where things will do things and with things won't do it so like like this one right I mean there are certain regions in the in the face space which is four-dimensional right there are some pockets where there's no chaotic behavior right but the point is that there is so I don't know the those what those pockets are I mean it's four-dimensional but exactly so that's and and and by the way you can actually do this again with your you know granted you have the time you can actually convince yourself that that you know this is not not like artifact I mean this is not just a fake right this is simply kind of plotting the solutions of those of those terrace of those of those valleys of theta and the velocities right at all times now are those the other's equations always matching the experiments no because in experiments you have friction you have even more complicated things right so in experiments things eventually we're gonna settle down right because your energy is lost through friction but here here is you know except that thing this is actually as close to reality as as you can as you can tell okay so the last thing was to actually show you that if you import this well export import in od solve so I wrote that code yeah well anybody can write it based on those equations okay but and you can go to od solve and load the system okay now it's it's a mess right but it's it's explicit right you can write it down so so this is and then you can you can plot this and again the point is to plot you see in this case it would really have to be x1 x2 you can try the things of course but okay and this is what comes out for this initial conditions which are not the most probably so you have to I think that's what you refer to that we actually try this with various initial conditions I think if we choose y1 and y2 equals 0 this means the initial velocity is 0 so you always just place the initial configuration at rest at some point but if you change those you can see this yeah yeah yeah it's like you're asking if you when you type on a on a URL in a in a browser and it's wrong is there something that tells you you're wrong no I mean I'm amazed how many times I mistyped Google and sometimes it just you know Google says it's not you know so I don't know I mean looks like typing is something you still realize on us yeah of course if you see a behavior that's totally different than what you expect well which this is but for for you know domestic for domesticated models where you know what you should expect then that's a way of checking but okay any questions on this so it's so this this is a this is a chaotic behavior of course in four dimensions but if there wasn't three dimensions I mean now it's like no no no surprise this is happening in more dimensions right still I think this is much more tangible than the Lawrence model right even though there was three-dimensional this is this is a pendulum you can actually manufacture in and and see okay so where's my so so so so anyway so that's that was kind of what I wanted to go through so similarly e plus e minus turn out to be stable but the main point the main point is the phase space analysis is key to longtime behavior in nonlinear systems okay three or more dimension especially especially okay I mean even in two dimensions even in two dimensions just looking at the at the state of states limit cycles it's it you cannot actually conclude what happens with every every solution a long-term behavior but it's a much it's much well the possibilities are much more restricted than in three dimensions or more okay so so again this is pretty much what what we can do at this point if you if you really want to dive into the like why this I mean how can you explain this and kind of a deeper at a deeper level I suggest as I said this last time I gave you that chapter from from this book and you can just see for the logistic equation how I forgot it was in chapter 3 logistic equation where you can actually write things explicitly even when you have chaotic behavior you can write the iteration explicitly to see why there is that kind of behavior or how can you explain this but it involves you know somewhat more complicated concepts from you know an analysis so it's a little bit beyond this this class and by the way there is a there's a little section in the same chapter 3 in this book following that discussion of chaotic behavior that talks about kind of an experimental setup where this chaotic behavior was observed in and in and I think a bacteria population or in some some sort of population model so it's not just that we discretize a continuous model for our convenience or for our numerical integration purposes and we observe this thing and it's wow right so so it actually happens in in you know in in biological in a population dynamics too so they're kind of interesting and then it has pointers to more you know more in depth studies of this okay so let's see I'm kind of ready to unless you want to talk a little bit more about this I'm ready to go to to chapter 7 start talking about probability how many of you had probability course I mean okay well they're supposed to have one so okay so we're not going to be using a lot of like a lot of hard concepts probabilistic concepts but models and so again most of you know things that we're going to be talking is it's going to be self-contained but you should try to make connections with what you've seen before and again what am I why are we adding this you know aspect to the till modeling course well simply because many many times when we make simplifying assumptions we ignore random you know random changes or random external effects on on our model so and so we until now we everything was deterministic right we said it's we know how the rate of change of certain quantity is depending on you know current state previous state whatever right it was it was very