Thanks for joining us. Before we get started, I have to show you a safe harbor statement. You've probably seen these before. Basically, we're going to show you some stuff that's in research, some experimental stuff that is no promise that they will end up in product. So I've got to legally say that. All right. My name is Evan Atherton. I'm a senior research engineer for Autodesk Research in the AI and Robotics Lab. And today, my colleagues and I are going to share some of the work we've been doing at Autodesk to leverage AI. And both the tools we make today, as well as looking at how to use it to change the way designers and artists are going to work in the future. So I want to start by setting some high-level context about where we see the role of AI in the design process. So this super scientific and definitely accurate graph shows the complexity of our world over time. So as our world becomes more complex, the problems we need to solve become more complex. And therefore, the tools that we need to solve those problems also become more complex. So to illustrate the point, imagine you've never seen a hammer and you've never seen a chisel. If you walked into a room and saw someone doing this, I think it would take everyone in here probably 15 seconds to figure out what the problem was and what the tool does, right?  So a super simple problem, super simple tool. Now let's say you walked into a room and you saw a wood lathe. So the problem is similar but a little bit more complex. So you still want to remove some wood but you want to do it on this cylindrical object that's spinning and maybe you want a more ordinate pattern. So you think, all right, I can do this. Let me put the cylindrical thing and the spinny thing and turn it on and take a chisel to it but then maybe wood goes flying, chisel goes flying, hit someone in the head. It's kind of a mess. So you have to start understanding things like grain direction and spindle speed and things like that. So maybe the machine takes more maintenance. So the problem you're trying to solve has become more complex and again the tool that we've created to solve that problem also has become more complex but it's still within our ability as humans to grasp that. Like if any of us spent enough time in a room with a wood lathe, we'd sort of figure it out. But unfortunately our ability as humans to manage this changing complexity, this increasing complexity is actually finite. So our brains unfortunately don't benefit from Moore's law for instance. So to take our analogy home, let's look at a CNC machine. So this is a massively more complex machine. It could take months of training to become proficient at. If you've ever seen G code or had to write G code by hand, it's pretty gnarly. But humans have developed software techniques that manage a lot of that complexity for us that write the G code for us and control the machine for us. So that lets us as designers and engineers to leverage this complex tool to make more and more complex parts. So this space between these two curves here is where we think the really profound impact of AI is gonna be. So there will come a point on this curve beyond which we won't be able to solve problems without it. So what we wanna do is use AI to create tools that understand what we're trying to accomplish and can then augment our abilities so that together we can solve problems that may be otherwise impossible for either of us to do on our own. It's hard to see, but that's a VCR clock trying to be set. I feel like I'm barely old enough to still make that joke. So the way we think this is gonna happen is in three phases. So the first phase is what we're calling smart tools. So these are tools that on the surface don't really seem that much different to interact with. And then the tools we're already using. So there might be single push button automations. But on the back end, what's going on is wildly different. So these are tools that use mountains and mountains of data to train algorithms that help solve the problems for us. And they're able to solve problems that are tedious, super time consuming, and also problems that we just to this day haven't been able to solve. And what's cool about them is they can continue to collect data as we use them and refine themselves and get better and better. So this is actually the stage we're in right now. The next stage, which I think is close, is what we're calling intelligent assistance. So these are tools that will actually learn who we are and what we're trying to accomplish. And we'll be able to reason about the actions that we need to take to help our goals. So if you're an architect, for instance, and you need to lay out an office, you don't need to be pushing polygons around, right? That's not the problem. The problem is, how do I balance the needs and requirements of the people in the room? So why not have a tool that lets you balance the needs and requirements of the people in our room? So because of that, the types of interactions we're gonna have with our tools fundamentally has to start changing so that we can represent what our goals are. And finally, the third phase is when our tools become truly trusted collaborators. So this is when they'll not just understand our goals but they'll understand the context that we're working in. So they'll help us achieve our goals, but beyond that, they'll be able to give us insights to make sure that the goals we're trying to achieve are actually the right goals. And again, the experience of working with the tools in this stage is gonna change. In this stage, we're gonna have interactions with our machines that are much closer to how we interact with other humans than how we interact with the tools we have today. So with that, I'm gonna hand it off to Will, who's gonna share some of the work that's already making its way into product. Thank you, Evan. So I'm Will Harris, product manager here at Autodesk for the Flame family of products. And if you haven't heard of Flame, it is a what we call a non-linear editor or a video editing tool that is also able to do visual fax compositing as a procedural node graph and also has a 3D scene environment for motion graphics and 3D scene compositing and also does color grading. So it's kind of a suite of tools that you might use as kind of like a Mac Daddy version of Adobe Premiere or Final Cut Pro if you're familiar with those. So with that as our background, we have been doing a lot of solving of problems in the world of visual fax compositing and trying to do image segmentation. And so our AI, if you like, has been focused towards image processing. So you might think of, you might, your first thought for AI might be to do with Google predicting what you're gonna search or picking which images in this authentication image are traffic lights or crosswalks. But our approach was very specific to the problems that we face and I guess complexity that's outside our reach, like Evan was saying earlier, in compositing and in working with images. So this illustration really brings in the idea that in a CG render world, you have as well as a beauty image of your spaceships that you wanna composite, you also have AOVs or render passes, arbitrary output variables that are more informant, they're rich asset types if you like or they're things that can allow you to apply depth of field, defocus through Z depth and normals to potentially relight those spaceships post render. That richness is something where we have nothing on our back play, right? So there's no such level of asset. There's no AOV for my camera shot scene shot on Sony or Arri or traditional cameras today. What if we could bring and generate that little detail of asset for live action? So enter AI, what we really need is we sometimes joke AIVs. And here you have some examples of what our research led to. So applied research is probably specific to this issue of image processing. We wanted to be able to generate depth for a live action scene so that we could brighten the dark in the background, add light in the foreground and we wanted to be able to generate normals for human faces, but not CG rendered human faces. We want the equivalent asset for live action content. So the power of normals brought to working with live action content. So that's pretty cool. Let's just have a look at how an artist actually uses it. We want to in some ways try to do VFX at the speed of grading. It's what we sometimes call this. And you can see how through the kind of mechanism that you might associate with doing color grading, like keying and doing a track on a shot, we can come right in here and quickly generate normals. And then from that, we can use a widget like this to then change the lighting on a face. So this is really a powerful new way that an AI tool, a push button automaton, as we said earlier, has brought us. There's nothing to do. There's no special knowledge needed, but there's no way you could paint or build the complexity of a normal matte for a human face. It would take months to build this asset, right? And from it, so you can quickly generate a matte that then you could use for color adjustment. We could also be able to defocus the background or change the lighting in the background as well, based on depth. So in this second example, what we have actually is the depth. So doing global analysis for the scene, it doesn't have to be a traditional camera trackable scene. It can be blurry, handheld. It's using the intelligence of our training to know things like the skies typically at the top of the image and at the back. The foreground is typically closer at the lower part of the image. And there's blobs of things that are gonna move around in the mid-ground from the hundreds of thousands of images that we trained it with. But this quickly can lead you then to be able to relight the background, add light in the background. Something that would have been historically, okay, I have to rotoscope all these people by hand or with maybe some traditional pixel-based keying. But through having this richness, you can start doing the kind of thing that you would do if this was a CG-rendered scene. Okay. So just to summarize, we took our applied research. We've built these tensor graphs that have then been computed into algorithms or train models. That was a big process that was done in part of our development. We then packaged that into a tool inside the Flame software itself to allow our artists to get quick and easy access to that, to be able to do some of the cleverness of VFX, but in, for example, here, some of the color grading discipline. We think that that's a great symbiosis, as we said. And we would just like to let you know that we think this is just the beginning of AI tools for us. We're already starting to think about other specialized trainings like being able to extract sky, just sky. So when you understand depth, be able to key the sky in different types of environments. There's a lot of our customers are doing, you know, replacing overcast skies with nice, puffy clouds or removing clouds or adding clouds or matching one sky from one shot to another. So having an auto keyer, an automaton keyer for that, we think would be good, but we're looking for feedback into what other specializations would be good as well. Do you need to be able to isolate buildings? Do you need to be able to up-res your footage? That there's AI that can do pixel up-resing, right? And then the last thing to mention, if you're a larger company, if you have teams that would actually like to utilize this framework and have specialized data sets, excuse me, give us specialized data sets that we could then build you a way