noc18-me62 Lec 42-Probability distributions

Good morning, welcome back to the course on engineering metrology and we have discussed the statistical part that is used in engineering metrology. In this lecture, I will discuss about probability distribution. I had given a brief information on what are frequency distributions, how do we construct the frequency distributions. In metrology, what happens, there are multiple distributions which are used or not in even in metrology like in statistics, the probability distributions are there. The contents I like to cover here would be the need of probability distribution or we also call them probability density functions. Then, I will discuss a few distributions, Poisson and binomial distributions, then non-normal, rectangular, triangular and normal distribution. In this lecture, I will just introduce these distributions and I will discuss more about the normal, binomial and chi-square distribution that I will discuss in the next lecture and also try to solve some problems to have the deep understanding of how the distributions are used. Now, what are distributions? Probability distributions as I said in the previous lectures is a function or a table or it can be an equation that shows the relationship between the out come of an event and its frequency of occurrence. So, what is the outcome of an event? It is frequency and it is an outcome here. How many times this specific frequency, specific event has occurred? Now, why do we need all these distributions? Why do we need Poisson, binomial, non-normal, rectangular, triangular, all these distributions? These are actually the Poisson and binomial, I will discuss from discrete viewpoint for discrete data and this will be for continuous data. However, there can also be continuous binomial distribution and on the other hand, we can have a discrete normal distribution as well but more commonly used distributions for the discrete purpose for the discrete data are the Poisson and binomial and normal is generally used for continuous data. Now, what happens? When we had made the observations, we have different kinds of calculations like we have go, no go gauge, we learn how to take measurements using various instruments, how to measure lengths, how to measure temperature, pressure, strain, force. Most of these values would fall in a continuous data. Sometimes, what happens in industrial measurements, the measurements are not continuous and not continuous, sometimes it is, it can be continuous but it is not just the way we look at the height length. For instance, we can say like in go, no go gauge, whether it is go or it is no go. So, I can say whether the product that I have produced is acceptable or non-acceptable, whether it is defective or non-defective, there is no in between, okay. So, it is kind of 0, 1, 0, 1, okay. We have binary kind of data here. Then, what distribution will fit here? Then, sometimes there is a, the interval is fixed, the time space or the observation space is fixed that out of, within a week, how many times the defects occur or I can say within an hour, how many times the defects occur, how many pieces per hour or how many pieces per 20 pieces. So, in that case, the time space is fixed. So, we have distribution for that. Then, sometimes we do not have a plenty of data. We do not, we are not very clear about the distribution. We call it, it is a lack of knowledge or lack of information. Then, in that case, we can use uniform or triangular distributions. So, we need to select distribution carefully based upon the type of observations that we are taking. Now, the first distribution I would like to discuss here is Poisson distribution. Now, here the events randomly occurring in time P x refers to probability of observing x events in the time t and lambda refers to x dash here. So, what is Poisson distribution? Poisson distribution here in event can occur 1, 0, 1, 2 or so many number times in an interval. Some event occurs in an interval. Let me say there is some noise, some chatter in the machine that we are using in manufacturing. How many times does the specific peak noise value or peak sound value occurs in an hour? Or how many times is the setup disturbed in between? So, we can have multiple examples in this. So, but the thing is that in interval, the space is fixed here. I can just put certain key points here. Then, average number of events is lambda here. That is lambda is average number of events in an interval. So, I can call lambda is event rate, ok. The lambda is event rate. Also, it is called rate parameter. So, the probability of observing k events in an interval is given by this equation. If we have to observe k events in an interval, the probability of observing k events in an interval is equal to e power minus lambda lambda by k by k factorial. So, this is Poisson distribution. So, the very first application of Poisson distribution were the soldiers who were accidentally killed by a horse kick, ok. So, this was an example. Now, there can be certain examples in Poisson distribution that number of specific crimes that happens in a month, ok. Then, I can put certain examples here. Then, we can have number of arrivals in a service center, number of cars that arrive in one hour for service, ok. Another example could be the number of defectives produced in a day by a machine, ok. Then, I can have like cars, we can even consider number of patients those walk in a clinic to get treated in an hour. Then, similar like in a specific time space in an hour or in a day when the specific number is there, this kind of distribution Poisson distribution can be used. Now, the Poisson distribution can take various forms here. Depending upon the value of lambda, the Poisson distribution can be maybe like this. So, Poisson distribution is highly skewed if the value of lambda is small. For lambda equal to 1, for lambda equal to 1, the