Good morning, everyone. Welcome back to another day in the life of a teacher vlog. So it is currently 6 24, and I am just getting done getting ready, and I'm getting ready to pack my lunch. I think I'm gonna take some leftovers. We had a barbecue yesterday. Mandy was here. Our friend Brittany and then our next-door neighbors came over as well.  So I think I'm gonna pack my lunch. I don't think we have any hamburgers left over. We have some hot dogs left over. So I might pack a hot dog, and I have some tortillas at school. I could totally make a little hot dog wrap with some tortillas. I think that would be delicious. And then I have some cucumbers and bell peppers that I want to use to dip into my buffalo chicken dip. Anyway, this week we are, ooh, the fridge has good lighting. Have you guys ever noticed that? You vloggers? You vlogger out there? You ever set your camera in the fridge and talk to it? What am I doing? Okay, I'm going to take these hot dogs. And I also need to make Cass's lunch. So I'm gonna do that really quickly, and then I'm gonna see you guys at school. Hey you guys, welcome back to another teacher life vlog. So today is Monday, September. I don't even know. 20. Wow, I don't even know the date. Today is September 13th, and I just walked in the door. I'm getting ready to get set up for the day. So I wanted to share with you guys what I have going on this week in science and social studies. So we are sticking with geography, and we're gonna stay with geography for the next two weeks of school. So it'll be a three-week total unit. And I'm doing pretty heavy on social studies this quarter, because I really want to hone in on the geography portion of my unit. Next quarter I will be working on history. Native American, North America. Native American tribes. Wow, that was a lot to swallow. So I have a plan for these next couple weeks. Today and tomorrow my students are wrapping up their key features maps and notes that we've been working on. So we've been learning about all of the key features of North America. So like the Rocky Mountains, the Great Plains, the Appalachians, the Canadian Shield, all of those really important landforms and key features of North America. All right, you guys, I'm totally going to interrupt myself right now and talk to you guys about my new glasses because every time I vlog at least 10 people ask me where my glasses are from. So these new frames are by Zimp Optical.  They are my all-time favorite online glasses shop by far the best one out of all that I've ever tried. So they have super high quality glasses, super trendy, super adorable. Basically, just like any other optical online place, you find the pair of glasses that you like. You input your prescription. You choose the type of lenses that you want. You can add bifocals, blue light blocking, which is what I always get. Progressives, all the different things that you would need for your eyes, you can do that. So then you place the order, they show up in a couple of weeks, and then you look fabulous. So these are the Browline Apollo Black Browline, I think is what they're called. I will link them down below. I don't memorize the names of them, but these are super cute, very feminine take on the classic like black and gold frame. And then another pair that I got. These are called Aphrodite Round Turquoise, I believe, and these are super cute. I actually wore these in my last vlog, and quite a few of you asked me in the comments which ones they were. So these are the Aphrodite Round Tortoise, and I absolutely love these. I love big frames. They make my face feel a little bit more feminine, especially with my bob haircut. My face just looks more round nowadays. So anyway, these are the second pair. I will link them down below as well.  And these are my Apricot Gray Hoodie Glasses, and they are Thaub Ulyss. They're so big, they make my face look small. They don't hurt my head because they're so massive and absolutely love them. So if you guys are interested in trying Zymph Optical, I do have a coupon code. I will put it down below. It is just TRS, and you can get 50% off of your frames and 20% off of your lenses, which really brings down the price. So if you guys are interested, they will all be linked down below. They're also eternally linked in my Instagram little web page that's in my bio. You could just click on it, and then you scroll down a ways, and all of them are linked down at the bottom, and the coupon code is listed at the bottom as well. So anyway, back to the vlog. So that next quarter, when we start studying the Native Americans, they have a better understanding of where these people lived. So I will try to link as many things as I possibly can in the description box below. Charlotte, when you edit this, please don't forget to put links down there.  And I'm so sorry. I just, when I'm editing, I'm kind of like in the zone. And once I'm done, I'm like, yes, I made it. I did another video. So anyways, something that I'm kind of leading up to, hopefully by the end of this week, we'll be able to create our salt dough maps. So that's going to be basically just like a 2D model of the maps that they made. So we'll get to paint the rivers blue, the mountains, brown, the plains, yellow. You get the idea. I want them to make really cool physical maps, 2D physical maps with their salt dough. And then like finishing up this week and into next week. We have a weird couple of weeks. That's why I kind of picked this week to be kind of like a project based couple of weeks. I have plenty of science grades that I can add into that, and which is why I'm not going to keep going with science because I'm getting ready to start rocks and fossils. And that's a really big, long unit. So I wanted to save that for next quarter when we come back and not take a break because we do have fall break this week. We have benchmarks next week. We have parent teacher conferences. So we have three half days. So I just didn't want to introduce a new scientific topic when we're not going to have a full week. So anyway, I decided to go ahead and start a National Park project. So I found this really cool PBL on Teachers Pay Teachers by Shelly Reeves. I'm going to try my best to show you with my weird tripod. So it looks like this. It's by Shelly Reeves. I will try really hard to link this down below. Try to remember. But if I don't look it up on TPT, it's by Shelly Reeves. And basically what they do is first they learn about all the different national parks in North America. Well, technically America, but they are in North America. And they do a little bit of information exploring those. So what I did was I actually found a bunch of national park books on getepic.com for them to explore. I went and found a couple of different websites that explain the national parks. And I linked all of those in my Google Classroom so that the kids could spend some time kind of researching. And then once they do all of that, they actually pick one. So they they try to research four of them and then they narrow it down to one national park.  