to recognize a certain kind of character. If you have a whole bunch of movies coming up with one type of character, that might be useful. And so this is a framework that we built that we think could be specialized. If you'd like to hear more about this, please join me tomorrow at 4.30. I've got a special guest, Stefan from A52, one of our most cool and innovative customers to show some of his real work from this. With that, let me hand you over to Seb. Sebastian, thank you very much. Thank you, Will. Cool stuff. So I'm Sebastian. I'm a senior software engineer at Autodesk and I work with products like 3ds Max and Maya. And today I'm gonna talk about this product that we call UVAI. So as a refresher, for those of you who don't know what UV mapping is. So traditionally, when you want to texture a 3D model, that texture is defined into the, and UV mapping is simply the process of establishing the correspondence between the 3D model and the 2D image. This process is usually done in three steps. In the first step, we're going to add seams to the model, therefore separating the model in different pieces that we call UV shells or UV islands. Then we're gonna unwrap each of those shells so that they're flat. And in the final step, we're gonna lay those shells out in the UV space, typically in one or more square UV pieces. This is an important problem because it's taking our customers 40 to 60% of the asset creation time. And it's been a top 10 customer request for Maya and 3ds Max for more than three years. And it turns out that we have pretty good tools to do automatic unwrapping and layout, but we don't have very good tools to do automatic seams. So let's take a step back to understand why do we need seams and why they're hard. Imagine you have a sphere where you wanna place a map of the world. You could add a single seam, a vertical seam, and you will get a nice square UV shell. But you will notice that around the equator, there's a lot of stretching, which is represented by red. And around the poles, you will find a lot of compression, which is represented by blue. Now you could also add more seams, therefore generating more shells, which will lead you to less distortion, which is why all these shells are mostly white. But then imagine having a texture of the world split like this. It might be hard to work on and hard to understand. There's more. So the first layout that I showed you is very efficient because it's taking all that square, it's almost leaving no gaps, whereas the second one is leaving many little gaps in between. And that's very important in video games, for example, where every megabyte counts. In addition to the geometric and optimization constraints, seams are also used to create semantic boundaries. And that's because these UV maps are going to be used by a texture artist down the pipeline, and it's useful if they can identify the parts of the model easily. So you can see here that there's a head shell, there is a torso, legs, arms, et cetera. So we've seen three main characteristics. So seams are used to reduce distortion, they define semantic boundaries, and they also influence an efficient layout. Unfortunately, when you add seams to a model, you're most likely going to add some visual artifacts around those seams. So it might be a little bit hard to see the visual artifact in the image to the right. I don't know if anyone can see it. Let's zoom in. So you can see that there's a discontinuity on both sides of the seam, and you might think, well, this is okay because I'm looking at this asset from afar. But on a video game where you can get close to this character or in a movie where you're filming this character from behind, this is going to break the illusion, which is why artists will most often play their seams in non-apparent places, such as under the arms or between the legs or on the back. Now, when we compare what an artist would generate where they would place their seams with our current auto-UV, auto-seam tools, we can see that our auto-seam tool is preserving a good distortion, but it's not preserving the semantic boundaries again. So it's very hard for me to see what parts of the model are represented where in the UV space. But artists have been doing this for years and years, and so maybe a geometric approach is not the best approach to produce automatic seams. What if instead we could leverage the knowledge of artists and use all these models that they have already created, feed that into a machine learning algorithm, and let it learn and let it produce seams? So this is exactly what we did for this early prototype. I wanna emphasize this is a very early prototype in Maya. And here the artist is just going to, with one click generate the seams, one click do the unwrapping and two clicks the layout. So first they click on AI seams, they wait for a little bit while the seams are calculated. Now that they get good seams, they're gonna select the model, unfold it, unfold all the shells, finally orient and layout. So in just three clicks, they got pretty decent UVs. How does this compare to what the artist had imagined originally? Well, we have a very similar distortion and a similar number of UV shells. So there's similar fragmentation. But more importantly, we're preserving again these semantic boundaries. So you can see that there's a shell for the head, there's a shell for the torso and the legs, there's two shells for the arms. But then you might be wondering, well, the layout there doesn't quite look as the layout from the artist, the ground truth layout. Yes, but that's because we're not predicting the unwrapping and we're not predicting the layout. Only the seams for now. And we're still working on perfecting this seam prediction, but we're already thinking of what's next for this project. So first of all, it's this idea of generalization. We have tested on a few categories like characters and vehicles, but there's so many more assets that are placed in a 3D world. There's vegetation, buildings, furniture, and we wanna support all of these assets in our system. There's style transfer, which is very similar to what Will mentioned at the end.  You wanna transfer your style. We want to enable this workflow. And inevitably, when you have an automatic system, there might be things that you don't like from the result and you might want to do some tweaks, but you want the system to remember those tweaks for the next time. And so that's the idea of personalization that we also wanna enable. And finally, one of the biggest complaints of machine learning is that it's very hard to control the output that it generates. So we want to give artists some guidance or allow artists to guide the process, rather. By, for example, adding initial seams and let the system take it from there or blocking certain regions that would usually have seams so that the system does not generate seams there. And with that, I wanna leave it back to Evan who's gonna talk about all the great work they do at Autodesk Research. Thank you. Hello again. So as promised, I'd like to share some of the work we've been doing in Autodesk Research. It's a little bit further out there, where we're really trying to explore what I mentioned earlier about leveraging AI to create completely new design paradigms and to help us start moving toward that vision of that trusted collaborator. The first project I wanna talk about is called Deep Form. And what we're trying to do with Deep Form is try to use machine learning to essentially parameterize the essence of an object. So we wanna learn what are the key features that make something an airplane or guitar or table. And the way we're doing that is by showing a network, in this case, using auto encoders, if you're interested, a bunch of pictures of airplanes, and then it's able to encode its airplane-ness, for lack of a better word, as a vector of numbers. So it's essentially compressing what it means to be an airplane or guitar or table into something, into a string of numbers that we can then represent as a 3D point cloud. So this is real output from that network. So in this case, what it thinks a laptop is. What it thinks a vase is. So it's really cool about doing this. When we get a basically a vector of numbers that can represent an object, we can then add and subtract objects the same way we add and subtract numbers. So this is what we call the Cheroplane. So here we're finding all of the objects that exist between a chair and an airplane. So that's kind of a cute example, but for something more practical, we can actually take two objects of a similar class and interpolate between those to find all sorts of design variations in between, perhaps variations that we might not have thought of. So here's real output from the network as we interpolate between a table on the left that has no base and then the table on the right that has the base. So you can see this computing. But we can actually go beyond just like a straight, linear interpolation between two objects and we can actually control individual features. So for a chair, a feature might be the style of the legs or the style of the back or whether or not you have armrests or the hole in the back. So again, here's some real outputs. This time we're taking a vase that has sort of a lip at the top, pretty curvy. We're subtracting a vase that doesn't have a lip at the top and then we're adding a vase that has straight sides and the result you see here is a vase that has sort of a lip at the top and straight sides. So what this approach ultimately lets us do is draw from the massive body of knowledge and design language that already exists. And that lets us push our designs further faster and it also lets us come up with, again, perhaps designs that we wouldn't have considered on our own. We can also use this approach to help build tools to help people learn. So we've been doing some experiments with Tinkercad. If you're not familiar with Tinkercad, it's an online web-based CAD tool that's free. So a lot of kids and hobbyists use it. And because of that, learning is a really important part of the Tinkercad platform. So it has millions of users who've generated a ton of public data that we've been able to mine. And essentially what we're doing is learning what types of models people are creating and then how they create them. So after a while, the algorithm is sort of learned to predict what people are trying to make and it's continually updating. So you can see down there, it's a continual update of what it thinks you're trying to make. In this case, it thinks the person wants to make a fidget spinner. And then it's also able to understand, if you're trying to make a certain object, it can understand the steps that you might need to take to get to that object and provide those suggestions to you so that you can then use those to learn, learn how you might design an object. We can see now just by adding the block on the back you get things you want to make an airplane. So switching gears a little bit. The last thing I want to share with you is something that we spend a lot of time on and that's simulation. So simulation is in all of our tools, super powerful, but it's really hard to use. And more than anything, simulations just take forever. They take a long time to set up and then they're really computationally expensive. So they can just take a long time to compute, which makes it really hard to iterate.  