 Okay good morning again what we are going to do today is build upon the last lecture when we found out that somehow this idea of drop size distribution velocity distribution and if I may even say temperature distribution these are all now becoming relevant quantities. So I need to get some mathematical framework in which I can talk of these distributions okay. So we will see the whole idea of this framework is going to be that from knowing some information of all the drops that have sampled thus far okay whether it is in a spatial sense or in a temporal sense. I have no idea what to expect from the next drop in a with a 100% surety okay. So if I sampled half the frame in the spatial sense and I am about to step into the second half I have some estimates but I cannot be 100% sure what to expect of the next drop I am going to sample. This is the idea that this is not deterministic but I need some sort of a statistical measure because there is some sort of a stochasticity there is some uncertainty okay. Now we will probably about a third of the class later on we will see what are the possible sources of this uncertainty but for now we will just say look I do not know what the next drop is going to be so I need some statistical measures of these distributions. So we are going to try and understand that those okay. And one of the fundamental concepts we need to understand is this idea of probability okay. We are going to go to high school probability for a moment and we will sort of try and see if we can build upon it okay. So we will take a regular six sided die so I have a six sided die I roll it and I get some I get an outcome okay. So let us say I get these as my these are my outcomes from these outcomes I can build a probability of the number which is the number of times one occur divided by the total number or I will use subscripts to this is a fairly obvious okay this is called a posteriori probability that is I only can build this after I have done all these trials okay. But then you say that this die has not changed the floor on which I am rolling has not changed. So this can also become my a priori probability for the next role so I can use this information to predict statistically to predict what could be the outcome for the next role okay. So these are some simple high school concepts what we want to do is see so I have this probability of getting one two dot if I have a perfectly fair die I know that these are all going to be one six if I have a loaded die and let us say if I have this die and this bottom part is all lead and this top part is all plastic. So this is the case of a loaded die then I am more likely to get the number four from the die role then other numbers that does not mean I will not get the other numbers at all I will get but I am more likely to get the number four. So I can now write a probability of getting any number I want to use the subscript notation as a function of I so the probability of one two three four five six whatever there is some probabilities associated with each of the six outcomes and the sum of all the probability has to be equal to one that is our basic you know high school in high school understanding of probabilities because there is no other outcome possible okay. Now if I take this six sided die and I make it a 20 sided die I can create a 20 sided die and it is got 20 flat facets so every time I roll one flat facet comes up to the top. So I have one of 20 outcomes possible and for every one of those if I do a sufficiently large number of die roles I can build an aposterior probability from each of the different sets of count of each of the outcomes right. I can do this for 20 sided die can I do it for a 40 sided die sure I roll it one faces on top the only limitation I can go from six to eight ten twenty forty hundred I can keep going as long as I do not get into the point where there are an infinitely many outcomes as long as I have somehow countably many outcomes from this die role this theory works. So if I say that there is only one two three four two twenty thousand those are the integer outcomes possible in this experiment in this trial I can do if there are twenty thousand outcomes I have to do a very large number of trials but I can build this with some patience I can build this probabilities of each of the outcomes right. But if there are infinitely many outcomes okay now I want to talk of a case where there are infinitely many outcomes what does this mean. So this works first of all what do I mean by infinitely many so this works this theory works okay I want to understand what the so what happens if this is violated because if I have an infinitely many outcomes the probability of any one outcome is nearly zero okay we will take a simple example of such an outcome case okay I will take a hoop so instead of instead of a die role like we saw we are going to now roll a hoop a hoop is around bangle kind of thing. So I am going to mark it just like the dial on a clock okay so I am going to say this is twelve o clock this is three o clock this is six o clock that is nine o clock okay this is a hoop okay I am going to roll this on a floor where the floor is soft okay I need to so in other words if I roll a bangle it will never come to rest as long as the floor is floor does not provide enough friction okay just a simple understanding of how to I need the role to stop for me to count the outcome okay. So I roll this hoop and it comes to rest so let us say I am very lucky very first outcome is where twelve o clock is on top okay and just for the sake of argument I will mark that as my zero degrees and count an angle phi going zero to two pi so my outcomes are and my outcome is the point on top okay that is my that is what I call the result of this trial. So I roll this and let us say if the number three was exactly on top then my outcome is pi by two we will first just drop on our intuition briefly before going into the theoretical aspects if I roll this hoop ten times okay I do not know the initial condition I am just sort of randomly rolling it just