So like sites to see at the park, snacks to pack, getting the car ready, what to pack in there, what games are going to play. States are going to travel through. This is an important one because I have to use mapping skills, places to stay. So they're going to have to research hotels or campgrounds in the area and then what they're going to pack for their trip. And then of course things for the car. And then there's a math component here. So there's actually a bunch of math questions that I'm going to give to my math teacher to go through with them. So I've actually shared this with him. And then there's a place for them to choose pictures of them in the parks. And I also thought it would be kind of cool because I know on Google Meet, you can make your background different. So I thought it would be really cool to get these guys on Google Meet and have them change their background to their national park and take a selfie and actually put it here in their little phone. So that's something we'll do. And then pictures of the national park. A postcard so they get to write about their favorite thing that they saw in the park. You know, they're actually not going there, but and then some messages and there's all these little tasks in here like you want to tell your friend about the park you're visiting, write down a texting conversation that you have between your friends. So they just add speech bubbles and Instagram post. So again, they can either draw for this or they can take a picture of them in the park somewhere. Here's another Instagram one campfire songs playlist. So just a list of their favorite songs that they'd want to sing by the campfire. Their favorite spot in the park. And then their favorite location, which was the most beautiful must see spots things to do. Word search. And then they actually make a final project. So they have to map their route to the park from where we live. And then there's a whole rubric that goes with it. There's a brochure that they make again, another brochure. And then there's actually a writing assignment that Mandy is going to do with them next week. So all of the work that they do this week and next week for me will lead up to their writing assignment that they're going to do in ELA. So this is kind of like a thrown together PBL and then something super, super amazing. I got a message from one of the associates at the National Parks Foundation and they are sending us. Well, they didn't, they aren't sending. They already sent. I just have to find where I stuck them. The entire fourth grade gets a National Park pass for the year for them and their family. So it's going to be a really fun unit. Hopefully I'll be able to get them their passes before they leave for fall breaks. So hopefully this project will take them this week and then finally end next week with them getting their National Park pass. And I think it's going to be really fun. So I also have, they know how to use Google Maps. So they're able to go and explore their National Park through Google Maps. And then we also have 360 cities, which is a near pod thing. It wasn't working for their computers though. So I'm hoping that they fixed it. If not, Google Earth is just fine. And also YouTube is a great resource. So I'll be able to send them a few, like I'm thinking about making a playlist of all the videos for the National Parks. So once they decide which one they're going to do, I might go through and make them a playlist of like videos about the National Parks that they can go see what it looks like. But I'm really excited. So hopefully this will be fun for them. Today though, I'm doing my first first grade science buddies activity with my friend Patrick. He is a first grade teacher. He's amazing. And we are doing our first science buddies project with his class. So my third period, my homeroom kids are going to be their official fourth grade science buddies. And so maybe once a month, we're going to get together and do some sort of like science project that is all like first grade level, but the fourth graders get to go and assist in the project. So today we're going to be doing the five senses. And so Patrick and I have put together five stations, which we're going to have to all probably be working through my lunch today because I have almost everything ready, but him and I have to kind of get together to do a couple things. So basically what they're going to do is use their five senses to explore some stuff. So we have four bags here. There are different things in them. They have to put their hands in without looking and then they have to write an observation or prediction what they think is in the bag. And then same thing with this one. They have to just shake the cups and try to infer or guess what is in them. And then of course, so the fourth grader would write down what they think it is. And then on this one, we're going to soak the cotton balls in different flavored extracts. They have to smell them. We have a couple of different things for them to taste. So we got saltine crackers, Hershey's Kisses, cocoa powder, and lemons. So they'll put a bunch of stuff on plates that each have an individual fork or spoon so that they can taste all of the items and tell whether it's sweet, salty, sour, or bitter. And then they have four wrapped objects wrapped in wrapping paper that they're going to be looking at to see what they think is wrapped in each one. So there are five different stations, one for each sense. And the fourth graders are going to kind of just help them go through each station, make their observations or predictions. And then the next time we meet, we will talk about the five senses, how we use them to explore our world, make observations. It's a generation genius lesson. So if you guys haven't checked Generation Genius out yet, you definitely, definitely need to do so. I use them to springboard almost all of my science lessons. At some point, I use their lesson, whether it's in the beginning or in the end or if it's the video or just the stations or the exit ticket. Either way, Generation Genius is one of my all-time favorite resources. So yeah, so I'm going to get to work. It is now 7.04. I do have duty in about 40 minutes. So I'm going to get hot to trot on my work so that I can be ready for station's eyes. So it is now 3.45 and I am pooped. Our science buddies was so much fun. And I'm really excited to do it again. The fourth graders had a blast leading the first graders through making observations using their five senses. Super fun. You can kind of see what's left over here. So tonight I have quite the task for myself. And that is making 80 helpings of salt dough so that I can create salt dough maps with my students tomorrow. We're going to be creating salt dough maps using these little cardboard cutouts. And then I have to go pick up a few pizza boxes from Borrow's Pizza tonight. To donate them so that I can use those for the remainder of the salt dough map. So I'm actually going to go home tonight and make one on here to be able to show the kids what it should look like. And then I have to make, like I said, 80 helpings of salt dough. I am going to separate all of them into ziplock bags just so that they're already pre-portioned and ready to go. So once I create my own map and see how much is actually needed to cover that whole thing and do a good job. Then I will portion out all of the other salt dough and I'm thinking I'm going to have cash help me because he would probably love it. And maybe the little girl next door. So anyway, today's vlog is done and I am exhausted. So I'm going to sign off, but before I go, don't forget to check the link and all of the description box below for all of the information about my Zimp optical glasses. You can also check my Instagram website all the time. It's right in my bio anytime you need. Tini, are you sad that I'm leaving? You want me to go home? No? Night, baby. You look sad. Do you want smooches? I have to go pick up cash, but I have time to give you smooches. This is not the cutest, baby. Hi, sleepy. I love you. I love you, mama. I love you, baby. I love you. Okay, you girls have a fun night. I'll see you tomorrow. And I will see you guys in the next vlog. Don't forget to follow me on Instagram and subscribe to the channel. And I will see you all in the next video. Bye.