So this is true whether or not you're doing computational fluid dynamics of flow through a pipe or an explosion for visual effect. So here's an example of a finite element problem which is often used in mechanical product design to calculate the static stress on an object. So this is a couple of cylinders. On the left we have a boundary condition. So the left of the cylinder is attached to a wall. The green arrow here is basically a force vector. And what you see on the right here is we're using AI to approximate the simulation calculation. And we're able to do that in around four milliseconds. So what this enables us to do is adjust the design in real time and see the updated simulation result as we do that. So we can have a more interactive approach to designing and simulating at the same time. I'll also add that as geometry gets more complex, typical numerical approaches to simulation tend to increase the time it takes to calculate exponentially. This isn't true of these AI techniques. So as this geometry gets more complex it should stay computing on the order of the milliseconds. We've also done this with computational fluid dynamics.  So what you see here is actually again an approximation in this case of laminar flow over this object that's updating in real time. You can see 60 frames per second up there as the object is being manipulated. So these are pretty like hardcore engineering problems but they're essentially the same problems that we face with effect simulations and tools like Maya. You can also see here on the right just the amount of parameters an artist has to play with. So especially if you're a new artist you might not know the effect that some of these are gonna have so you can burn a lot of time up front trying to set up a simulation that won't end up giving you the result that you're looking for. So I'll leave you with sort of a what if. What if we could just show Maya a ton of simulations and then it could use all that information to essentially learn how to use itself and form some intuition so that it can do what artists do all the time which is take some piece of reference photography or footage and then use that intuition as form to perhaps suggest the bifrost graph you might need to get started to get you that simulation recreated in 3D that you can then manipulate. So again the goal is not to replace the artist rather it's to augment their abilities so that they can start further faster so they're not faced with sort of the blank page and they can spend more time on the artistry of it and less on the setting it up. I've never met an artist who just finished their sim early and was like chillin' right? And so with that these are really complicated problems. So I wanted to leave you with just three main challenges we see in leveraging AI for this type of work. The first of which is identifying problems suited to the technology and I mean this in two ways. One is often in pop culture AI is presented as something that is just like magical that you can sprinkle on something and just call it a day. But there are certain types of problems that are better suited to these AI techniques, machine learning techniques than others. And then the second way I mean this is that we wanna make sure that we're solving problems that our users care about and that can help them do their jobs better and faster. Second major challenge and this is true for any AI practitioner is getting good data. So in fact if you have a inferior neural network that's trained on more and better data it will often outperform superior networks that's trained on worse data. So the higher quality data the more of it you have the better your algorithms will be. Which finally brings me to we don't believe any one group will be able to solve some of the toughest challenges with these design problems using AI. We have ball and researchers but no one understands your workflows better than you do. No one has better data than the companies that are using our tools day in and day out to create magic on the screen. So we're looking to partner with groups and help people solve these really challenging problems. So if that sounds rad and you wanna talk to us about it maybe get some more information about what we can do shoot us an email at ai.autodesk.com come talk to us after and yeah that's it. I think we'll take a couple questions and thank you. Thank you so much it was really inspiring and when you talk about like airplane-ness of the object you can see the scene analyzing like this whole thing for example emotion-ness or composition correctness or some following some rules of the scene analyzing which is cause some render recommendation how to light this scene to make it more impression or neutral. So not the only object analyzing but scene in self or for example you can analyze the render logs and predict light setup or render time or something like this a more global level of render analyzing. So it looks like life analyzing itself but it's really very inspiring your presentation. Thank you. Appreciate it. No question. Any other questions? Go ahead. Okay yes looking farther ahead do you see a space for AI to collaborate with humans on things that are at more higher semantic levels like the story and the emotion that the artist is trying to communicate and this might get into automating the blocking of the characters the movement of the camera the composition of the camera shots. Do you see a need for AI assist at these higher storytelling levels? Absolutely and I will say that I actually have an intern right now as a PhD researcher who's in the lab right now trying to figure out how we can use machine learning to map cinematography styles. It's really the early stages of that. So I think when we start to talk about things like emotion that's a little bit harder but we can start by giving creatives tools so that they can explore these spaces a little better and maybe you do have a camera shot and maybe if trained on enough data we can start to see clusterings of camera movement that feels to us more anxious and then they could pick one of those camera sort of features and add it to their camera. So I absolutely think that's where we're going and where I'd like to see us head. We can all make it look like J.J. Abrams. Yeah basically. And that's not to say that I certainly don't want to replace like cinematographers and I don't think that's what a tool like this would do. But again just having an assistance that can help you out and explore these different types of shot styles for your specific question. Okay, thank you. Yeah. Hello. I was wondering for the UVAI project the type of models you were using to handle the 3D data. We're still sort of developing our models so we cannot talk too much about them yet. So unfortunately. Okay, thanks. Is there any kind of rough estimate when technology like this will become available for artists to use? Like just rough like is it coming maybe in a few years? Is it coming sooner than that? I think we showed like the three, you know, the three futures kind of so flame. Yeah, the stuff that's already shipping is what I was showing there. And then I guess yours is sort of an active research. Yeah, I think like. Looking. Yeah, Autodesk stuff aside, I think we're already starting to see certainly in the last year more papers being written on tools for creatives. Seeing more examples. I think there's one from Google that's music related. So music generators and things like that. So I think it's in some ways like we're already there for the further vision like the stuff that I showed like manipulating objects. I think we're a little bit further out. I always hesitate to give time ranges but five to 10 years, somewhere someone on that order. Very exciting. Thank you. We'll stick around if anyone has individual questions for us. One more question. Yeah, one more. We just can't see him very well. There was an interesting research about Vimeo. They took it in. They feed the comments about videos to the neural network and they got like recommendation how to make video better or how to make like most emotional impact of the spectacular. And it looks like it's kind of manipulation start to be. So what do you think about that with developing these tools? We can easily manipulate spectacular in some ways. It's like we can push him to feel something or to go in some direction. So what is your personal opinion? It's good that we're not in the social media business. Yeah. We just make creative tools for people to make cool stuff. Yeah. I think certainly for stuff like maybe nonlinear editors there's room for, if I have a ton of footage that we shot and I just want to see a quick assembly edit can we show that tool a script? Have it understand like sort of the feeling that's in that particular scene and then just cut together a quick trailer or cut together a quick sequence from that footage. Well, we've had customers request us grouping shots. So like all the shots of this actor that need the beauty work we wish we could scan our entire movie and find the 500 shots that have that person in. Would be another scenario we might use. This feels related but just this morning I think I read or saw an article that someone trained an AI to watch all of the Queer Eye for the Straight Guy episodes and then write an episode based on that. So you see that kind of stuff like happening already. Yeah, that's semantic at that point. Yeah. Cool, all right, no one's popping up so maybe we'll hang out. We got a couple minutes we can hang out and answer some questions if you're interested. Thanks for coming. Thanks for sticking around. Thank you.


NATO, trying to intimidate Russia, Moscow's top spy. Western speculation about a potential Russian attack on NATO is aimed only at frightening Moscow and forcing it to abandon its goals in the Ukraine conflict. Sergei Naryshkin, the head of Russia's Foreign Intelligence Service, has said. In an interview with Riya Novosti, Naryshkin described the warnings of numerous Western officials as part of an information war being waged against both Russia and the populations of their own countries. This type of warfare seeks to justify the West's existing aggression against Russia, he said. According to the Intel chief, by conducting hybrid actions, the West is trying to intimidate Moscow by raising the possibility of a full-scale direct conflict between Russia and NATO. The Westerners emphasise their belligerence, their mobilisation readiness, hoping to influence Russia so that it abandons its plans and goals of the special military operation he explained, adding that he considered the attempts futile. Naryshkin's remarks came after Admiral Rob Bauer, the chair of the NATO Military Committee, warned the US-led military bloc that it must brace for a major clash with Russia within the next 20 years, one which would require the mass mobilisation of civilians. While NATO Secretary General Jens Stoltenberg has said that the bloc sees no direct threat from Russia at the moment, German Defense Minister Boris Pistorius has warned that his country should be ready to repel a possible Russian offensive and urged NATO members to arm themselves for such an eventuality.


I am Dr. Sumit Talwar. I am the Chairman of Bariatric Surgery of Manipal Health and Prizes Private Limited. Regarding the diet after bariatric surgery let me start from after three months say about after two to three months most people will be on a normal or a near normal diet. Immediately after surgery for the first week or so you will be on liquids for the next two to three weeks that is up to a month you will be on thick soups or thin porridge consistency something which is which you can visualize of as a blended sort of diet then gradually you will progress to semi solids over the next month or so and finally in about two to three months you will be able to take full solid means like you used to take earlier.