like I did I do with a die what is the chance that any two numbers in this would be exactly the same exactly the same it is almost zero okay I only have to say almost zero because I can never say zero but I do know I have an infinitely many outcomes because all the real numbers between zero and two pi are possible choices of outcome and between any two integers there are a set of real numbers in this case between then the real number zero and the real number two pi there are an infinite infinite number of real numbers. So I am now I have to somehow construct the same a posteriori probability for this hoop okay so let us say this bottom part of the hoop has got some lead in it so this bottom part is lead and the top part is plastic this like before so I have created a biased hoop so I will now go back biased hoop role so every time I roll if I know that the bottom part is got lead then I know what to expect see but the idea of using statistical description is that I do actually do not know what part of the hoop has lead through my trials I want to understand that okay so if I do this I can sit and count numbers coming out from my outcomes they will all be real numbers irrational some rational a few rational many irrational I have to if I have a way of actually counting the angular position I will get as many numbers as as I as the trials I do and chance of any two numbers being the same in a finite set is all most zero so how do I construct my a posteriori probability or for that matter some information about what the hoop is made of ultimately that is what we are after okay so what do I do here so I create this list and then I have to go to this histograms. So I am going to say instead of assuming that every individual real number outcome is the same I am going to go through a process called binning so I am going to now say all outcomes from zero to thirty degrees are alike so I am going to not distinguish between twenty nine point nine and even thirteen I am going to put all of them in one bin now I can do a count because even if two outcomes are not exactly the same since I have sort of created this equal this similarity between outcomes I can put them in the same bin moment I put them in the same bin they are all the same I just need a count okay so this process I can they are alike so once I do this I create this bin in the in this outcome so now I have a I only have twelve bins the way I have done this the last bin is three hundred and thirty this is my last bin so I have one two three so I have taken a hoop and created a twelve sided die that is essentially what I have done okay I can now go back to my old theory and find the probability of bin one probability of bin two dot dot dot create the same histogram just like I did with the six sided die okay now remember that these divisions are entirely artificial the hoop does not have any distinguishing feature between twenty nine point nine and thirty point one correct but I have created that distinguishing feature by putting this line at thirty degrees so I have an obligation to check if this distinguishing feature that I have put in there has a consequence so in other words if I do this and I find the probability of one probability of two dot dot dot probability of twelve if I recreate a different set of probabilities following a different binning sequence so this is I will say binning sequence one if I do it following a different way where my this will be the thirty sixth bin right thirty fifth or thirty sixth think it is a thirty sixth so essentially I have now a P one P two I will call these primes just to distinguish between the other ones I have these thirty six P primes it is the same exact data set so let us say I sat did this dire old twenty thousand times the my biased hoop roll twenty thousand times and I got twenty thousand real numbers and I did binning in one way in the first part binning in another way in the second part the histograms will look completely different right but clearly it is the same data set I have to convey the same statistical information in both how do I do this okay in order to make an equivalence between these two I have to define what is called a probability density okay so if I take this probability density say for example in the first case of in the first binning example I take three of these bins here I will write down the third one also just to three bins in case two map to one bin in case one so all the count that goes into three individual bins and in this case two map to only one bin in case one so I can clearly say that the count in the three bins for 0 to 10 10 to 20 and 20 to 30 each one individually will surely be less than the count in the 0 to 30 which also leads us to sort of an intuition that the wider I take the bin the more count I am going to capture in that bin the narrower I make the bin the less count I am going to capture in that bin so if I did this 0 to 1 1 to 2 degrees the count in each bin for the same set of trials would be smaller okay so I have to have some way of finding not the probability or going beyond probability and finding the probability density around a given value okay so if I take if P i is the probability of the ith bin probability of finding an outcome in the ith bin I can now define another f i which is given by this P i divided by delta x i so if I take all the range of outcomes say 0 to 360 degrees or 2 pi if I take one value x and if I take all the outcomes in a bin that is delta x wide around that so this is the probability of finding an outcome in this delta x width okay I will call this P of x because I now depending on the value of x I choose the probability could be different because that is my idea of a biased hope if x is closer to the 12 o'clock position P of x is going to be higher okay so if I take this P of x as the probability of finding a drop in a bin that is with delta x around the value x so P of x so just to finish the discussion here f of i is what we will call is what leads us to the idea of a probability density so rigorously we will define we will talk