So, let us start our discussions. So, in the last class, we started discussing about converges of sum of random variables, right. So, we said Sn is sum of n random variables and then we were interested in the case where Sn by n converges in to what value? So, we notice that, so if I have a sequence of random variables which are IID, then this value of Sn by n converges to expected value of the random variables which is expectation of X1. So, notice that here what we are saying is the limiting random variable is a constant and what is that constant? That constant is the mean value of the random variables. So, this we called as weak law of large numbers and we could also show that this is indeed converges also in almost sure sense, but we proved it only under the special case that the fourth moment is finite and this version we called as strong law of large numbers. So, these limiting distributions are very important in applications. So, can somebody think of why it is so, so why these results are, I am saying they are very useful, can you think of an example why this should be useful to predict the future, what future, but what distribution you are talking about. I am saying it is an IID sequence, right. Every time it will have the same distribution, what to predict there. What we are saying is what is Sn by n, this is Sn is the sum of the random variables and n. So, like he gave an example of fertility rate, right. Let us say number of children per woman, we have it for many, many women and we want to calculate what is the average number of children per woman. What you can do? You can just count, take many, many samples, talk to many, many women and take a survey and just and then average, then if you have taken sufficiently many samples, maybe this is going to converge to that value. And another simple case could be, let us say the voting pattern, right. So, everybody has preference for one party of the other, right. Let us call party A and party B and now you want to see that what is the, with which probability each guy is going to prefer. Let us say all of them has a same kind of preference for like, if I have preference for party A with probability P and preference for party B with probability minus one minus P. Let us say that if it is same across all the people, how I am going to calculate that value P? I will just ask everybody, okay, tell me what is your preference? You prefer party A or party B and then just I will take large an example and then just do the average, then I am going to get this value. So, you might have seen, right, like all these after elections, many people will just say, okay, this guy is going to get this many seeds and this many, this and they are going to say this is going to happen with certain probability, how they will do? Maybe they will just mostly some variant of this results they are going to apply. And usually when they say this, they are going to say what is the sample size they use to make this lick. They say, will that, okay, we talk to 100 people and based on this, our survey says that this party is going to win this with this probability. So, and most of the times these surveys are believable if their sample size is large, that they have got this input from more people. If their survey is like, okay, if they have just gotten talk to 100 people and from that they are concluding we may not take it seriously. But if they have talked to maybe like 10,000, 1 lakh people maybe that is more available because our result says that this average converges to this and this convergence is more accurate if my n is large, okay.  How you are going to do, how you are going to find that?  How you are going to do that? You just keep throwing it and see that how many times head comes, map that head to let us say 1 and map tails to 0, just count how many times you get 1 and then divide it by n, then this is telling that that should go close to as you have take more and more samples that should be go to that value p, okay. So whenever we have such things to be done, whenever we have kind of parameter estimations you are going to do this. So, suppose let us say all my random variables are exponentially distributed random variables, X1, X2, this sequence is IID but each random variable is exponentially distributed with parameter lambda.  What is that average will be? Let us say I have an experiment which has a distribution which is exponential. Every time I am going to repeat it many times, each time I am going to repeat it independent of the previous one. Whatever the sample I got first time, let me call that as X1, whatever the sample I got it second time I will let us call that X2 like that, like this I have 10,000 samples. If I just all add them 10,000 samples and divide it by 10,000, approximately what is the value that you expect? 1 by lambda, okay. So, in this case suppose if you a priori know that my distributions are exponential but you do not know what is the rate then what you can just do is do this experiment many, many times and then take the average you already got close value of lambda, right. So, for example all this you might have studied in class 12, right, like life of atoms, they will be like kind of exponentially distributed but you do not know the rate at which they are going to decay then maybe you collect many, many samples and doing this you can estimate their rate, the lambda rate, okay. Today we are going to see one more important results of this convergence of sum of random variables called center limit theorem. First let me write this result and then we will discuss. So, what it says is let us say the sequence of random variable that are IID and they have now this mean and variance sigma square. If I look at their sum like this and then look at their normalized sum. So, I am going to call this normalized sum because what is the mean of Sn n mu. So, I am basically subtracting mean already from them in a way it is centered because each of them are now it is a zero mean random variable Sn minus n mu and then I am normalizing it by square root n. So, this is centered and normalized random variable here and we say that this goes to whatever your sequence is whatever distribution it has if it is IID then it goes converges in distribution to normal distribution with variance sigma square, okay. So, what does this result say? Now, say earlier when I did it in Sn by n just took the average it always converged to a fixed constant, but now here the limit is not a fixed constant. It is a distribution which is Gaussian with parameter zero and sigma square. So, let us try to understand why this result is true. So, to this true result the characteristic function comes to a help and we just use it here. Let us say so again I am going to prove this result assuming mu equals to zero. If mu is not equals to zero you can straightforwardly still conclude this