We're not the commercial. Good afternoon. My name is Anne Pigram. This is Sharon Yates. And this is our story, A Promise Kept, Developing 21st Century Thinkers and Learners Through Culturally Responsive Teaching. My name is Sharon Yates. I'm a lifelong career educator with 41 years of dedication and service to one locality and one school district. However, my AHA moment and story began in 1965 after the Civil Rights Act of 1964 was passed. I was in ninth grade just beginning my high school journey. It was a time of racial upheaval and uneasiness across the country and Denver D. County was no different. Schools were under attack and were being forced to provide a plan to integrate public education. Denver D. County followed the freedom of choice as its means of complying with federal laws. Of course, only a handful of black students chose to go to Denwoodie High, the all-white high school, and no white students chose to attend Southside, the all-black school. But after being encouraged by my algebra teacher, I chose to leave the county's only secondary school for African-American students in high school. I left an educational environment that was familiar, comfortable, and where I related to my teachers and fellow classmates to go to where I was unfamiliar, uncomfortable, and yes, unwanted. Though there were other black students who chose freedom of choice, there were so few of us that often I was the only African-American student in the class. I did not speak up, volunteer, and always sit very quietly trying desperately to be unnoticed, unless the teacher singled me out which was always terrifying. What's the experience all bad? No. For I learned some things from this chapter of my life, and one of those lessons was to try and never put my students into a situation to always provide an instructional environment conducive to learning, one that was comfortable and that my students could relate, and a place where they could respect diversity. So I know firsthand the importance of a culturally responsive instructional environment. So you may ask what is a culturally responsive teaching? It is pedagogy that integrates a diverse curriculum and an instructional environment which reflects the student demographics. How does it manifest itself in the classroom for teachers? Here is the self-checklist in providing a culturally responsive educational environment. Know the cultural backgrounds of your students. And the lessons. Introduce lessons with what students already know from home, community and school. Find ways to bridge the social language and academic language. Allow opportunities for students to talk about elements of their cultures. Provide assignments that encourage students to share experiences. Ensure that classroom visuals are representative of cultural groups. Encourage interpersonal interactions and a sense of community within the classroom. Find ways to engage all students in each lesson. Vary the use of culturally connected instructional approaches such as storytelling, affirmation, imagery, call and response and mnemonics. So how does this keep the promise of developing community learners? A culturally responsive instructional environment provides the skills needed to be productive citizens and work and function in a global society. Those skills intentionally provide opportunities for communication, collaboration, critical thinking and creativity. How can stakeholders create a culturally responsive environment? This is where my story begins. It begins in the mid-1980s when I moved from teaching English in a rural school setting to teaching English in Fairfax County, a large suburban school district in Northern Virginia. I was afforded the opportunity to work with students of all ages, ranges, age ranges, abilities and diverse backgrounds which brought with it successes and opportunities for professional growth. However, as my classrooms became more ethnically diverse, I now had students who brought with them different types of language challenges. This led to my foray into working with a group of students who proved to be the most awarding of all of my experiences in my teaching career. These students were ESOL students, also known as English language learners. Armed with the advice that had been passed down to me by one of my mentor educators, don't allow ESOL students to stay where they are, bring them up to where you want them to be. I set out to expose these students to literary masterpieces and creative compositions. But I quickly learned that my ESOL students could not comprehend or even relate to my zeal in teaching them the intricacies and beauty of Shakespearean language. For in my attempts to bring them up to where I wanted them to be, I was overlooking the cultural experiences and backgrounds that they carried with them on a daily basis. Experiences that they brought into the classroom with them every day that we would meet. This then translated into a way of letting them know that I recognized the importance of the wealth of cultural experiences that made them who they were, along with celebrating the diverse backgrounds that were part of our classroom learning experiences. At that point I decided to use a variety of learning strategies at various times throughout the course, which allowed students to showcase their cultural and ethnic backgrounds to each other and also to me. One of the most effective of these strategies was a variation of a vision board, which I like to call if I can see it, I can be it. In creating this visual presentation students would be encouraged to not only show pictures and images of their future goals and accomplishments, but they were also asked to include something on the board that would give others a snapshot into their cultural or ethnic background. As a classroom teacher these images helped me to learn the importance of the quinceanera to the Spanish culture or the significance of Ramadan in the Feast of Eid to my Muslim students or to the importance of the centuries old Asian custom of animal birth signs much like our zodiac. I came to enjoy hearing stories from my native African students about their naming ceremonies or admiring the beautiful pictures of hijabs and saris that would be illustrated on these vision board presentations. As each student took a turn in sharing this important representation of themselves I noticed how each one became more comfortable and supportive of each other. As the year progressed with these vision board presentations afforded me the opportunity to frequently check in with the students using the board information as a conversation starter. Posting them in the classroom demonstrated the importance that I saw in their representations and eventually it helped me to make some real life connections in their boards and many of the academic topics that we covered in class. I also enjoyed sharing with students any attempts that I made in creating my own visual interpretations of topics that related to cultures that were of interest to me. As you can see from this example I recently had the opportunity to summarize my visit to Montreal Canada by putting together a representation of images along with words and phrases that are entirely in French. In doing this I created an environment where students view my way of connecting to a culture or a language that is not inherently part of my background but then be used to make universal connections which can extend far beyond the classroom. So in conclusion our story ends with a promise kept developing for our first century thinkers and learners. However, this could be the beginning of your story and keeping the promise. How can the information presented through this brief presentation apply to you as a promise keeper of public education reflect on your role as a stakeholder in education and briefly share a statement or thought with your shoulder partner as you talk about maybe some of the important points that we have brought forward. It's just the importance of recognizing cultural teaching as an important component of 21st century learning. How does this connect to you? Think about it, share about it Ann and I are going to come down in the audience and we'd just like for you all to just talk a little bit about some of the points that we have presented to you this afternoon about cultural responsive teaching. I want to keep that going. We would like to leave you this is the official end right now. We would like to leave you with a quote by Johann Goethe which says there are only two lasting bequests we can hope to give our children one of these roots the other wings. Thank you so much.


