Welcome to the course on dealing with materials data. In the previous session we introduced what is known as random variables and from this session onwards we would like to talk about its expectations, its moments and many other properties of random variables. Let us review what we did in the past. Random variable we defined as a real valued function with probability space omega and p as its domain and then we said that there are there is always a cumulative distribution function capital F of x is attached to any random variable x and that is defined as f of x is equal to probability of x less than or equal to small x. If x is a discrete random variable then we defined a probability mass function which is nothing but a small f of xi which is equal to probability that x random variable x takes on a particular value x sub i, i varies from 1, 2, 3 etc. please remember it is a discrete random variable and therefore it takes countable values. In case x is a continuous random variable then there is a probability density function attached to it and it is defined with a small f of x as f of x is greater than 0 it is a positive function, the area under the complete curve of f of x is 1 that is integral from minus infinity to infinity f of x dx is 1 and then the probability that the random variable x lies between a and b is nothing but the integral from a to b small f of x dx. We gave some examples in the previous session and now we move on with the further properties of random variable. So, in this session what we plan to do is give a definition of expected value of x which is given as e of x for both discrete random variable and continuous random variable. The expected value of a function of a random variable gx we will define that also. We will define what is known as a moments of a random variable and then we will give you some more details on the measures of skewness, kurtosis and excess kurtosis of any random variable x. So, to begin with let us talk about what is expectation of a random variable. See when you have a data or you have a random experimentation or a random event which has a certain output which we called a random variable we would generally be curious to know that what do we estimate? What do we expect from this data? In other words what is the expected general value of this data? Like you remember in the discrete sorry in the case of descriptive statistics or exploratory data analysis we said that x bar or the average value of x gives a general location of where the data lies. Similarly, expected value of x is also a general location where the random variable x lies when it takes different values. So, expected value of x is of a random variable is like taking a mean value of the random variable it is actually defined as a mean value of a random variable. However, in this session we want to introduce expectation as a function and therefore we start that random variable which is a discrete random variable has a probability mass function small f of x attached to it and we find that x1, x2, x3, xn are the values observed values of this random variable x then the expectation of x is defined as expected value of x e of x is equal to summation of xi multiplied by f of xi you remember f of xi is a probability mass function. So, you can see that this is something like a weighted average of xi because summation of f of xi is 1. So, this is like a weighted average of an xi and that is called expected value of x. In general for a function of x say g of x an expected value of g of x is defined as e of g of x this is a notation is equal to summation from i to infinity g of xi multiplied by f of xi this is again the probability mass function of the random variable. Let us talk of its moments. Now expected value of a random variable to the power k, x to the power k remember that this is also a function of random variable. So, expected value of x to the power k which is summation of xi to the power k f of xi is defined as a kth raw moment of random variable x. Please remember now we are going to have two kinds of moments. We go back. I want you to remember that this is called a raw moment of random variable x. If k is equal to 1 then the first moment of random variable x is nothing but expected value of x and it is also called it is also called a mean value of random variable of x. If expected value of x is mu mean then expected value of x minus mean to the power k is called the kth moment about mean. This is the difference I want you to remember something is called a moment about mean and the other one is called a moment about sorry it is called a raw moment. And the variance of x is nothing but the second moment about mean of a random variable x. So, variance of x is expected value of x minus e expected value of x to the power square and therefore it is this. Please recall this exercise in a different format we have done it for the descriptive statistics when we define when we define the variance of a data x 1, x 2, x 3, x n. This is for a generalized definition for any random variable x. Let us move on. Expectation of random variable in continuous case. So, please remember in general the discrete case we have a probability mass function in continuous case we have a probability density function and whatever the definition we have for the probability mass function generally in the case of continuous random variable it gets replaced by the integration instead of summation. So, expected value of x is defined as integral for minus infinity to infinity x of f of x dx which we again call the first moment as mu it is a first raw moment. Expected value of a function of a random variable x can be defined as integral minus infinity to infinity g of f of x dx. The kth row moment can be defined as expected value of x to the power k and therefore it is the integral x to the power k f of x dx. The kth moment about mean can be defined as expected value of x minus mu to the power k which is again the integral minus infinity to infinity x minus mu to the power k f of x dx and therefore you can also define what is known as variance