by saying that. So, if mu, mu are zero this is Sn by n, right. So, if mu are not zero I could as well write the numerator as Xi minus mu i 1 to n, right. So, I have just replaced Sn minus n mu by Xi minus mu. So, this is by definition. Now, each of this random variable Xi minus mu is I can treat this as a new random variable, right Xi minus mu. What is the mean of this random variable? Zero. So, then what I am basically adding is n random variable where each one of them has zero mean and still their distributions are same. So, because of that I can as well assume that the mu's are all zero and try to proceed to prove this theorem. Now, so because of that I just need to worry about Sn by mu n and let us try to find out what is the characteristic function of Sn by n at a point u. So, this is a characteristic function of h u, right. So, this is Sn by n is sorry square root n and this is a characteristic function of Sn by square root n. Now, if you just expand this guy what you are going to get? So, Sn is sum of n random variable where each of them is independent, right. So, I am going to get it as product of J Xi square root n of u, okay and this i from 1 to n and then what I am basically doing now next is. So, now I am going to treat it as characteristic function of Xi computed at value u by square root n, right. That is my argument here. So, I can write it as I want to add expectation of J Xi and u by square root n. So, which is nothing but right and now because these are all identically distributed I could write this as well as phi of x 1 u by square root of n to the power n. So, notice that if they have identical distribution, their characteristic functions should be necessarily the same, right. So, that is why we have used this property. All of them should have the same characteristic distribution of that of x 1, okay. Now, let us try to understand how this guy behaves. So, now I am going to use this characterization. So, how many of you know Taylor's expansion of a function? So, let us say I have a function f of x and I want to expand this Taylor expansion of this function around a point f of x naught, okay. So, then I can write it as, so in general I can write it as and what is this? This is to the power and what is the n indexing happens here? 1 to infinity. But there is an alternate characterization of this Taylor function which says that I could do this as instead of taking the infinite sequence, I will look at the truncated version of this. What does this say? Then the nth one is going to be, so what should be here? It should be i here, right. What is this y n? For some y n belongs to x and x naught. So, this is the truncated version of my Taylor series, okay. So, as you see this, so you understand what I mean by this superscript i here, this is the i-th derivative of this function f, okay. Now, this Taylor function involves the first derivatives, second derivative and higher order derivatives, right. But we will be interested only first and second derivative because I want to have, I have only first order and second order statistics here. And when I have first order and second order statistics, my moment generating function, I need to only care about first derivative and the second derivative, right. So, to write this, let us see what is the value of this function at 0, phi of x 1 at 0 when the argument is 0. Whatever the argument is, for any random variable x 1, this guy is going to be what? 1, okay. And what will be the first derivative of this at 0? So, what is, I am looking at the first derivative of the characteristic function x 1 at the point 0. What is this? So, if you differentiate the characteristic function and put, take the argument 0, what you are going to get? You get 0. So, how do you are going to get compute the mean of a random variable from the characteristic function? I cannot hear you. Expectation of, yeah, I mean I am asking how you are going to derive the mean of a random variable from, if you know its characteristic function, you take the first derivative at 0, right. So, what exactly I am doing? I took the first derivative and put 0 and I am, that I am assumed to be 0, that should be mu, right. And that I have assumed to be 0 here. And what is, if I take the second derivative and put 0, what I am going to get? Expectation of, expectation of x 1 square. But in this case, our mean is 0. So, its variance. But with what sign? It is going to be negative with a negative sign, right. Just check it. When you, second derivative and because of this complex term involved there, we are going to get it as sigma minus sigma square, okay. So, now, for this characteristic function, I know what is the mean value. I mean, I have been able to express this, the mean value in terms of the first derivative and the secondary area of the characteristic function. And I know these things. So, let us try to do the Taylor expansion of this only for n equals to 2, okay. Now, and I am going to do the Taylor expansion of this function around origin, around 0, okay. So, then what will be the Taylor expansion of this? So, it is going to be phi of x 1 of 0 plus u of minus 1 times, I am just writing expanding for u, but I could have as well skipped this. And here, it is going to be not 0, but a value y 2 here, which is going to be dependent on where that value of y 2 will lie, 0 and u by n, is this correct? So, when I wrote this Taylor series expansion, I kind of expanded it assuming that my function f is real here. But here, my phi function could be always real or it could be complex as well. It could be complex as well, right. So, when I have a complex function, all I need to do is this guy is here is wherever I have this phi i x naught, right. I have to make it as real part of f of i x naught plus j times imaginary part of x naught, when my function f is a complex function, right. So, this guy here I have to replace it like this, okay, because my function phi is complex, okay. Now, let us look at this. This phi x bar, so for the phi of x 1, its derivative at 0 is always, it is going to be 0, either it is real or both its real and imaginary parts are going to be 0. So, because of that, this guy is going to get 0. Now, what remains is this part. So, what we have is phi of x 1 of u of root n is going to be, again phi of x 0 is 1 plus this is going to be u square by n and 2 there in the, rather 2 there and this one I have to write it as real part of phi of x double prime 0 plus j of imaginary part of phi of x double prime 0. So, you see this, when I write this phi of x double prime, its value is minus sigma square. So, it has only real part, right. There is no imaginary part in this, right. So, because of that, I will ignore this part. What is this going to be? 1 plus u square 2 n and what is this part? This is simply going to be minus sigma square, okay. I have just substituted the value for this. So, what I could get is only this part, but what I am interested in is this quantity raised to the power n. So, that is what my phi of s n by n, right. So, now let us go back to that phi of s n by n at u is this whole quantity, this guy, the real part of this is what? Oh, right. It should be for y to write. So, just let me correct this. What is this guy is going to be? This is going to be yn here, which is going to be 0. Yeah. So, I want this to be exactly this, right. Okay. This is fine, right. So, and where is this y 2 is going to be u upon