मैं पूजाशर्मा योर इंगलेशाड्य। मेंगवालेगेगेगेटर लेक्कामस्योगेटर। में जद खोगा आज करनेवालेगेगे एक पूम आप यह ठा गद ली गुड देखा रहे हैं जो जो अपना वीगार करते हैं भारष्च करते हैं हम प्यार करते है, बारिष की उन बूंदों से, उस खृष्बूं से, सब चीजों से. लेकिन यहां हम आपने व्यूज नहीं बताएंगे. हर विया गोईं तु रीड अब दे वोईस अप दे रें. आप आप रें वोईस तु टेल आस. क्या है रें की वोईस? आब आप सुछ रोगे रें की वोईस तो होती नहीं है. रें का तो साँंड होता है. बर दाट्स दी में इंबाज्मेंद है, इन देस पूर्म. आप देस पूर्म के लिए आप बुक होंबिल और होंबिल की जो बुक है, प्री वोईस का मुत्टा हो ता ये बच्चा कोई पतिकलर आप ने राविंसकीम नहीं लिया है. इप ने जो प्तिकलर राविंस कीम को फोलो नहीं किया है, प्री लिए जो मन में आवं रेन लिखते गया है.  Poem क्या कहती है? And who are two? Said I to the soft falling shower, which strange to tell gave me an answer as here translated. I am the poem of earth, said the voice of the rain. Eternal I rise, impalpable out of the land and the bottomless sea. There are so many poetic devices used here, I'll let you know one by one. Look, what is written here first? Who are two? Who are two is said, Who are you? Two were used first for you. So here the very first question is asked, Who are you? Who are you? Said I, I said, Poet says, I asked someone, Who are you? To the soft falling shower, who asked, the soft falling showers, I asked you, Who are you? Which strange to tell gave me an answer. And I was not expected to get an answer. But strangely I got the answer. I got the answer. And I translated it here. If the answer is asked, what was the answer given by? So this was the answer that was translated here. And the answer says that I am the poem of earth. So who said this? This rain said that rain told that I am the poem of earth. This is the poem of earth. Now when we talk about poem, poem are the things which are well described, which has beautiful words, which has different embagement here. So all these things are very beautiful, you know. So who said that I am the poem of earth? Rain said that I am the poem of earth. Sent the voice of the rain. Now voice of the rain, first of all it is a personification. Because voice, as you know personification, what is personification when we give man trade, a human beings trade to any other object, so it is called voice of human beings. But here rain voice is saying, sound is not saying, that means it is a personification used here. Eternal I rise, impalpable out of the land and then bottomless sea. Bottomless sea is what? Bottomless sea is here, which tells that there is no end to it. Means there is no bottomless, it goes to the end. So it is called hyperbole. It is called as hyperbole. Someone might say hyperbole, but hyperbole is not the correct one. It is hyperbole. So in these lines we got to know that the poet asked a question to the rain and the rain give also the answer. So the answer is also given by the rain as well. And here what are the poetic devices? I told you everything. Moving ahead. Upward to heaven, vaguely formed all together, changed and yet the same. I descend to live the trots, atomies, dust layers of the globe and all that in them without me were seeds only, latent unborn. Now again here carry forward these lines by the rain. Rain says that these are the upward to heaven. That means he goes towards heaven. Vents means when, whenever when, vaguely formed all together, changed and yet the same. When all the things together are formed they are the same, but they are also changed. So these are the lines which say simply about the rain. It tells us about that how it is explained here. That the poet tells that it rises upwards towards the sky. Upward to heaven means rain rises towards sky. Rain rises towards where it goes. Heaven, heaven means it rises towards sky. Vents vaguely formed what is vaguely formed means vaguely means not clearly. Vents were vaguely formed vaguely means clearly not clearly sorry unclear, vaguely means unclear we formed all together changed and still the same. I descend to live the trots, atomies, dust layers of the globe here the eye is used for eye is used for the rain. So here it also says that although it gets changed it still remains the same. The rain comes down to wash the draught. Draught means bar. So it says that here sorry, it is dry. I am sorry. So draught means dry. So it comes to remove the draught. Rain removes the draught. It removes the draught completely. Then it removes what? It washes even dust and small particles as well as and provide what? Provide water. To what? To the latent and unborn. Latent means it is hidden. Means the seeds. If we talk about the water cycle. If we talk about the things which are fully dependent on rain. So you talk about seeds. So the seeds which we have sewed. After that how will they grow? How will they grow? When will they get rain? When will they get rain? So rain says that because of me the draught goes away. The dust particles are gone. I also vanish. It gets removed. As well as all that in them without me. Without me, if I talk about the seeds. The seeds will remain hidden. They will remain unborn. If only I come. Okay? This is all about that. Moving ahead. And forever by day and night I give back life to my own origin. And always I originate my life back to start to end. Means rain starts and ends. So this is their cycle. And make pure and beautify it. Along with that what I do. I make everything pure and beautify it. When there is rain. Everything is clear. And you feel it. Everything is beautiful. For song issuing from its birth place after fulfillment wandering. Rect or unragged. Which is like touched or untouched. Duly with love returns. Duly with love returns. So when we talk about this part. When we talk about this part. It means we can call rect or unragged. Cured or uncaired. So forever by day and night I give back life to my own origin. I give back life to my own origin. And make pure and beautify it. These are written in brackets. These are written in brackets. It can be written as the above lines. They are not written like this. Because this thing is a little more interesting . The hidden meanings which the poet wants to deliver to us. These are the words which it says. It says for song issuing from its birth place. That I am the poem of the earth. That song which is issuing from its birth place. Where it is born. Where it is beginning. After fulfillment. It is wandering. Wandering without care. Wandering from here. With love returns. The love which we give we get it in return. These are the lines which are written here and I hope that you get it very well. It is a life cycle for itself again. Life cycle is preparing for its life cycle. Very important part which is written here. Life cycle. This is the first line for this paragraph. So the earth tells about the life cycle. What did it tell about the seeds? It told about the water cycle. How essential the water cycle is for the earth and for the seeds. So there are two cycles here. Moving ahead. So this is the end part here. Where we can see how seeds turns and flourishes and how they grow. If only the rain touches it. We know that moving ahead. We will talk about the rhyming scheme. There was a small poem. There was a lovely poem. I hope you didn't have much to think about it. You have understood it with love. This is the voice of the rain. This is the voice. It is said that this is the voice of the rain. Because in this poem poet tells us about the songs originated from the heart of the singer. Travels across to fulfill the aim of love for the singer only. Okay. There is no rhyming scheme. As I have already told you. These were written in free words. Then poetic devices. Personification. Metaphor means metaphor in which you compare quality or trait of a thing to the other directly without using like and as. Hyperbole use I told you already and imagery means what? Visualize things. Like we can see how rain is told here. Our mind creates an imagery. Our eyes create an imagery. So for that reason imagery. When you can use imagery through any lines through any poetic lines you can visualize it. You can create a picture of it. So that is called as imagery. So you have to clear all this. Moving ahead. So this is the idea of this poem. So write it like this. The poem the first of the rain is by Walt Whitman and it signifies the eternal role that the rain plays in nurturing. Nurture. Quenching and purifying the various elements of earth. Purify. Already the words which we read these are the only words which are going to help you to write your answers. So you nurture, quench and purify what the elements of the earth which is really important to write in your answer. If you get any question and you are writing this then atleast you will get some marks. So always go for the theme part. If you know the theme then you can answer it very well. Moving ahead with questions. I hope that you liked it. How this poem was told. It is a lovely poem. If you like rain then you will understand it very well. The second thing is if still you haven't filled that form then first fill that form. You can see it. Questions I hope. Question is the poem. The voice of the rain is a conversation between between poet and rain between poet and mountains. Rain and trees and birds and rain. I'll let you do this part. Only A is the correct one. You can eliminate it. Elimination is the best method. You have eliminated it. You will get the rest. Elimination is the best one for you. Next is what this wrecked or unrecked means. I am not going to tell you here. You are going to tell me in the comment section. Enrichment or no enrichment cared or not cared for to purify or not to wash or not to wash. I will tell you in the comment section. What hyper believer was used in the poem. What was hyper believer. I am the poem of soft falling shower bottomless sea and voice of the rain. I have told you the answer. I have explained one word for you. I have told you poetic devices and meaning. There is no question. I am not giving assurance. There is no question of poet or poem. I have told you poet. I have told you poetic devices and meaning and central theme. There is no question. Now you have to work. You have to do your work. That means you have to go through by. You have to see it 2-3 times. If you have not come once. And then make your own short notes. Make your own short notes. I don't think it will be necessary. If you have to write so much. This is pdf. You can write it by your own. You will be helpful. You will not be helpful by taking screenshots. We also study today. Before that we make our short notes. I have to go through these lines and I have to tell about these. When you do this. It helps you a lot. When you go for your further things in life. So this question is completed. What does? To what does Whitman compare? Whom does Whitman compare? The formation of rain and why? Why does he compare? So Walt Whitman compares rain to a song. Whom did he compare? The voice of the rain. The song of the rain. The rain forms in the sea and the earth. It returns with love just as a song. Issues from the singer and returns to him with love from the listeners. I told you this last time. When the poem ended. That it is like a song. When the singer sings. Then the listener listens to it. And reaches back. So this is the whole process. That love just is as a song. Issues from the singer. How does it reach them? Like you attend a music concert. And Rijit Singh is singing a song. He stops and we sing that song. He feels so good. That he is getting love from me. He remembers every word, every lining. So that is the way here we are talking about. Next question. There is a parallel drawn. Drawn between rain and the music in the poem. It is told about rain and music. Rain and music are similar. Because they return to the place of their origin. Because the way music goes back to the singer. Through the listener. The same way rain is its origin. It starts and ends there. So it goes to the origin of their place. And spreads happiness all around. Rain returns to the earth. Where does rain come from? Then it comes to earth. It brings new life. Like they originated seeds. And happiness spreads. The pleasure created by music. Enriched the singers with love. The admiration of the listener. This is the same part. So here we are done with this poem. This poem is the voice of the rain. Always remember, don't get confused. You must have read this poem in class 9. Don't get confused between the poet. Don't get confused between the words. The linings of them. Just I want to say it again. That you must have given this in the description box. That you have given in the comment box. A google form. Click on that google form. Fill it with google form. There is no problem. It's just to bring us together. We have been asked to bring one. The questions are simple. Like what is your name? What is your class? What is your stream? What is your email id? Do you have any questions from your teacher? Like that. You will be able to write your answer. And definitely submit with before. That you should go like this. And after submission, do let us know in the comment section. That we have filled the google form. Ok. I will check it out. Definitely I will check it out. And lastly, do like it and share this session with your friends. Bye bye. Have a great day ahead. And have a smile.