of x I do not think it needs any definition. If you recall that as such this formula which is shown here applies to both continuous and random cases. So, please note that. So, then we come to another important measure which we have briefly introduced in the descriptive statistics. Now, we put it in here again. If a data is positively skewed it has a long tail on the right side and if the data is negatively skewed it has a long tail on the left side. This is a symmetric data in which we do not have long tail anywhere and it is very nicely symmetric bell shaped curve. The skewness in terms of expectations is measured as expected value of in other words it is the third moment about mean divided by the appropriately normalized variance that is variance to the power square root of variance to the power 3. So, that this becomes a unit less quantity this is defined as a skewness. If skewness is less than 0 it implies the negative skewness. If skewness is greater than 0 it implies the positive skewness and if skewness is 0 it implies the perfect symmetry as in the normal distribution. So, going back to the previous slide here the skewness value as defined as a ratio of the third moment about mean divided by the variance to the power 3 by 2. This will have a positive value because it is a positively skewed this will have a negative value because it is negatively skewed with a perfect symmetry skewness will be 0. Let us move to another quantity which is called kurtosis. Kurtosis measures the flatness of the tip of the frequency curve.  This is a sharper tip which is greater than the normal tip and this is a flatter tip which is less than the normal tip and this is defined by the fourth moment about mean. It is expected value of x minus mu to the power 4 divided by variance square. Please remember this variance is already a square term it is a second moment about the mean. So, you have to match the units and therefore you have the different powers of the variance so that you have a unit less quantity to compare across the data in order to check the kurtosis of the data. So, kurtosis for a normal distribution is always 3 that is if you go back this is a perfectly normal curve the black line if you can see here this is a black line and this is a kurtosis equal to normal and that kurtosis is generally 3 it is 3 not generally 3 it is 3 and therefore there is also another definition called excess kurtosis in which you take this measure of kurtosis and subtract 3 out of it. If the excess kurtosis is greater than 0 then tip of the curve is sharper than the normal distribution. If the excess kurtosis is smaller than 0 then the tip of the curve is flatter than the normal distribution. So, here you will have excess kurtosis greater than 0 here you will have excess kurtosis less than 0 this is excess kurtosis will be 0. Let us look at certain properties of this expectation these are very simple properties because you can imagine that expectation is nothing but an integral value and therefore you can show that for a random any random variable x a and b are 2 constant please remember I am not now going by having a discrete and non-discreet case this is I am saying any random variable x and a and b are 2 constant then expected value of ax plus b is nothing but a expected value of x plus b this is very obvious it can be shown very easily by taking either the discrete case or a continuous case the variance of ax plus b is always a square times variance of x. The moment generating function this is another quantity is defined as mx of t which is expected value of exponential to the power tx remember this is a function of x. So, we know how to deal with the expected value of a function of x the beauty why is it called a moment generating function because if you take a kth derivative of this moment generating function and you put t is equal to 0 you get the kth row moment kth raw moment of the random variable x and therefore it is called a moment generating function. This name is called a moment generating function because it generates the kth raw moment of random variable x by taking the kth derivative with respect to t of the moment generating function and putting t is equal to 0. This is I have shown a simple example by working it out if you take the first derivative of moment generating function t it is derivative of expected value of exponential t to the x which is expected value this is interchangeable. So, it is expected value of d by dt of e to the power tx which is expected value of x e to the power tx which is ex when you put t is equal to 0 here the n e is missing please note here the sorry e is missing. So, it is when t is equal to 0 similarly I leave it to you you can try and show that the second derivative gives you the second row moment of the random variable x and you can generalize it to show that the kth moment kth raw moment will be derived from the kth derivative at t equal to 0 of moment generating function. So, let us summarize what we learned right now we introduced a function called expectation of x and we defined it for a discrete case as an expectation of x is a summation i is equal to 1 to infinity xi times f of xi which is a probability mass function of x, x and then we defined in the case of x is continuous we defined it as an integral from minus infinity to infinity x f of x dx. We defined the kth raw moment of x defined as expected value of x to the power k we gave a measure of skewness and kurtosis these are all also called coefficients of skewness and kurtosis and we showed with respect to normal distribution what values they take skewness less than 0 indicates negative skewness skewness greater than 0 indicates positively skewed data kurtosis less than 0 indicates that the tip is flatter flatter and when kurtosis is greater than 0 it says that the tip of the curve is sharper. We also learnt that an expected value of a function of an x can also be defined as a in case of discrete as a summation i is to 1 to n infinity g of x i f xi otherwise you integrate in case x is a continuous function. Thank you.