divided by n, right, because that is what our, that is what we said here y 2 belonging to 0 to u by n and this is going to be still the real part of phi of x double prime y 2, okay. Fine. Now, and there is a power n here. So, now can you look into the sequence and try to see, you know, are you familiar with the sequence as n goes to infinity, what this sequence converges to? e raised to this guy, right. But this itself is changing. This is as n, if you let n go to infinity, right, this itself is changing and what this y 2 goes to 0 and what is the value of this phi of double prime at 0 is minus sigma square, right. And then what is the, can you, then what as this n goes to infinity where this limit will go? e raised to, e raised to, it is going to be e raised to u square. So, you people understand why I got minus sigma square here, okay. Now treat, so yesterday we have discussed that if I have a sequence like this, it goes to e to the power z where z n converges to z. If I have a sequence like that where z n converges to z, then this sequence converges to e to the power z. Now look at this, look at this part, think it as z n, u square by 2 remains like that as n goes to infinity, this guy goes to phi x prime, double prime at 0, right. And at that value, what is its value? That value is exactly minus sigma square and that is why we get u square by 2 is minus sigma square. Now this characteristic function has converged to this characteristic function, this function and what is this function? What is this? This is a characteristic function of a standard normal distribution, right. That means this has 0 mean and variance sigma square. So, okay, this is converging to a distribution which has mean 0 and variance sigma square. So, that is why we are saying that my s n, so if that is the case, we know that. So, one more terminology, so in general we are going to call, if it is variance is 0 and mean is 1, then we call it as standard normal. So, in this case, it is just going to mean 0 and variance sigma square. So, is it clear? Now, so we will come to see like, okay fine, what is this? Why this center limit theorem is important? So, in a way it says that if you have aggregated large samples, okay and if you center them and normalize by square root n, then we can, that aggregation you can treat it to be like Gaussian distributed if your n is sufficiently large, right. So, whatever be the x i distributions, I do not care, x i all that could be exponentially distributed or all of them could be uniform distributed, whatever they are, then if I am going to look at their aggregation, normalized aggregation, then for, if I take sufficiently many samples to aggregate, they will behave as if they are normal distributed, okay. So, in a way I kind of already know how this with sufficiently many samples, how that distribution going to behave like, okay fine. One last thing I want to cover on this topic is Jensen's inequality and the Chernoff bounds. Let us quickly discuss the Jensen's inequality. So, how many of you know Jensen's inequality? What is Jensen's inequality? Expectation of Fx, what is this F convex function? What is a convex function then? You know what is a convex function? Yeah, define it. So, let us take any lambda between 0, 1 and let us take two values. So, if I take, so I am defining a convex function now. So, if I am going to take lambda x plus 1 minus lambda y, what is this going to be? So, if a function satisfy this for all lambda and x y, where x y are in the range of the function or sorry domain of the function, then we are going to call this as a convex function. So, how does the convex function look like? Let us take two points x here and let us say y here, okay. So, what is this guy here and what is this guy here?  So, this is a linear combination of the point f of x and f of y, right. So, f of x is here, f of y is here, this is going to be the linear combination of them and what is the function value at this guy, the curve falling?  So, what is my function value at this point, this value and what is my lambda f of x into 1 minus lambda f y? It is going to be on that line and what is we are saying? Whatever the value of this, this is, let me make it, this value I am going to get that is going to be smaller than the linear combination of these two points. So, if I have a convex function, it is always the case that my function value at the linear combination of those two points is going to be smaller than the linear combination of the function itself as those two points. Now, the question is, if my function f is like this, why it is, why it is the case that this is true? So, now we understand how the convex function looks like, right. So, that is the consequence of this definition. So, another, so there are properties one can think of for any convex function for all x. So, why this is true actually? So, I have a convex function like this. So, if you start looking at the slope of this function, so the slope is somewhere here at this point and if I look at the slope it will be like this and at this point slope is going to be 0, at this point the slope is going to be like this, at this point the slope is going to be like this. So, what is this slope here? It is negative here, this is again negative, but this is having a larger negative slope. At this point what is the slope of this and what is the slope of this guy and what is the slope of this? More positive, right. So, what is happening? The slope itself is increasing. So, if the slope is increasing means what? There is double debilitative is increasing. So, that is one property. Another property of this convex function is at every point I can draw a tangent that touches this function at one point. So, I can draw a linear line at any, give me any point on this function. I can draw a linear line that acts as a tangent to this function f. So, from this property actually we can right away write this. So, let us see why that is true. So, take a point x, some point. Let me take this x, let me take this y. So, this is x and this is my function f of x. At given point y I can draw a tangent here that means this line here, here is a lower bound on my function, right. So, what does this mean? I think maybe this much space is enough. So, f of x is going to be greater than or equals to l of x and this is true for all x, right. Because this line is, take any point this line is going to be light below this function f. Now, let us take expectation of this function and in particular I would be interested in taking this y to be expectation of x. So, expectation of x is going to be somewhere in the middle of my domain, right. So, let us take that point to be f y here. And on this point we know that this function and this function both have the same value, right. So, what is this l of x? It will have some slope and it will be like this, right. Now, let us take an expectation of this and this is going to be a plus b and expectation of x, right. Is this true? I just applied expectation to both sides, right. And what is this? At this point this is the line a plus b x, right. This is a plus b x. Now, if I compute this function at the point y which is expectation of x, at that point this value is going to be same as my function f of x. So, this is my f of x, right. So, on this, this is simply same as expectation of x. So, I am applying this property because this line at the point. So, what does this mean? a plus b to expectation of x means this is nothing but this linear function computed at point x which is nothing but the expectation of x, right. So, this is a linear line which is computed at different point a plus b x. But now if I am saying this is a plus b expectation of x that means this function being computed at expectation of x.  