of what it is P of x is the probability okay f of x is not equal to P of x divided by delta x but really speaking it is the limit so I will rewrite that so as I come closer and closer and closer to that point x the value of the probability of finding an outcome in x minus delta x by 2 to x plus delta x by 2 becomes smaller and smaller actual probability and the width itself is becoming smaller but the limit is a finite value that limit is what we will define as our probability density so the idea you have to sort of understand this idea of density it is like I know the probability of any one number is 0 but how dense is the outcome is the outcome distribution around that point that is all I care about so if I go back to this f of x now f of x has units of probability has no units this is a point that you have to understand f of x has units which is the same as per radian or per degree look at what is in the denominator I should I should not say delta x with a it has units of x basically in the denominator so for the case of a bias group role the probability density has units of per radian or per degree depending on what we choose to plot as the independent coordinate so per degree this is the this is the density of outcomes possible so let us find some simple argument so if I say I will rewrite what I wrote here I will rewrite this statement in a more correct way if f of x is a pdf then f of x times dx is the probability of finding an outcome x plus dx so in an infinitesimal neighborhood around the value x or near the value x you look at how many outcomes or what is the probability of outcomes that you have so essentially let us draw this in a graphical sense so if I have f of x is now a function okay so I have defined a probability density function so I can draw a graph of it something like that let us just say then at some value x going from x to x plus dx f of x dx f of x is a value at that point f of x dx is essentially the area of that infinitesimal strip that infinitesimal strip the area of that infinitesimal strip is the probability of finding probability density a probability of finding a value in the limit x to x plus dx in the range x to x plus dx likewise you can see that the probability of the same width here would be higher so if I take all the range of values of my hoop going 0 to 360 degrees the idea that no other outcome is possible other than values between 0 to 2 pi or 0 to 360 tells me that integral 0 to 2 pi f of x dx sorry whatever is my probability density function f of x f of x dx is a probability in a thin strip and that integral of that f of x the summation over all the areas of these thin strips which is what we call integral going from 0 to 2 pi has to be equal to 1 that tells me that is just simply coming from the criteria that no other outcome is possible other than values between 0 to 2 pi okay so let us see how we can use this and let us apply this information to first let us continue our hoop discussion and we will finish it so let us say I know it is a biased hoop around 12 o'clock I am seeing like more values come up near 12 o'clock so I postulate a model okay so I have not yet done all these experiments I have done like 10 experiments found that 12 o'clock is coming up or points near 12 o'clock is coming up more often than the points near 6 o'clock I jump to a model okay so I say f of x is of the form 2 plus cos x so if you go back look at our definition of what I want to define a model for phi not x sorry where did I come up with this function 2 plus cosine phi that is because if I draw the graph of 2 plus cosine phi instead of using this f I am going to use this function I will call this function g just for the sake of differentiating it from f which I will use later on this would be the graph of 2 plus cosine phi so at least graphically it captures the idea that phi values near 0 which is also the same as near 2 pi are more probable than phi values near pi it sort of graphically captures that information and I am happy to start with this model okay what do I know from here on is g of phi a probability density function no not yet all I have done is sort of postulated a function that seems to capture my imagination that in itself does not make it a probability density function you need to make sure that integral g of phi d phi gives you the area under the curve so if I do this for this case integral of cosine phi is sin but the limits are 0 to 2 pi so that becomes 0 so the value of this integral is 4 pi so area under the curve g the way I have drawn it is 4 pi so if I now define a new function f of phi equals 1 over 4 pi now this is a PDF this qualifies to be called a probability density function because the area under the curve in the range of values expected is actually equal to 1 so if I create a plot of this the value of this is 2 over 4 pi and that would be so the value in the previous case also would be 3 for a maximum value for g right so same here would be you will go from a maximum probability density the maximum probability density is 3 by 4 pi the minimum probability density is 1 by 4 pi this is the case of a biased hoop so if I had a perfectly fair hoop where all outcomes are possible you can see that the area under that curve would have to be equal to so area of the curve going from 0 to 2 pi of some constant value has to be equal to 1 for that to be the case the constant this would have to have a value 1 over I am sorry 2 over 2 pi 2 over 4 pi or 1 over 2 pi I want to write it as 2 over 4 pi or 1 over 2 pi so this idea of a probability density of the unbiased hoop is also a number just like a probability of an unbiased die is a number like probability of any outcome is 1 by 1 over 6 probability density is also is a function probability density function but the function takes on a constant value equal to 1 over 2 pi so if I know so I started to reconstruct this from some model like I said you know I have this model of a biased hoop that is biased towards the 12 o'clock comparison in comparison to the 6 o'clock and this is where I ended up if I want to start