While it's official, the 2018 Minnesota State Fair has set a new overall attendance record. More than two million people visited the Great American Get Together, which concluded its 12-day run on Labor Day. While reaching that benchmark is a huge milestone, state fire officials say, state fair officials say, there's no magic number that's too much for the fair. So it's not really possible to know at this point. We have so many different gates and not all those people are here at one time. So at this time we feel pretty good that that that number is fine and safe and people are still here enjoying themselves. The fair set an all-time single-day attendance record on Saturday with more than 270,000 visitors. If you enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland PBS.


Now, we looked at some discrete probability distributions last time, in particular we looked at some properties of the binomial distribution and then I pointed out that the binomial distribution went over into the Poisson distribution in a certain limit well defined limit. We also looked at simple physical example of the binomial distribution which is density fluctuations number density fluctuations number fluctuations in an ideal gas classical ideal gas okay. If you recall the formula we had for the binomial distribution was P of n was n n n is n where n runs from 0 up to capital N here that was a binomial distribution and this is the distribution of a set of Bernoulli trials n Bernoulli trials capital N Bernoulli trials each with the probability little p of success and then I pointed out that if you took this limit in which by the way the average value of n was equal to n times p and if you took a limit in which n tends to infinity and p tends to 0 such that this number is finite at some value mu then the probability distribution tends in this limit n tends to infinity p tends to 0 n p tends to some value mu some constant value mu this tends to the Poisson distribution which is e to the minus mu mu to the n over n factorial and the mean value remained of course at mu and this distribution now n has the sample space 0 1 all the way up all the way to infinity alright. So we needed a simple example of this distribution and many many about Poisson distributions abound in nature we will come across many many Poisson distributions Poisson processes in fact as we go along but for the moment let me point out that if you did this elementary nuclear physics example of taking a large sample of a radioactive sample and asking what is the average number of decays that take place nuclei that decay in some given in a at till some in some given interval of time you discover it increases linearly with the time with some mean decay rate lambda. So you discover that the number is distributed exactly as in a Poisson distribution with an average mu which is equal to lambda t where lambda is the is a mean rate of decays so that is a classic example of a Poisson process. I also said I give you an example based on a number statistics exactly as in the case of the classical ideal gas and this is afforded not by black body radiation which had a geometric number distribution of photons if you recall but rather by coherent light. So ideal single mode that is one single frequency one given wave number wave vector one state of polarization if you look at ideal single mode laser light it turns out that the number of photons in this radiation field is described by a Poisson distribution once again precisely a Poisson distribution for that let me go back a couple of steps and do a little bit of quantum mechanics to tell you where this emerges where this comes from at least in the kind of non rigorous way and this has to do with the idea of coherent states in quantum optics such a state is described by a state which is expandable in a basis set which comprises photon specific photon number states and if I call the state n a state of the radiation field in which there are n photons exactly n photons of a given frequency and a given state of polarization then the so-called coherent state is built up of a superposition of these states and it looks like this it looks like a summation from n equal to 0 to infinity e to the minus half modulus alpha squared I will explain what these symbols mean alpha to the power n over square root of n factorial where alpha is any complex number. So you give me an arbitrary complex number alpha of finite modulus and I construct this superposition of number states of specific numbers and this state here is denoted by symbol alpha label by this complex number alpha as you can see when you change alpha you get a different state and so on and it is called a coherent state. Now there are technical reasons as to why it is called a coherent state what is so specific what is so special about this state and so on you will not get into that at the moment here because it is not the focus is not on quantum optics but I merely want to point out that in this coherent state it is not very difficult to show that you get precisely a Poisson distribution for the number because after all if you want to ask what is the probability amplitude that in this state alpha you have precisely some fixed integer number say r photons that is given by this quantity here or an alpha and I presume those of you who are familiar with quantum mechanics will recognize this as a scalar product it is a complex number in general and it is a probability amplitude that in the state alpha you have exactly r photons okay and when you do that and you use the fact that so put that in here and this is equal to e to the minus mod half alpha squared summation n equal to 0 to infinity alpha to the n over root n factorial and then you have an r with an n but you see these number states are all orthonormal to each other they form an orthogonal basis the orthogonal generality of the basis implies that orthonormality of the basis implies that this quantity is a chronicle delta it is equal to 0 if n equal to r and it is equal to unity if n is 0 if n is not equal to r and it is unity if n equal to r just a chronicle delta which helps you immediately to write down what the sum is because the sum fires only when n is equal to r and therefore this becomes equal to e to the minus half mod alpha squared alpha to the power r over square root of r factorial that is the probability amplitude and the rule of quantum mechanics is if you give me the probability amplitude for something to happen the actual probability is not a complex number it is the modular squared of this complex amplitude and therefore the probability that in this state you have r photons is given by this quantity r alpha whole squared and that is easy to figure out because you get another such factor here so it is e to the minus mod alpha whole squared and then you have since you want the complex conjugate is alpha on the left and r on the right is the complex conjugate of this number here this is a real number that is a real number this is a complex number so you have to do alpha star to the power r that makes it mod alpha squared to the power r divided by r factorial because there are two of these factors the square root goes away and of course you immediately recognize that this is a Poisson distribution with mean value equal to mod alpha squared right so the photon number distribution in ideal single mode lays a light of a given state of polarization and frequency is a Poisson distribution very drastically different from the geometric distribution that we had earlier because we know what the variance of a Poisson distribution is we know that if this is the distribution we know that the mean value n equal to mu and we know that the variance of n is also equal to mu the mean is equal to the variance that is fairly easy to derive because we know the generating function for this distribution and therefore it is trivial to compute that show that the variance is also equal to the same is the same as the mean that is one of the properties fundamental properties of a Poisson distribution now compare this with so so what does it mean if I have a number distribution that is a photon number distribution that is Poisson so if you have ideal single mode coherent radiation the P of n is equal to e to the minus some average value mu mu to the power n over n factorial on the other hand for thermal radiation the P of n was a geometric distribution very different from a Poisson and we saw that this thing here is 1 over 1 plus mu and then there was a mu over 1 plus mu to the power n so this was Poisson this is geometric the variance in this case equal to mu itself the variance in this case here so this immediately implies the variance is mu the standard deviation is square root of mu the standard deviation divided by the mean is 1 over square root of mu and therefore delta n over n equal to 1 over square root of mu that is in that case on the other hand here the mean value equal to mu and the variance not difficult to show this is mu times 1 plus mu that is trivial to show from this expression here and this will immediately imply delta n over n equal to well the mu there is a square root out here and then this is going to give you a square root inside so this is equal to square root of 1 plus 1 over mu and as mu increases when you have a large number of photons this scatter becomes extremely small on the other hand this scatter is always bigger than 1 this ratio is always bigger than 1 showing that in thermal radiation you have a huge scatter about the mean whereas relative fluctuation is very very small in coherent radiation so one of the defining properties of this coherent radiation now these results are very easily derived if you use the generating function in each case we know the generating function for this we know the function for this is a trivial to do this so I urge you to do this as an exercise so in some sense the most coherent radiation you can have is ideal single mode laser light and the least coherent one you could have is blackbody radiation thermal and what is happening is that you have distribution which goes all the way from here to here it would be nice to have a family of probability distributions which interpolate between these two extremes okay and we are going to see very shortly that there is such a family it is called the negative binomial distribution which is going to interpolate between the geometric distribution on the one hand and the Poisson distribution at the other extreme here in fact one can use that family of distributions to analyze photo photon counting statistics and see how much of thermal light is at mixed is mixed with the with the coherent radiation in an actual experiment but before we do that let me take up another question and that is the following what happens if you have a sum of two Poisson distributions if you have a sum of two random variables each of which is Poisson distributed okay with different means in general what would happen let us suppose that you have two species of radioactive nuclei and you are trying to count what is the total number of decays in a given time the rate at which one of them decays is lambda 1 and the other one is lambda 2 so each of them as I said is a Poisson distribution at any given instant of time and the question is the question you can reasonably ask is what is the actual distribution of the sum of these two processes of these two random variables okay so let us suppose that you have one random variable n which is distributed p1 of n in a Poisson manner you have to the n over n factorial and you have another m which is distributed also in a Poisson distribution but with a different new and we would like to know what is the probability distribution of the random variable s defined as m plus n and I would like to know what is p of s now what would you say it is how would you approach this problem these are independent of each other completely independent of each other so fundamental of basic first principles way of writing down the probability distribution of s is to say that this is equal to you try to make up s in all possible ways by summing m and n and first of all what is the sample space of s so what is the sample space of s also 0 to