So, that is why expectation of x is going to be greater than or equal to f of expectation of x. So, this is exactly what we said as Jensen's equality. Yeah, L is the tangent. At the point I have chosen this tangent to be at a particular point which is the expectation of x here, okay. I know, give me any point on this curve. The convexity property tells that at this point I should be able to draw a tangent. What I did is I chose a point y which is the expectation of x, right. And on that point I have a linear function like this and I know that this linear function always lower bounds my function f of x. So, that is the first property I used here. And then just make sure that this f and this L function they have the same value at expectation of x. So, that is why Jensen inequality holds, okay. So, because of this you can derive some of the properties very quickly. For example, I want to find the relation between expectation of x square and expectation of x whole square which is bigger. Expectation of x square is better than this. Why? Can you use Jensen equality? So, what function you are going to use here? x square, right. So, I can think of this as f of x here where f of x is x square. So, if I have to do this, then I know that this is nothing but expectation of x and f is nothing but square function, right. So, that is there and as you said that we already know that is true because variance is a non-negative quantity and variance is the difference of expectation of x square minus expectation of x whole square. So, like this comes handy to prove, this Jensen equality comes handy to prove many inequalities of this part. Some of them are already there in your exercise or in your assignment try to solve them.


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Can these Asian basketball players be successful in the NBA? We are here for Asian basketball watch. Let's talk about it. Oh, yeah. Don't you know? Have you guys ever heard about maple? Yeah, and we are talking about division one basketball players potentially transitioning to the NBA. But also, why does everybody doubt them so much? And can they beat the doubters? So make sure you like, subscribe and turn your notifications into but you know what's an Asian thing that people are not doubting right now.  Check out the Instagram. Lots of people are reposting it. I just put it in my hot cocoa yesterday. And let me tell you this. It worked. It was delicious. It's not for the faint of heart. It's got a nice kick though, but check it out. A lot of computing news right now, Andrew. Here's an article saying you're going to lose anyway. Asian-American athletes face stereotypes across sports. However, Andrew, on the other end, seven-four, half-white, half-Chinese center, Zach Eadie from Toronto, Canada, just led Purdue to a win over number one ranked Arizona and NCAA Division One basketball. And he had 22, nine and five. So it's almost like there's some articles coming out about why people doubt Asians in these like big, high-contact sports. And there's also some examples of people breaking it. Right. Well, we're going to talk about some of the top D1 players that are Asian, that are in also some players who are in the G League. There are also Asian right now. So, David, first of all, let's go to Zach Eadie. He won National Player of the Year last year. He almost went to the NBA. He went to the Combine, which is the tryout. But he decided to go back to Purdue for one more year. He might win National Player of the Year again, but on the projected NBA drafts, people have him going at the end of the second round, barely getting drafted. Right. What's going on? All right, long story short, they say that his style of play, because college basketball still runs off old-school basketball inside out. The NBA has switched to an outside in style, and they're saying that his archetype, while it still may be dominant in the old-school college game, may be more relevant overseas than it is in modern-day NBA. Right, because the NBA is very heavily guard-focused, a lot of long, quick wingmen that are 6'6", 6'7". Well, everybody's like 6'9". It's like Lamello's the point guard, and everybody else is like 6'10". I mean, would you say the biggest worry about Zach Eadie? First of all, he is really good. He's very skilled. He does kind of play like Yao Ming, and I'm not just saying that because he's half Chinese. I mean, play the highlights right now. It looks like Yao Ming. He literally kind of moves like Yao Ming, but here's the thing. He's slower than Yao. Yes. And he is just slow, and even Yao... I think he jumps a little better than Yao, but he's slow. And Yao played 20 years ago.  There's a lot of like quick big men. Big men are starting to shoot more. Can he shoot? He can't really shoot that well. He's definitely more of a, you know, low post threat. So I guess that's the worry, but it's sad because David, as an Asian, I'm like, yeah, listen, man, I mean, he's 7-4, and he's national player of the year, maybe two times in a row. I would say he's Asian presenting. How could you not... Yeah, how could you not give him a chance, though? Yeah, I'll say this. He only picked up basketball seven years ago. He was primarily playing baseball and hockey, because I think, you know, his dad is white, his mom's Chinese, so they probably like didn't... You know what I mean? Like the dad's sports were more dominant in his life. So it means that theoretically, Andrew, his IQ and his skill sets have a lot of room to grow. So some people, you know how like they look at a guy who picks up something really late in life, and you can either look at that as a pro or a con. You can either look at it like, man, he's never gonna get the IQ reads of somebody who's been playing their whole life like Yao Ming, because Yao Ming actually had a very high schematic IQ, or you can look at it like he's got all that room to grow. Right, right. I mean, I think ultimately, I believe he will win national player of the year again in Division I basketball, but, and I think he will get drafted, and it depends if an NBA team can find places to put him like a Boban Marjanovic, where Boban can't really shoot either. I mean, I'm not saying... I think Zach can shoot more. I just think the schemes that produce runs, he's not allowed to showcase that. Right, right, right. So I guess ultimately, Andrew, I took a look at the chart versus him, versus a lot of other seven, three players. He's still having the best D1 season, versus like Taco Fall and Hashim Tabeet and everybody else who couldn't stick in the league. Right. He's ranking analytically higher than them. Yeah, no. I mean, my prediction is overall for Zach Edie, he's going to get drafted. It is going to take a particular situation on a team for him to get a lot of burn. I don't think every team wants him. What, like how much, what would you say, out of the 30 teams, how many teams would not? Out of the 30 teams, I think there's like six or seven teams that are going to be like, yo, we could use a Zach Edie. We'll work with him and see what happens. Right, to see if he can get the other team in the foul trouble in a mismatch or something. But there's going to be other teams who are just like, ah, big man, that's slow. I just don't want it. Right. And that's okay. But I think Zach Edie, he's going to get a shot in the NBA. And if at worst, he can't make it in the league, there's a whole bunch of other leagues in the world. I'm sure he will be able to make millions playing in Europe where there's no three in the key.  