 So, next we are going to look into something called, the first thing is, so I am sure like all of you already heard Gaussian random variable right many, many times. So, let us formally define that Gaussian distribution we are going to denote it as with the symbol mu sigma square where mu and sigma square are the parameters like earlier we had P n as the parameter for the Gaussian distribution mu and sigma square are the parameters and here mu is a real number and sigma square is a positive quantity and it could be also take zero value. Now this is we are talking about continuous random variable right. So, we are now going to define the probability density function of this random variable for a Gaussian random variable we are going to define its density as of x and this is provided my sigma square is strictly greater than zero and if it is sigma square equals to zero then it takes a degenerative format in that case it is simply going to be of this form 1 if x is mu. So, this is if sigma square greater than zero otherwise we are going to it takes a degenerative form where it will have only one possible value at x equals to mu and zero and its characteristic function is given by function exp. So, it so happens that for this Gaussian random variables with parameters mu and sigma the mean turns out to be the parameter mu and the variance turns out to be the parameter sigma square. So, we are just going to call a random variable Gaussian with the mean something and variance something. So, the mean is nothing but that parameter mu and sigma square is the other parameter ok. So, here if you look into this, this is a pdf and this function is valid for all r as for if for all r for all x now let me know if sigma square is not equals to zero then my cdf is going to be defined like this for all x if not sigma square is zero then it has this format and it is character's functions are given like this. So, let us just because this is one of the most popular distributions that is used let us see so let us say this is my mu here and this is my x and this is my x. So, just looking at this function here in variable x is this function symmetric in x about what about mu just by looking at this you can see that because this is a square term here it will be symmetric about x. So, whatever and it goes on like it goes on like it is defined for it does not look symmetric but fine it is symmetric and suppose let us say this is for some sigma square and mu. So, suppose I increase this variance sigma square how do you think this lobe will look like I will keep the mean same suppose I want to have another random variable Gaussian random variable with the same mu but let us say it has a variance which is larger than this it is going to be does this lobe it become fatter like and it will like maybe like something like this maybe this sigma 2 square is going to be larger than that but now if you have a some sigma 3 which is going to be smaller than sigma 1 how it is going to be so in this case it could be like it is going to get thinner and eventually if you trying to make it sigma square to be 0 it will be just this just this a delta at x equals to mu this is going to just take the value of 1 and it is going to be 0 ok. So, if it is sigma square equals 0 this height is just going to be 1 that is the only point where it will have positive mass and 0 everywhere fine. So, I have a Gaussian distribution like this where you want to use it you know it is very popular right where it is used or where you think it is going to be of natural use or you feel that I cannot imagine anything where this can be used IQ distribution of humans animals humans ok. So, what is the range of IQ but range kaya 0 hota hai kaya negative hota hai 0 maximum kuch bhi is there a so is there an upper mode on IQ or anybody somebody can have infinite IQ theoretically infinite. But then one side is fine right like negative to nahi hore hae so 0 to infinity. So, this one is by the way I have defined it has also negative values here. So, if you have only positive values maybe you do not want to use it right what could be other example yeah, but again if you just if you want to begin with you know that the height is going to be some in the range let us say some 2 feet to 6 or 8 feet and it is never going to take a negative value right, but still yeah you want to cut it off and yeah you can condition, but before conditioning I am talking about yeah yeah fine so maybe that is one possible thing like let us say you are like a big, big, big angel investor maybe you can make billion, billion dollars or if your investment goes dish like you may make negative billion dollars and so you have a big, big range right maybe. Other application is in fact tolerance design for the company. Tolerance design ok. So, that can you make this more concrete example. So, suppose we are manufacturing natural products that we have set up the mean value to be something suppose let us say 15 microstats so we just do not want to you know increase that tolerance yeah taking that to be point one or point four that is if I need to be that there is not the manufacturing effect or something like that. So, another example I can think of is like let us say some kind of deformation with temperature right like I can apply temperature there can be both positive and negative right in some critical applications temperature can have a wide swing. So, if you are going to apply this positive temperature how your deformation or whatever the related quantities look like and when the temperature goes below negative how it looks like. So, there are again like it is just about as you can see that it finds many, many application, but it has this need to have both positive and negative value that is not a