infinity also 0 to infinity so that way there is no change in the sample space so what you have to do is to say let us suppose the first variable the distributed n has a distribution p1 and then a p2 of m and you sum over all possible values 0 to infinity summation n equal to 0 to infinity subject to the constraint that m plus n is equal to s and that should by definition give you the probability distribution of s remember that m and n are dummy indices here they summed over so the answer is a function of s and it fires only when m plus n is equal to s for any given value of s so that is the definition and what we got to do is to put this in here and then do the summation but you cannot independently sum over m and n because there is a constraint here so you should use that constraint you get rid of one of the sums but it will constrain the others for instance n neither m nor n can exceed s because they are the sum of two non-negative integers s is the sum of two non-negative integers for any given value of s m can at best go up to s and so can n at best go up to s so while you can do the summation in principle the sum will get constrained here and you got to be a little careful in doing this on the other hand you could do the following you could say alright let us suppose f of z let us call the generating functions for this and this f1 and f2 of z and then we could do the following we could say what is the generating function of the sum that is equal to this by definition z to the power s but this p of s I put this in here and you can see that this is equal to a summation over n equal to 0 to infinity summation n equal to 0 to infinity and then I plug this in I multiply both sides by z to the power s and sum over all possible values of s okay so there is a summation s equal to 0 to infinity this guy and then you have p1 that is e to the minus mu plus mu and then what so remember that yes over m factorial n factorial here and then you have e to the power you have z to the power so mu to the power this is a better way of doing this and this is got to be a much better way of doing this mu to the power n mu to the power m z to the power s but subject to the constraint m plus n equal to s so all I have to do is to put that in here this becomes z to the power n mu to the power m sorry z to the power okay and that is it so what is this equal to now I have used the constraint here I finished off the sum over s because I have used that constraint to replace s by n plus m okay and what does this become mu n mu z to the power n mu z to the power m and each of the sums can now be done completely there are no further constraints on it therefore this gives me e to the power mu times z minus 1 plus mu times z minus 1 and all we need to do is to take out the coefficient of any power that we want to find p of s we need the coefficient of z to the power s in the power series expansion of this quantity in a power series in s okay and therefore this immediately follows that p of s equal to e to the power minus mu plus mu mu plus mu to the power s over s and what is that distribution it is a Poisson distribution with the means added up therefore the variance is also added up variance of s is the same as the variance of this plus that incidentally one can show I have been sure that if you have two random variables which are independent of each other statistically independent of each other their variances must add up their means must definitely add up that is a trivial thing to show but the variance is also add up why is that why do I say that the variance is add up because the only place they need would not add up if you try to find the mean square value of m plus n for example the sum of two random variables you would have the mean value of m squared the mean value of n squared and then the mean value of m times n right but that factors that factors because these are independent variables since it factors it cancels out when you subtract the mean whole square immediately so it is trivial to see that the variances have this additive property which the mean square value of a random variable does not so if a random variable is a sum of two or more random variables which are independent of each other then the mean value is just the sum of the means of the individual components and the variance also is a sum of the variances of the individual components a property which is not shared by this mean square alone or any higher moment so this invariance is very crucial and we will see that it is actually a reflection of property of what are called cumulants which also add up in additivity of cumulants is very very fundamental notion we will come back to that okay. So our first lesson if you have two Poisson random variables which are independent the sum is also a Poisson variable added up in this fashion I urge you to show that in this instance for example a slightly more general property suppose a and b are any real constants positive say then or negative it does not matter real constants consider the random variable let me use a symbol for this random variable let us call it u this is equal to a times n plus b times m this random variable does not have necessarily integer values in sample space because it is m and n are integer valued but a and b are any constants real constants then what is the mean value of mu u what do you think is the variance of its a squared because you are not finding the variance so it is a squared variance of m plus b squared variance of the other way now you know this property that the sum of two Poisson variables again a Poisson random variable follows very simply from the fact that the generating functions essentially multiply and the generating functions multiply and because they are exponentials because of this property here you get an additive exponent out here so you can see clearly that this is going to happen anyway but now I could ask a slightly more complicated question what about the difference of two Poisson random variables what do you think is happened going to happen there so let us go back and ask what happens if I consider not the sum but the difference let us call it r equal to m minus n what kind of distribution would this have what would this do what is the sample space of r minus infinity to infinity now not 0 to infinity all integers all integers are allowed right and now we try to find out what is the generating function of this thing here well we do exactly the same as we did here except we now have to write r equal to minus infinity to infinity p of r r to the power s and it is the same as this except that you are going to have to write let us let me put the symbols back here p 1 of n p 2 of m and then we multiplied by sorry what did I do here z to the power r z to the power r and if I plug that in here I get a z to there is going to be a delta function of m minus n comma r and I am confusing n and m all the time so this is p 1 of n that is correct p 2 of m let us call n minus m so that I do not confuse between 1 and 2 I call this 1 and this 2 and then I have a delta function of n minus m comma r and if I multiply by z to the r I replace that by z to the power n minus m so this becomes z to the power n z to the power minus m so this guy is equal to f 1 of z and f 2 of 1 over z because this power series with z to the power m is going to give me the generating function f 1 but now it is 1 over z to the power m and what is that equal to for a Poisson this becomes equal to e to the power minus mu sorry mu times z minus 1 e to the power mu times 1 over z minus 1 which is equal to e to the minus mu plus mu and then e to the power mu z minus plus mu over z and what we seek is the coefficient of z to the power r in the expansion of this quantity in powers of z but then all possible powers positive as well as negative powers are going to exist and it is not so easy to first you expand this and then you expand that and of course for a given power z to the power r an infinite number of terms are going to contribute because there is an infinite series in positive powers here and negative powers out here so we need a little formula what we need is something called a generating function for a special function so we need an identity which is the following e to the power one half t times some xi plus 1 over xi when expanded in powers of xi positive as well as negative this guy here is a summation r equal to minus infinity to infinity xi to the power r and it is multiplied by t this here is the modified Bessel function of the first kind and order r it is called nice interesting properties I will come we will come back when we study random box we will come back to this special function I tell you all about it but it satisfies a certain second order differential equation called the Bessel's modified the modified Bessel equation it is a nice function it is what is called an entire function in the sense that it has no singularities whatsoever for any finite value of t as a complex variable that was a nice entire function it can be written as a power series in t which converges absolutely for all finite values of mod t converges even faster than the exponential does the power series in t but whatever it is we can write down what the answer is based on this from this guy here because this here we can cast it in that form what the trick is how should I cast this in that form I need a xi and a 1 over xi but I got a mu z and a new over z so what should I do I want to make it some variable xi and its reciprocal take out mu that does not help root mu take out a square root of mu yeah so take out a square root of mu nu and then you have square root of mu over mu z plus square root of mu over mu 1 over z outside the square root you have this but you need a 2 out there so you put that 2 in by hand write it in this form and that will immediately tell us the p of r in this case is equal to e to the power mu plus mu square root of mu over nu to the power r because I want the coefficient of z to the power r in the expansion so that is got this guy sitting with it and then i r of twice square root of mu this factor here depending on whether mu is bigger than nu or nu is bigger than mu is going to be dominated by the positive or negative r one of the other so it is just a power factor there the Bessel function itself i r has an interesting property it says i r i sub r for any positive integer r is equal to i sub minus r it is got this symmetry property so the bias is entirely in this i r of anything of xi i r of t so it is not at all not at all a Poisson distribution some more complicated distribution than that by the way this guy has got a name this thing here is called a skelem if you like it is the generalization of the Poisson distribution to all integers to a sample space which is both positive as well as negative integers and 0 so this is the skelem distribution here and the difference of two Poisson variables has a skelem distribution okay. Now we already know the generating function this is the generating function for this distribution so from here it is possible to find all the factorial moments fairly straight forwardly without any difficulty for example the average value of r will turn out to be mu minus nu that follows immediately if you differentiate this once with respect to z and set z equal to 1 you get this immediately what about the variance what do you think will happen to the variance well remember we looked at a distribution of a n plus b m and we said a and b are just real numbers it does not matter whether they are positive or negative so you could set one of them equal to one and they are equal to minus one and of course immediately it follows that this guy is equal to mu plus mu so when you have the difference of two Poisson variables the mean of course is the difference but the variance is still add up in fact we will see a little later that the cumulants this is called the first cumulant this is called the second cumulant and then there are higher cumulants and so on we will see that the kth cumulant of the difference of two Poisson variables will be the first cumulant mean plus minus one to the power k times a second so every other guy will going to be this and every odd cumulant