 Welcome to dealing with materials data, in this course we look at collection analysis and interpretation of data from materials science and engineering, we are specifically looking at probability distributions and we have looked at several discrete and continuous probability distributions, we are continuing with continuous probability distributions and one of the important distributions which is important in material science engineering is called Lorentz distribution or Cauchy distribution. Lorentz or Cauchy distribution has two parameters, there is a peak position or mean which is given by mu and there is a width w and the probability distribution function is given by this, so 1 by pi w 1 by beta x and 1 plus x minus mu by w whole squared whole inverse. At x is mu plus or minus w the function is half its maximum height, so that is the full width at half maximum is 2 w. From this terminology peak position width and full width at half maximum etc., you might have guessed that it is important in spectroscopy measurements and also when you do retail refinement of x-ray peak profiles for example, it is not uncommon to see a combination of Gaussian and Lorentzian to fit the peak profiles, so that is the reason why this is a very important distribution and you can use Cauchy, D Cauchy, P Cauchy, Q Cauchy or Cauchy in R to get the probability distribution function, cumulative distribution function, quantile function and random deviates and we will generate plots for the Cauchy distribution with the mean position at 25 and the width of 1 for example and as usual we will plot and see what happens. So we are going to generate 3 plots, the first one is the probability distribution function, the second one is the cumulative distribution function and the third one is a quantile plot and in all cases we are going to use a mean of 25 and width of 1 and so this is how the Cauchy distribution function looks, so it must remind you of the spectroscopy or x-ray diffraction peaks that you typically tend to see, of course we can just plot probability distribution function to see this better and so here is the plot. So it looks very familiar I am sure because it looks like the peaks that you would see in some of these experiments and of course let us generate some random deviates and plot the histogram that should also look very similar to this and so here you can see maybe we should generate more deviates. So you can see or maybe we should need more, so you can now very clearly see the peak and it is at 25, it is expected at 25, so you can, so you can see that the data does follow nice Cauchy distribution function and there is one more exercise let us do, so how different is this distribution from normal distribution, so let us understand what this script is doing. So we take data points from 15 to 35 and we generate random deviates and we plot the data but we normalize the data so that the maximum will be 1 and that is what is done here, so random deviates and the probability distribution function using those random deviates after sorting so that when you plot it will give you a nice line and we plot it with a red line and then we also generate random deviates from the probability distribution from the Cauchy distribution and notice that the normal and Cauchy has the same mean and here it is the standard deviation, here it is the width, here again the Cauchy data is also normalized so that it will also peak at value of 1 and we are going to plot both, this is just to show you the deviation of Cauchy distribution from Gaussian. Like I mentioned in read world analysis it is a combination of both which is used to describe the X-ray peaks so it is a good idea to know how different they are from each other so we will plot it and see. So you can see the red line is the normal distribution so that is what y is from the normal distribution and so away from the peak the Cauchy still has some tail compared to the normal, near the peak Cauchy is much more narrower than normal. So this is the difference between the two so typical X-ray peak profiles are best described by a combination of two. There are other functions also that is used in read world for refining the fitting of the peaks but it is very common to see a combination of normal and Cauchy to fit so it is a good idea to know how they look and that is what we have done. So to conclude Lawrence or Cauchy distribution is one of the common distributions it is very important in material science engineering because many experiments spectroscopy experiments and like X-ray diffraction and such diffraction experiments the peaks that if you want to fit then Lawrence or Cauchy distribution is one of the important distributions that describe the shape of these peaks. So we have learnt how to use R to generate random deviates as well as plot the probability distribution, cumulative distribution function and quantile functions as usual. So the distribution is called Cauchy it is not called Lawrence in R so that is just a technical point that you have to remember. Thank you.