well it was sort of very explicit right so so let's see so in this chapter we're going to talk about two types one is discrete probability models and continuous probability model now the word discrete and continuous actually has a different meaning than well somewhat different meaning than until now until now we talked about discrete in time or continues in time right so if the if the variables depend or change continuously odd with respect to the time okay so so let's see what does it mean in this context discrete so we started discrete probability models well clearly the I mean that experiment that of the throwing die dice is on everybody's mind when we talk about discrete probability models so so let's just think about this experiment of throwing two dies and and record the outcome okay so possible outcomes are as follows you could have numbers 1 1 numbers 1 2 numbers up to 6 6 appearing on top right so so this forms a set right which you know for now and we're going to talk the sample space whoops you're supposed to look upside down now the sample space okay and it simply consists of I mean literally of this are of this values right so you can think of them as some discrete set okay so it's the screen set of possible outcomes well even even if it's not finite even if it's infinite it could it could still be discrete right the point is that it right but it so the question is you know what would be a continuous probability modeling you would have to have the sample space to be continuous right in some sense whether it's space or it's you know other things but it should be kind of you should be able to move in that space continuously right yeah yeah I'm thinking that it could also be uncountable or no I mean yeah probably not probably you want yeah you want to stay comfortable uncountable yeah of course you can have infinite uncountable but can you have an infinite uncountable set instead of be discrete I don't know well there are other types of uncountable sets but anyway it's a good it's a good thing to imagine that I can count this numbers right and I can list them countable set means you can list them even if it's infinite you can list it find a listing of that so so yeah so oh it's coming up let me save this before it's okay okay so okay so just just a little bit of terminology so I want to I want to kind of write this down so we again with your with your experience prior experience and we can we can speak the language so a set so an event a is just simply a subset of the sample space okay so again thinking of it as a finite or discrete sample space you know if I only take a subset of this and I call this a then this is actually called that I'm calling this an event right and I can give it a name and of course how many subsets I mean what kind of subsets can you have well you can have the empty set this would be the impossible event and the other extreme you could have the whole set right this would be certain event okay so what would be what is probability of an event so once you kind of say choose your favorite subset of this sample space you can assign a probability to to this event okay so so if every outcome in s has equal probability of occurring then the probability of an event a is simply given by account right so it's something we write p of a to be and that's the number of outcomes in a divided by the number of possible outcomes so if it is a finite set it could it would simply be the ratio of the size right the size of the of the of the subset a divided by the size of the whole set but again if if the set is infinite then you don't have that right and again it's important that is this count you know that each outcome in this set has the same probability or it's equally likely right so this is a determination you have to make prior to to talking about probability of an event you have to talk about probability of a of an outcome okay from occurring so alright so so those you probably know so for example what's the probability that the sum the two dies what's the plural is it die the to die is is it dice okay thank you is like eight right so when you say what's the probability that's that some event is occurring it actually means count the positive outcomes in that event or the favorable favorable event outcomes right so these are these are all the possibilities now I think there's one one issue here is like do we count two and six and six and two is two different outcomes and the answer is well I think it should be right because the two dies or have their own independence they're not touch each other so so you could one could be red one could be blue so so if the red comes to and the blue comes six that's a different event that if the red comes six and the blue comes to something like that right so so in that sense I have the probability of this event is going to be five over the total possible total possibly which is 36 by the way should we count four and four is two different ones no right so so you don't because that's just you know it's just one one outcome right so all of these are equally probably I mean equally likely to occur right each die is um is you know fair something like that okay so okay so the next thing which is kind of the most yes please I don't know I mean look these are these are questions that you have to ask yeah so again what order I'm just trying to die is if you have a blue or red die and order batters and you always count blue first then you're going to have two or two different combinations of four because order matters no look I'm not a problem it's okay so all I'm I mean all I'm thinking is it's just what I think it's true but I think it can be you know formalized to the answer is you know that's just one one event equally likely from the other ones okay but let's just I just want to introduce one more thing before we believe and there is random variables to me the the concept