is going to be this in this case for the sum of course all the cumulants are going to be exactly the same we will see that let me talk about cumulants here okay so this is an instance of what happens to the difference of the two just one final remark about a physical example once again of such a situation we will see extensively that the solution to the simple random walk problem in continuous time is precisely a scalar distribution so if you did the following if you took a coin and you had an infinite linear lattice started at some origin and you tossed a coin and went to the right with some probability little p and to the left with probability little q depending on whether you got heads or tails and you did this at random instance of time with some mean rate according to a Poisson process then the probability of landing up at the point r on this lattice on the lattice point r positive or negative it does not matter it is precisely this scalar distribution okay so this will be the difference of two Poisson processes in other words steps to the right and steps to the left would be regarded as two Poisson processes and then you want the difference of the two and we will see all the various places at which in which this random walk problem various problem map on to the solution to this random walk problem we will see how that comes about okay we should probably take a little gap here and then any questions yes if alpha and beta are not integers that is not true anymore so his question is what happens if I did that which is precisely what we had written down so let us call the variable u equal to a times n plus b times m a b real constants and then ask what is the probability distribution of u itself okay this is not Poisson it is not or anything like that because the sample space is no longer integers right it is any a times integer plus b times another integer so it is again a set of points but the question is what is the probability distribution in this case one way to do this is to go right back to what the the ab initio definition of this quantity and instead of writing a chronicle delta you write a direct delta function of u minus a n plus b m okay and then write a representation for that direct delta function perhaps as an exponential which factors n and m and find the generating function of this u in terms of the generating functions of n and m exercise for the reader to do this okay so that is a nice interesting exercise to find out what is the probability distribution so do the following write this thing in terms of so use use a delta function of u a n plus b m minus u put this formally inside the summation for n and m put this guy formally but you want to factor out this is additive in n and m you want to factor it out and make it into some product of something or the other when you got a sum and you want to convert it to a product what will you do you exponentiate you exponentiate right so you would like to find the representation for this delta function which converts it into an exponent okay and use a thing like delta of any real variable x is integral p to the i k x you write a Fourier representation for it so put that in here you are going to have one more integration to do but that is not very difficult to do and then you will end up with the generating function okay so that is when interesting exercise so recall the binomial distribution which was p of n binomial distribution was p of n equal to n n p to the power n q to the power n minus n but the negative binomial distribution also has two parameters one of which is a positive integer capital N and the other is a little p exactly as in this case but this distribution p of n is actually equal to n minus n plus n plus n plus n minus 1 and then n here p to the power n and q to the power little n in this case where n equal to any positive integer and the sample space here is n equal to 0 1 to all the way up to infinity unlike the case of the binomial where 0 less than equal to n here it goes all the way up to infinity it is immediately clear that if you put capital N equal to 1 for example this is 1 that cancels there and this symbol becomes 1 then of course it becomes p times q to the power n which was precisely the geometric distribution so it is clear that n equal to 1 corresponds to the geometric distribution the question is what is it in general what is it okay general capital N what does it look like well the thing to do is again to find the generating function so f of z is summation p of n z to the power n and this time n runs from 0 to infinity and I plug in this in there so this is equal to little p to the power n n equal to 0 to infinity n plus n minus 1 little n and then you have q z to the power n that is an infinite series on this side however it is not very hard to show that this is equal to p to the power n over 1 minus q z to the power n it is precisely the binomial expansion of 1 minus q z to the power minus n and that is the reason for calling it the negative binomial distribution as you know when the index is not a positive integer this binomial series is an infinite series and it simplifies if you write it in terms of gamma functions and so on it simplifies to this guy here okay so once we have this it is easy to see what is the mean value of n equal to if I call this mu say definition of the mean value is mu this is equal to the derivative of this with z set equal to 1 if I differentiate it I am going to get out here p to the n and then there is 1 minus q to the n plus 1 so there is going to be a p in the denominator and an n which comes from there so it is equal to n and q by p because this is going to become n plus 1 when I differentiate it so that is a p to the capital N plus 1 and it cancels against p to the n and a q comes out on differentiation the minus sign goes away so it is this that is the mean value now of course we know already that for the geometric distribution when you put this equal to that 1 it is q over p you know that already let us rewrite this a little bit and let us write this as mu equal to n times 1 minus p over p which means that mu plus n p equal to n or p equal to n over n plus mu and q of course is mu over n plus mu 1 minus p is mu over n plus mu then if I put that into the distribution into this guy p of n equal to n plus n minus 1 factorial divided by n factorial n minus 1 factorial that is the job combinatorial part and then a p to the n that is n to the power n over n plus mu to the power n that is p to the capital N q to the power little n that is mu to the power n over n plus mu that is what this two parameter distribution looks like in terms of n and mu instead of p and q I got rid of p and n I got rid of p and use mu instead now look at what happens when capital N goes to infinity keeping mu finite at some value what does it do so p of n this is going to give me an n to the power n plus n to the power n plus n dominant term by sterling's approximation this is going to give me an n to the power capital N this guy here is going to look like 1 over 1 plus mu over n to the power n if I divide through by this n to the power n it is just this fellow here and then you have a mu to the power n and you got an n to the power capital N n to the power capital N and n to the power little n this gave you n to the power little n plus capital N so this leading term cancels against this and that and leaves you with just this and you cannot forget this so that is sitting there and the limit of this of course is e to the power mu so that goes up on top and you get e to the minus which is precisely a Poisson distribution so we have this family of negative binomial distributions which tends n equal to 1 geometric and n tends to infinity so that is the great advantage of this distribution and it is got a very nice very simple looking generating function we wrote it down just an algebraic function here so we can write down all its moments and so on okay. As I said one uses this in the study of photo electron photo photon counting statistics for admixtures of thermal light and coherent light among other things many other places in population dynamics etc where the negative binomial distribution is used very very frequently just one final remark and then we will take it up from here we have talked about the generating function throughout okay and while it has advantages it also has disadvantages so our definition of the generating function was equal was that if you had a probability distribution p of n of a random variable n then I defined the generating function as this summed over all allowed values of n that was my definition of the generating function on the other hand I could also ask the disadvantage of this was that if I want various moments of this distribution with various powers of n here I have to go on differentiating this but if I differentiate this k times then I actually end up with plus 1 this average was dk f of z over dz to the power k at z equal to 1 so I actually find the factorial moments rather than the moments themselves okay to overcome that disadvantage its advantages to define what is called a moment generating and that is defined as m of u I will use a different variable u instead of z so I have not to confuse the two this is equal to the average value of e to the power u n but the average value of e to the power u n is summation over the allowed values of n p of n e to the u to the power n e to the u to the power n is e to the n u so this is all it is and therefore we have our relation which says the generating function and the moment generating function are related by m of u is f of e to the power u so instead of z to the power n I got e to the u to the power n so if I know f I knows m and vice versa the great advantage of this is that if I differentiate this after all if I take this and write this as summation n equal to 0 to k equal to 0 to infinity out here u to the power k n to the power k over k factorial expectation because that's a random variable n here then immediately you see that n to the power k is the derivative the kth derivative of m of u over du k at u equal to 0 so directly by differentiation generates the moments themselves and not just the factorial moments so that's why it's called the moment generating function and it directly gives you the various moments so that's an advantage and I will stop here but we have seen we have seen that the moments themselves are not that helpful first of all there's an average which you got to remove first of all so it's not advantageous to use the moments but it's much better to find out what's the average value of n minus n average to the power k and take its expectation value these are the central moments these are just the moments but these are the central moments the moments about the mean value so any systematic drift or shift is got rid of for example the second part of these follows when k is 2 is precisely the variance as you can see even this is not very useful when you have many random variables because the sum of the central moments of independent random variables is not the sum of the kth moment of the sum is not the sum of the kth moments central moments so we need to do exactly what we did for the variance get rid of those extra contributions and make it purely additive and that's where the cumulants comes in this is a little bit like if you are used to statistical mechanics it's a little bit like going from the partition function to the Hemmholtz free energy so what you have to do is to ask can I write can I write e to the power u n as equal to e to the power some function of u which is in a power series here in general okay so this like saying I take the partition function and then I take it's you know free energy is defined as a log of this guy this partition function can I do that trick or not and that k of u will be called the cumulant generating function in fact that is the basic idea in statistical mechanics when you are writing the free the partition function you are writing a generating function actually for some distribution and when you find the free energy you are writing the corresponding cumulant generating function and that's what is additive so we will talk about this next time okay.