And let's not enter. Let's be honest. The CBA. Dude, he would dominate in the CBA and dominate in the Asian basketball leagues. We got to keep it real here, Andrew. Last question on Zach Edie. Do you think NBA teams will take an extra long look because of the potential to market him in China? I think it depends on his ability to speak Mandarin. I think he speaks some, but I don't know how good it is. It's very rudimentary. Yeah, you know, and probably more than average, right? His mom's actually a CBC. She's born again. I don't think, I think it's going to be a consideration, but not entirely. Okay. They're not going to be quite like Yao Ming because Yao Ming is from China. He's from the Chinese system. You know, it's not like Utah from Japan because Utah is from Japan. Roy Hachimura from Japan. Yeah, even Roy Hachimura, even though he's only half Japanese, blood-wise, he's like from Japan. So obviously it's a little different. Moving on, Andrew. We're talking about Xavian Lee. Whoa, guys. You might have not heard. Shout out to Amazing HQ. Follow them for all the updates on Asian Hoopers and Asian just athletes in general. Andrew, I got a new nickname for Xavian Lee. What? Kinesian Reeves. Because he plays a lot. He's Canadian Asian and he looks like Austin Reeves when he plays almost to a 100 out of 100 match. David, do you want to say his stats right now for Princeton? First of all, he plays on Princeton.  They're not bad. And they're in a tough conference. I mean, they play tough teams. 18.15 rebounds, three assists with a pretty good field goal percentage for a swing man. Guys, I believe the Princeton record as of now is like nine and three. They're good. That's a good team. He's actually considered one or two or possibly the best player in the Ivy League right now. So we could be looking at a Jeremy Lin situation. I believe Xavian Lee is raised by his mother.  I believe his dad is half white, half black. Some people think in 2024, Andrew, Xavian Lee has surprised everybody. And next year, if he continues to extrapolate his upward arc, he will be a 2024 second round draft pick. Do you believe in him more? Like some people believe in him more because his game looks like a modern game. Yeah. So here's my thing about Xavian. Super skilled. I think he's one of the most... He's a really skilled Asian guard. He's 6'3", 171 pounds right now. That's pretty light. He's going to have to get more muscle over the next year. At least 15 to 20 pounds, right? Yeah. I mean, realistically, 20 is a lot, but at least 10 to 15 pounds. He's going to have to put it on. A lean muscle, right? A lean muscle. And he's going to need to, I guess, strengthen the joints to support the newfound muscle. And I do think it's going to come down to defense. At 6'3", you're not... That's not tiny for the NBA, but that's not tall. That's an average point guard size. That's Steph Curry, Chris Paul 6'1", Steph Curry 6'3". He could still grow. He could still grow. Yeah, yeah. Two inches would be huge. I'm not betting on him growing at this point.  I think that at the very least, even if he's not drafted in 2024, it could help him to come out and not get drafted and get a training camp contract so he can start getting put in a pro system. But obviously, nowadays in 2023 with the NIL deals, Andrew... Oh, wait, I was going to say, him and Zach, it's hard for them to monetize because they're Canadian. The NIL deals are mostly for Americans. Oh, interesting. The Canadians have different rules. Oh! So it actually limits their money, making potential a little bit. I just realized... What, Canadian citizens? Yeah, it limits it. They have to wait until they are out of the country. I don't know, they have weird rules around it. I'm not too clear on the exact extrapolations of it. But shout out to Xavier and Lee, Andrew. Oh, he's balling. He's balling out. Do you think that he's going to be really inspirational because a lot of people want to play like him? Whereas, versus Zach Eadie, it's almost like he's more like an Asian Greg Ocertag mixed with Yao Ming. It's tough to relate to it because he's 7-4 even though everybody's rooting for him. We actually know a lot of people in Canada that are on his team. Andrew, number three, moving on, we got Jeon Suk-yu from Korea playing at Gonzaga for his first year right now. They call in Gonzaga the Duke West. They're getting a lot of really high recruits. And they actually get a ton of players with interesting backgrounds. The only Mexican player in the NBA right now, Hami Haquez, came from Gonzaga as well. Yeah, yeah, yeah. They also had a Roy, came from Gonzaga. Oh, Roy Hachimura. So they love Asians there. Yeah, yeah, yeah. And also, Kai Soto was signed. I believe Fan Bo Tung committed to Gonzaga, but decommitted to go to the Julie. Jeon Jung Lee went to... Oh, wait, he was from Davidson. Right. Anyway, yeah. Anyway, this Jeon Suk-yu, Andrew, he's kind of like a modern-day 6'8", like guard-like swingman, right? I think a lot of people got high hopes for him. He performed really well on the international level. He still played against people like Chet and Wen Benyatta. He did play against them and he performed okay, you know? And overall, he's a 6'8", mobile guard, 220 pounds. He's got a little muscle on him. He can shoot. I mean, really, to me, what's going to hold him back, obviously, you got to see how he plays out at Gonzaga. That's going to be huge. But he has good potential to make the league.  Maybe the only downside right now is he doesn't really stand out in one area. You know what I mean? Even though he hits all the pre-wrecks, he might not be going way over the pre-wreck threshold in any particular area. I think it's going to be key that he stands out at Gonzaga and like, what are you going to need him for? But I will say this physically, he can hang in the NBA. It looks like he can hang. Jeon Jung Lee, Andrew, he just left Gonzaga. He went to go to the G League, to the Santa Cruz Warriors, but then he left the G League to go get pro reps in the NBL in Australia, which is obviously the Australian pro league. Famously, Lamello Ball went there for a year. Yeah, he's from Davidson, right? So he went there partially because of Steph Curry. Yes, yes, and I would say at Davidson, he played a little bit like Davis Bertans. Yeah, not Steph. He's not, sorry, he's not the Asian Steph guys, to be honest. He's more of the Asian Bertans, but he's pretty good. Sharp shooter, kind of tall, lanky. So he's got the length. He kind of has a high release point, so that's helpful. But I think in general, a lot of people are wondering of foot speed and can he play defense? And I think the Asian Steph is more Kago Kamenaga, who played on the Japan national team. Yeah, yeah, yeah. That guy's more Asian Steph. Shout out to Kago, shout out to Kago. We'll see where he goes. But David, yeah, Jeon Jung Lee, I think that, so he left G League. He's now playing NBL. Going to develop over there. We'll see what