with just simply going through the process of doing multiple trials and then reconstructing these probabilities what do I do if I take to the experimental route and say I have done this trial like 30,000 20,000 times a large number of times I have rolled this hoop over and over and over again noted the real number that showed up on top okay so if I take that list of real numbers go through the binding process and create an f of x i so the first thing is to take using the old terminology find a g i which is equal to p i divided by delta x i so I take all the outcomes find the set of outcomes that fall in a certain bin x i to x i plus delta x i this is just the count so this is n i n i is the number of outcomes I want you to also note one difference between what I have just written in the very first version that I wrote of this divided by delta x thing where delta x itself can have a subscript i in other words I can now have the ith bin here b of one delta x and the x i x j to x j plus delta x j can be a different width so I now have a count n j associated with that jth bin so I am not restricted to somehow delta x being uniformly spaced delta x being equal for all the values in this binding process I can choose whatever bins I want and I can place the outcomes into these bins depending on the actual value of the outcome once I do this n i divided by the total number which is essentially sigma n i gives me this number p i so p i is the probability of outcome in the ith bin if I now define an f i which is equal to this p i divided by delta x i this automatically gives me a p d f of x i p d f at x i so it is like a it is the value of the function at x i so if I had a model see the problem with experiment is I only can compute these at discrete points of x I can do this at some at value 0 degrees 30 degrees 60 degrees etcetera so I take all the values in the range between let us say 60 and 61 put them in the bin and then find the probability of values in the range 60 to 61 that gives me and divided by that one degree that gives me a probability density okay now so if I do this and I plot these functional these values f of f i and x i so I will do this for the for the case of a die where I am plotting this going 0 to 2 pi let us say if I do this with nine different bins this is these are the nine values I get these are the actual values of f of i at each value of x i so at each value of phi i so this is phi axis at this value of phi i this is the value of f i how do I know that this is a p d f remember our condition that for probability density the area under the curve has to be equal to one because we ensured by this definition of probability that the probabilities will add up to one the p i is nothing but f i delta x i which is like the area of the strip around x i that is of with delta x i so in a in the sense of a numerical integration we have approximated the area to be equal to one as far as that as far within the accuracy of the binning so the smaller the delta x i values the more accurately this number approaches the real probability density value okay so if I use if I get these x's through some bin one remember our using 30 degree increments or some using 30 degree increments I get these x symbols if I do the same thing with the bin two which is 10 degree and if I do that and plot circles my expectation is that the circles would fall something like that so I have developed a better approximation of the actual function f of phi by taking finer bins and as I go towards finer and finer increments of these bins I recover a better and better approximation of the actual analytical function but that point once I reach this point I need a I need to do a model I need to find a model so like fit an equation or find a model from some dynamics and then come see how this data fits it but our idea of pdf can be approached from both performing several trials and coming to a point where you can reconstruct this graph this graph which does not depend on this is the key thing remember that was the point that was the problem we tried to solve we had two different binning sequences that gave us two completely different set of probability numbers how do I reconcile the two by essentially figuring out this idea of probability density and if you can plot the probability density at a given phi location phi I which is f I whether you do it in one binning fashion or another binning fashion would only lead you towards the same answer so if I did this in one degree increments just for the sake of argument if I did this in one degree increments but took one degree on either side of my chosen value of phi I so I go from 0.5 degrees to 1.5 degrees as the phi I for f I to compute f I at one degree and 1.5 to 2.5 etc etc or whether I do it in one degree increments going 0 to 1 1 to 2 etc they will give me exactly identically almost similarly the same answer in other words the bin width is the only parameter that determines how closely converged I am to the real value of the real probability density function this is going starting with trials and trying to recover an analytical function if there is as though I have done an infinite number of bins that is the delta x tending to 0 so if I say take 30 degrees take the probability from of finding an outcome from 25 to 35 27 to 33 29 to 31 and then come to smaller and smaller compute f of I this will converge to one value at that point if I have a sufficiently large number of trials to start with that is the only problem okay alright let us quickly recap what we have done we started with the idea of probability which is usually only valid for integer outcomes and then develop this concept of probability density which could be extended to real outcomes okay. So if I go back to the spray arm I drop on an integer axis or a real axis if I ask the question clearly the answer is I certainly not on integer axis which means I only have to assume that there are an infinitely many outcomes therefore we have to go to PDF therefore we need to understand this basic mathematical framework okay thank you.