constraint like you can always you can adjust about your mean and take care and. So, as long as you feel that there is something heavy concentration about a particular value and then things slowly died down around it maybe this is a good thing. Even in case of a population or a height distribution like height let us say this is like about 5 feet there is maximum number of people around 5 feet right most of them like then other skies maybe like just died down maybe if I go here itself like around 5.5 and if I just take it like maybe 4.5 here already significant reduction is happening. So, this is in that way at least in this region it is still like looking more like a Gaussian distributions right you can still continue to use that, but provided that you appropriately. So, you mean right away when the Gaussian should come to your mind whenever you see that there is something like there is something going to be symmetric around some value and there is a going to be like as I said diffusion or dispersion about it both on the left and right hand side then you will do. So, fine you may feel that okay this is a good, but I do not know how concentrated this lobe should be this should be thin or it should be big that you find it from your data. You do lot of experiments and see that which what how you going to fit your data so that you get the right parameters mu and sigma square. So, fine so this is one thing now we are going to see another distribution called exponential and the PDF of this distribution we are going to take it as f of x goes to lambda times e to the power of lambda x and this is going to be for all x positive and it is a characteristic function. So, this distribution is going to look like if you give me lambda. So, this is going to start at x equals to 0 this is going to be what simply somewhere it is going to start from lambda and where it is going to go it is going to go to 0 as x increases this is my f of x suppose in this I increase my lambda let us say this is for some lambda and if I increase my lambda here what do you think how this another function is going to look like. So, at least I know that if I am going to increase so let us say this is for some lambda 1 and let us take a lambda 2 which is bigger than this lambda 1 at least I know this parting point was lambda 1 the other point is going to be lambda 2 it is going to fall much faster because this is coming in the exponential term. So, maybe it will start faster, but it will decay much much faster okay fine. So, I have a such a probability density function which is taking positive values I mean which is defined on positive real half. So, where do you think such a and you see that as x increases this is going to like fall rapidly it is falling actually exponentially fast. So, where do you think such a distributions can be helpful what kind of things you would like this to be used to model for example let us say some component inversion what machine like component what let us say I have a component I want to predict the lifetime of this machine like fine let us say I have a bulb and I want to see how long this bulb is going to survive before it breaks up. So, it may last let us say it is count let us say let us count life in terms of second it may last 100 second 200 seconds or maybe 1000 seconds and maybe like 36000 seconds or whatever like after that it may break right. So, you feel as you see that if you are expecting it to survive longer and longer that probability is going to be falling rapidly right. So, that is what in that case maybe this kind of exponential distribution is going to be much useful and in that case if you have two let us say bulbs one is of a better quality than the other and so which one you want to assign higher value of lambda the better one or the poorer one, poorer one right because with that it is going to decay much faster. So, another good thing about this exponential distribution is it has the memory less property we already discussed. So, we discussed memory less property with respect to its distribution geometry distribution geometric was a discrete distribution. So, it so happens that this is a continuous distribution, but it also satisfies that memory less property. What does that mean? What does that mean suppose you have a random variable x is going to take value larger than x plus t, given that it has already taken value larger than t. If x is memory less what do you expect this quantity to be? You expect this to be just probability that greater than or equals to s. So, do please verify that if you have CDF like this it is indeed like if x has a CDF like this this satisfies this property ok. Next is uniform. So, uniform I am going to denote it by d where the CDF of this is all a b to m b and I am going to 0 otherwise and its characteristic function is just 1 by e to the power j u b minus g u a by g u minus b minus and again its mean is going to be u, you can work out that its mean is going to be simply the average of the points a and b and its variance is b minus a whole square by 12. So, what it is saying is like suppose if I have a, let us say I have I want to assign, let us say I am interested in some interval a and b. I want to say that anything in this is equally possible ok. That means what it is saying that for all x in this range all of them are going to take the same value. So, that is this the function u here is going to be simply this quantity where this quantity here is 1 upon b minus a. So, this CDF is like a flat in the interval a and b that means if you interpret in terms of the intensity at every point the intensity is kind of the same here and anything outside we are not assigning any probability. So, you can just see that the mean of this is going to be simply the