of random variables well of course this is kind of the key no probability right and with enough kind of exposure to it is is no longer anything mysterious should be anything mysterious right because it simply is a function so when I say that x is a random variable all I all mean is that x is called a random variable x is is simply a function with some properties with some additional properties which I'm not listening here I mean it has to be measurable so but but again if s is like a final or discrete set all of that is kind of is thrown out of the window it's just a simpler function so you make an assignment and by the way this is this is a real value random variable right so you assign to every outcome a real number right and think about s as being discrete so yes s is finite or countable or whatever discrete means right this is a think about is a discrete okay so for example here are a few examples so some of two dies that is a random ver a random variable so if I'm calling the two dies the red and the blue whatever right so the red and the blue each goes between 1 and 6 and by the way we oftentimes will use omega maybe or you know that's customary to use omega as a possible outcome so X of omega or X of RB which is as a sum of the two that is a random variable okay it's just a function okay so so X can take different values anywhere between 2 and 12 for this one right this for this example right each with different probability so when you know in this simplest cases in this simple example the question of what values can the random variable take says is it possible for instance that the random variable takes a value 8 well if that's happening that's an event right so looking at this is actually an event so the event that X is 8 is simply saying collect the outcomes in that sample space for which the value of the random variable is 8 right so this is a subset it's an event right and what's the probability of this event oftentimes we just ignore the accolades so it's just say what's the probability that this event occurs well it's just counting things and we said right it's five over 36 okay so in that sense we say that the random variable can take different values with with with certain with each with a certain probability okay so if you do so similarly can the can the for instance can the well that's probably that the random variable takes a value one well that's the probability of the end of the null event or of the impossible event and by that count just by the count this is zero right there's no favorable outcome so what you what do we do here we actually write well it's it's a kind of a midpoint to to do this for every single value but you can see that the value to there's gonna be some probability right so the event you know is when when both are one one and again that's one occurrence one over 36 and so forth right probably that X is three is now two over 36 because you can have one two and two one yeah yeah right so I mean I mean how do the random variables show up well you have to make an observable so it's whatever you observe be beyond just the outcomes so based on the outcomes you have to make a measurement right in this case would be the sum of those two things but whatever measurement you make you assign a value to that outcome right and that's like a one run of that experiment yeah so it's and it's it's it's again yeah so that's I think that's kind of well it's part of the modeling process you have to determine what you need to what are you observing what are you besides just the experiment right but but yeah it's kind of early on in the in the process certainly so what I'm doing here is simply I'm saying that I'm looking at all possible values in fact it doesn't even have to be like one two three could be two point five right you could say what's the probability that X takes that value obviously zero right so in this case there's going to be just a few a discrete number of of possible outcomes so yeah you're going to have a discrete values for X and those values are with 2 to 12 and with this probability so so this computation is simply building kind of the probability distribution if you want of that random variable and it's typically it's just kind of writing it on displaying it as a histogram so histogram is basically saying how many possible well so let me put it the probability distribution either way you want to look at it it basically says the values to be between two and 12 are the possible values with what probability well this is probably one over 36 the next one is two over 36 right and so forth I think you're going to have the maximum well you have to figure out basically what the maximum probability it's going to be I think when when you only have seven that's six over 36 so seven is kind of in the middle right the eight which we computed was five over 36 and then it comes back down to one over 36 right okay so so it's obviously it's not a well it's not any sort of curve like this but but you have this distribution of this probability distribution of this random variable right so this is sort of what we like to do with any random variable that we were given right so the distribution of X is given by these numbers which are the probability that X takes value K and K goes between two and well you could all exactly start with one right the only thing is when X is when K is one that's zero right okay so so there are other things but we're out of time so we're going to continue this on Wednesday if you are really bored because you've had this I would say you know you can look at this the chapter in the book there's not much I mean so we're very quickly go to continuous models talk about the density function and all this and I don't know I mean again you shouldn't be bored because you have to do on that the other homework which is okay but use this as a kind of a review if you want okay thank you