happens. Do you think he has a shot back at the NBA? To be honest, I don't think so. Because he had already a shot in the G League. I think it was tough for him to keep up with his foot speed, but he could be a dominant player in Asia. Yeah. Do you think he'd even be a, like, you mean like maybe in Korea? Yeah, in the KBL, for sure. For sure he'd be really good. Yeah. Moving on, Andrew. Number five, and like I said, there's going to be some players that we leave out here that are half or like that, you know what I mean? Let us know who we left out. Yeah, I don't have a breakdown on Kago Kamenaga. Shout out to him, though. I watched him in FIBA. Number five, Andrew. We've got Roy Yuen from Stanford, who is a compside major and a walk-on. And recently, he got posted on SportsCenter because Stanford was up and he did a hezzy pull from deep and then said he got ice in his veins. All right, so this is interesting. Let's be honest, he's not going to the league because he's a walk-on at Stanford. Even though he was nasty in high school, he's very undersized. Yeah, he's really good. Six-one, he's six-one. He can shoot. He's got some moves. He has a YouTube channel. If you want to check it out, he's playing like 1v1s with people. Compound plays, I would say like Peyton Prichard with less driving. So he's not going to make the NBA, but he could play pro. He can definitely play pro in Asia. In Asia, for sure. Andrew, let's just take a look at some of the comments because that sort of goes back to our original article that we're talking about. Like, why do so many people doubt these players? Did you know that, all right, so Asian-Americans make up 7% of America. They make up 2% of NCAA Division I athletes. And in basketball and football, it's less than half a percent. We're talking about less than half of a percent is Asian. Somebody said Jackie Chan out here hooping.  Andrew, actually, he has a 3.9 GPA and he's a comp side major. And he's an intern at a top level. He's going to be, dude, if he decides to go into Silicon Valley, man, all the bros are going to love him there. I want to say he interns at ChatGPT, to be honest. Somebody said getting balled on by the water boy is wild. He's got rice in his veins. Man, rice gum is really out here getting buckets. Now he's got COVID in his veins. One ton broth is flowing smoothly. Actually, this is Zoom CEOs, Eric U.N.'s kids. All of you guys are going to be using WebEx now. That was a deep cut Silicon Valley joke. Andrew, why do you think so many people doubt these players? Because even guys in the NBA right now that are half or a quarter, whether we're talking about Jalen Green, Jalen Williams, Johnny Juzang, obviously, Uten Watanabe's in the NBA, Rui Hachimura. They're from Japan. There are so many articles about people doubting them because whether they're the half Asian side, people think that makes them weak or not good or not competitive or they came out of the Asia system. So people are questioning the quality of the basketball training in Asia. Everybody is like, there's so many articles about every part Asian player or full Asian player that's like, they beat the odds. Nobody believed in them. I'll say this, man, and this is going to happen for years to come until there's so many Asian or part Asian players that it's not even something you need to talk about anymore. But right now, it is still rare. So everybody's getting talked about. Well, if they're another race and they're this good, they're not getting this much coverage because they're just not going to make it and no one's going to think about them. But these guys, since they are on the cusp, and it is true that there's a lot of eyes on Asians in these leagues. Like, how many more Asians are going to, we had Jeremy Lin, we had Yao Ming, we had some successful Asian players, but like, are we going to get more? But it's like, because all the eyeballs are on them, that's why it feels like there's a lot of Asian guys who are on the cusp of the league, just because we're talking about them all. Right, right. But you're saying there's so many guys that are on the cusp of other ethnicities, but nobody's even talking.  Or even white, they'd be like, yeah, that's what happened to white players. They're good in college and they end up on the cusp in the league. Yeah, but I think it's just because Asians are now breaking through into the whole like, hey, Asians can play basketball. Right, they're in the discussion. Yeah, they're in the discussion. They can, like, don't overlook them. They're not all going to be Michael Jordan, but like, and they're not all going to be Jeremy Lin, but like, some of them are going to be solid. For sure, for sure. I mean, there's a lot of girls playing D1 basketball right now that are having really good stats. But obviously, the truth is, very few of them ultimately go to the WNA. I mean, I just want to shout out to the Princeton female team. They got three Asian American girls, one half and then two full Asians. And they're playing minutes too, right? Yeah, no, some of them are like the starters. Shout out to Ashley Che, right? Ashley Che, Kaitlin Chen, Sky Belker. So that's, I mean, shout out to Princeton for, I guess, taking all that Asian, I guess the Asian players want to play at Princeton. I guess as a final takeaway, Andrew, should Asian parents push their kids in a basketball or should they put them in a small ball of sports such as tennis and golf, swimming, you know, things that quote unquote, like Asians have had more conventional success? I mean, what's your goal, man? What's your goal as a parent, man? If they can play basketball at Princeton and if that's your goal and they can do it, they haven't played basketball.  Or I mean, get the Princeton education.  Well, now you got the NIL deals, you can make money in college. No, I'm just saying even just the Princeton degree though, that can't be bad for your life. Yeah, I mean, if you're good, it can help you get into college.  So some of them are still somehow paying for school partially. Yeah, that's true. I think they figure out a way to help them out. Yeah, yeah, yeah, of course. Yeah, they're paying for like a third or fourth of it. Anyway, guys, let us know what you think of Zach Edie, Xavier and Lee. Let us know what you think about all the Asians, you know, potentially going to the NBA. I get it. I'm not saying there's not a ton of other Asians being successful at other sports. We happen to follow basketball very closely. That's what I personally like, just feel very passionate about. And yeah, I know it's a height issue. I know it's a young coaching issue. Sometimes the parents, you know, Asian parents, they want to push their kids into academics more than sports. Like, and you only have 24 hours a day. And I'll tell you, a lot of guys that we didn't even mention, I know some other Asian guys who are starting their first college, their D1 college careers, you know, but we just, it's because they're just starting. So, you know, we can't put them in the same category as Zach Edie. But I'm just saying there's a lot of Asian Hoopers. Now, is basketball the best sport? I mean, there's a lot of great Asian soccer players right now. Right, they're doing really well. Yeah, maybe all Asians should play soccer. I don't know, you tell me, man. Let me know in the comments down below. Guys, check out SmallLah Sauce. Happy holidays, and until next time, we out. Peace.