Hi Nessar just how satisfying was this afternoon's performance considering the strength of that White Sox lineup? Yeah I mean very respectable lineup you know went in there and got the job done and get gave some length again and then came out with a W. You've been a guy that's shown a lot of different looks throughout the course this season whether you're dropping down or you're changing your arm angle just how difficult is that to learn for you and to do seamlessly on the mound throughout the course of an outing. Yeah I mean I've been doing it for a couple years already I want to say five or six years and you know it's based off of feel and it's worked for me so far so hopefully I can continue to do that. We'll take the next question from Bob Lappish. Hi Nessar we're just asking that Luke Voigt about what it's like to face a picture like yourself. He says it's just it's just different because everybody else throws so much harder you have a very unique approach to the craft so my question to you is if you could wave a magic wand and tomorrow start throwing 98-99 and pitch like everyone else would you do that or would you continue to do what you do from different angles and different sequences and just do your thing. You know I would love to throw a 97-98 wave of wand around and throw a 97-98 obviously but as long as I can get your job done and keep getting out which is the most important thing you know I stick to what I got. And the way you at times drop down to three quarters is that something that you decided to do at the very last second as you're going to wind up or is there some planning in that. Yeah I mean through the game plan me and Hagee or whoever is catching me on the play obviously you know you know we like to go about things accordingly and I think you know when the time comes and the opportunity arises we try and do that. Usually on two strikes. Usually on two strikes but I like to mix it in every now and then so nobody knows that you know we can do it at any count. Any time right. Okay thank you. Brian Hoke you have the next question. Hey Nestor in general what's the clubhouse vibe after this series taking two out of three from the White Sox and nothing really came easy for you guys. Yeah big serious I mean you know coming into this roadside I'm sorry we knew where we were and what we needed to do and I think we took good care of that and you know clubhouse five is good right now got the serious win and hopefully continue that. Did this feel like the kind of series that could be a postseason match up for you guys. Yeah definitely I mean White Sox are a very good team they're first place right now in their division and you know it's probably somebody that we could potentially face you know down the line. Thank you. Joe Sherman you're the next question. Nestor a lot of this success over the last month or so has been fueled by guys like you who you played with in Scranton or seem to lose jobs over the last few weeks. I wonder what you think that says about your group. Yeah I mean we've done a really good job of guys coming up and holding the fort you know some guys have gone down with the COVID I out and some injuries unfortunately but the guys coming in like I said from Scranton you know they're hungry and they want to keep this team afloat and we're battling right now.