average of these two points and the variance you can compute like this. So, where do you want to use such a distribution? Let us say in terms of the temperature right like often like what did you we can say that the temperature of a day may be like uniformly distributed in so not necessarily like a temperature. Let us say anything you want to apply like let us say pressure on something the amount of the pressure that this guy can sustain could be any uniformly distributed between the two extremes. Let us say some minimum amount of pressure and some maximum amount of pressure it could be taking equally like it can break at any of this amount of pressure equally likely. So, this is one possibility other anything other useful things you can think of like maybe like fine you have grades like a, b, c, d these are discrete but think of instead of this think of these are numbers between 1 to 10. I can think of like the distributions could be uniformly like any the distribution of the marks could be uniformly in between 1 and 10. Can I use that as good? So, you want to be uniformly distributed the marks between 1 and 10. Okay fine that is and then comma and then alpha well and real so the PDF of this. So, actually when you have this Bernoulli toss right we have only head and tail. Suppose you assume it is a fair coin that is taking head is half and also tail is half. So, then we can say that both my outcomes are equally likely right or my head and tails are like uniformly distributed that is in the discrete version. So, what we are now talking about is uniform distribution in the continuous domain where any value in this interval is like equally likely. So, PDF is alpha to the power n and find that its characteristic function is. So, another useful distribution is comma distribution. So, it has again defined on for n integer and for any x which is non-negative valued gamma and this is a gamma function where gamma is defined like this. So, what is the useful of this function? So, it so happens that gamma function is related to exponential distribution in what manner if you are going to take gamma to be n alpha this is related to exponential distribution through it so happens that this distribution can be expressed as sum of n exponential distribution with parameter alpha. So, if you are going to take this gamma to be alpha and add such n exponential distribution then you will end up with gamma distribution. If needs another requirement that these are independent further we will define what we mean by independent today later in the class. So, if we add n independent exponential distributions with parameter gamma then we will end up a distribution which is gamma n alpha. Last one is yes. So, it is characteristic function is bit complicated I am not expressing here. So, how what is this Rayleigh distribution? So, this Rayleigh distribution I think are not going to use this much but this may be useful to this communication network people it so happens that it comes very handy when you have to deal with multiple Gaussian distributions. It so happens that this gamma Rayleigh distribution with parameter sigma square is nothing but the annual of k. So, it is going to be so first let me explain this. So, if x and y happens to be Gaussian distributed with parameter mu and sigma square where mu is set to 0 that is like the sigma square is going to be same as this sigma square. So, then it so happens that the distribution of this turns out to be this Rayleigh distribution. So, suppose you are dealing with let us say two quantities random variable x and random variable y which are both Gaussian independent mean 0 variance sigma square and if you need to happen to deal with such a quantity. So, instead of looking at each of the separately you need to end up with so this is like what this is x square plus y square will give what to you kind of circle and you want to take a square of that and you want to understand this is kind of a radius of this circle. You want to understand if at all it has a constant radius this radius could be varying you want to understand how that radius is distributed it so happens that that is can be characterized as Rayleigh distribution. So, it so happens that for a it can be thought of as instantaneous of a 0 mean narrow band noise. So, okay now just quickly coming back to this uniform distributions. So, uniform distributions when it is going to be useful uniform distribution you want to use when among the possibilities you do not have any prior information for example let us say I let us say tomorrow Mumbai's temperature is going to be between 20 to 30 degree Celsius. But I do not have any prior information like what it is going to be like let us forget I do not have any I just born today and I have to choose a what is going to be the temperature tomorrow and I was told that it will be something between 20 to 30. So, if I do not have any prior information what I am going to do is I am going to think that everything is equally possible I do not have any prior information right. In that case when you have to deal with a situation when you have to model some randomness about which you do not have any prior information you would go with uniform distributions. For example in this case exponential like when I take a lifetime I pay bulb I know that as time progresses the probability that the bulb will break down is going to increase right. But in such case like suppose I do not have any such information like let us say I have a bulb and I do not for like it can break down any time within a 0 to let us say 100 seconds. If that is the case then I would go and like to model it as a uniform distribution. If I do not have any prior information maybe that uniform distribution comes very handy to us.