I am delighted to welcome our undergraduate and graduate students for fall 2021 and what will be a full Husky experience. Many campus traditions are underway as we welcome new students with Husky's first four activities and convocation. Get ready for Main Street where you can learn about and engage with more than 200 groups and organizations on campus. Getting involved is a key to your success as a university student. Experience the deep connections of our Husky community through the festivities of homecoming. Soak up fall colors with a stroll around Lake George or along the banks of the Mississippi. Keep an eye out for some snowflakes and our Husky's first snow celebration. Replenish in our state-of-the-art recreational facilities and show our Husky spirit as we cheer our student athletes on game day. And remember, help us to protect the pack by staying informed of campus safety protocols for masking up, testing and vaccinations. Thank you to our faculty and staff for preparing for what will be an excellent academic year for you. I'm looking forward to seeing you at student presentations, competitions and hands-on demonstrations resulting from your class projects and research. So get ready for a great fall and go Huskies!


They lie to us. They lie to us all our lives about what soldiers do, about what cops do, about what journalists do, about what presidents do. They lie to us all our lives about how laws are made, about how kings are made, about how culture is made, about how norms are made. They lie to us all our lives about what our minds are for, about what our hearts are for, about what our neighbors are for, about what foreigners are for. The world is confusing, and they make it more confusing, on purpose, because they thrive in confusion. And now we watch as the world turns to ashes, carried on the back of confusion, made by newsmen and corporate creatures who eat ashes. And we stumble in confusion, eyes darting behind lenses made of confusion, trying to understand with minds made of confusion, trying to feel our way through with hearts shackled by confusion. Searching for an angel egg we know is there but can't seem to find, because who turned out the goddamn lights? And why can't I figure out which way is up? And oop, off we go chasing distractions again for the next news cycle, the next election cycle, the next spin cycle, the next amnesia cycle. They lie to us all our lives, and then when we catch them in a lie, they say, ah, the person who gave you that information is bad, or ah, that came from Russian hackers, or we made a mistake, but we don't do that anymore, or don't be a conspiracy theorist, or hey, look over there, or ah, we already knew that, it's no big deal, and then everyone forgets, and the confusion turns on. But the angel egg is there in our mind's eye, in our heart of hearts, haunting our dreams, calling us home, the embryo holding our original eyes for whenever we are ready to claim them. They lie to us all our lives about what our country is, about what our government is, about what money is, about what nature is. They lie to us all our lives about where the answers are, about where our friends are, about where our enemies are, about where our claws are. They lie to us all our lives, and they make the world confusing, and beneath that confusion we are watching them with our original eyes, with our primal eyes, with our natural eyes, with our uncivilized eyes. Watching them, like lions, watching a gazelle, and waiting with primordial patience until it is time to make our move.


Hi, Mac. How are you? I'm good. Thanks. You look nervous. Well, it's going to be mine and Lindsay's fifth year anniversary since we started dating. I'm trying to plan a special evening for us. Five years? Congrats! Are you proposing? Haha. Not at the moment. Oops, I need to get back to work. Have a good one, Cindy. You too. Wow, $10,000. That might be a bit rich for my blood, but maybe it's time to get saving. Oh well, time to move on to other things. That's weird. So, have you been thinking about getting hitched? Cindy, can you mind your own business? Well, I saw that WeddingRings.com and Tuxedo World started following you. That's strange. I checked a Wedding Ring website, and since then I keep getting all these emails, texts, and now Twitter followers about wedding stuff. Did you have to create an account on the website you used? Yeah. And you read the website's privacy policy. No, I figured it wasn't a big deal. Well, just so you know, you should make sure to read the privacy policy next time. The site may have written in language to share your information, which is why you're getting those messages. Oh, I'd better go back and cancel that account before Lindsay starts seeing all these wedding ads. Uh-oh, Mr. Timmons wants to see me in his office. Well, good luck, and remember to read the fine print. Hi, Mac. I called to my office to discuss something. What would that be, sir? I just want to remind you that while at work, it's important to stay focused on your job. We monitor everyone's internet activity, and I saw that you were doing some personal shopping today. Just make sure to save that for your lunch break and use your own device. Okay, don't worry, Mr. Timmons. It won't happen again. Alright, sounds good. Oh, and by the way, can I be invited to your wedding?