distribution is somewhat like this. So, I have discussed about skewness. I will take up the skewness in detail when we discuss in normal distribution. So, it is highly positively skewed distribution when the value of lambda is 1, ok. If value of lambda is greater than 1 or it is let me say the value of lambda is somewhere equal to lambda is equal to 4. If lambda is equal to 4, the distribution can take some value like this, somewhere like this. So, this is actually I am presenting k, this is k here and I am taking this out of 20 like the value here is 0 times and 20 times, ok. So, this is for lambda is equal to 4. For lambda even greater than 4, let me say lambda is equal to 10, it would be somewhere like this. It will be close to the normal distribution, ok. So, this is the characteristic or the behavior of the Poisson distribution. So, what are we bothered about here in the probability density function or in the probability distributions? So, what is the mean value of the variable based upon a finite number of measurements and how well this value represent the entire population? So, do the variations found ensure that we can meet tolerances of population as a whole or how good are these results? What essentially is happening? This distribution is for the whole population like somewhere did in a multiple experiments and were able to develop these distributions like have by conducting maybe thousands of experiments, but if we have a limited sample, we pick a sample like number of defectives in an hour, ok. There is a distribution for that. We pick for 80 for 8 hours, for 10 hours, ok. If we have some sample, ok, number of let me say 5 defectives per hour, maybe number 1 is 5 defectives per hour, number 2 is 2 defectives per hour, 3 defectives per hour, something like that and we pick a sample of maybe 10 items like this, ok, but the population is this, the probability distribution, Poisson distribution. Now, how well our sample fixes with the population? How well the curve fitting happens here? Curve fitting if you know the term, how well the curve fitting happens here? Based upon that, Poisson distribution is best suitable for the events that occur in a specific time, ok, when the interval is fixed. So, this is happening in probability density functions. We choose a distribution based upon this criteria. So, what is the target? Ok, I will put the target for probability density functions here. The purpose of probability distributions, purpose is a single representative value that best tells us the average of the measured data. We need to have number 1, a single representative value, representative value means that it should be close to the target value. Then, this representative value that provides a measure of variation in the measured data set, then representative value to provide a measure of variation in data set. Then, third purpose is we need to know how well the average of the data set represents the true average value of the entire population that is just I said. How well is the average of sample representing the true average of entire population, ok. For point number 3, we need to establish an interval within which the true average value of the population is expected to lie and this interval quantifies the probable range of this random error and this is known as random uncertainty, ok. So, we need to work on that. For that, I have just mentioned the application and the use and the applicability of the Poisson distribution. Now, next is binomial distribution. Now, this describes the situations when the number of occurrences n of a particular outcome during n independent tests with the probability of any outcome p is same. So, how do we represent the binomial distribution if probability of the random variable x is equal to k has to be determined, this is equal to n C k probability of k and 1 minus p, this is actually P, this is small p, the probability 1 minus p power n minus k. So, p is actually success and 1 minus p is no success, we can even call it failure. Say as we say n, small n belongs to capital N and it is some natural number. So, small n cannot be some number with decimal places, small n belongs to capital N, capital N is actually natural number, ok. I will put it here, it is a natural number and small p belongs to 0 and 1, probability can vary from 0 to 1. Then, the variable x or the random variable x is representing a binomial distribution with n number of occurrences and probability of success as p. So, the probability of getting exact k success is, this is actually probability to get k success, you put it small k here, ok, this is equal to this. Now, it is anotherly been stressed that in an industry, it is not always possible to measure the quality. In such situations, quality could be said to be either satisfactory or non-satisfactory. We use go, no gauge, no go, defective, not defective. We cannot just quantify the data all the time, the measurements which we need to take. If we are taking measurements, that can be a continuous data, that can be a specific value. If we are just telling that whether it is success or failure, these are the most common terms I can use. Success can be accepted, not defective, then go, no go is ok, ok, both are going in. So, this can be like, if this kind of data is there, binomial distribution is used, ok. So, in these cases, although it is possible to measure the quality, but there is not much time, equipment, personnel to make the measurements. I like to, in other words, I can say that the resources are limited sometimes and we just need to see whether yes or no. So, like go, no gauge, go gauge, as I said. So, these are known as attributes. So, it is actually most applicable in attributes kind of inspection. Number one is attribute inspection, ok. Now, inspections, inspection by measurements is known as variable inspection. This is attribute inspection, we will do this in control charts. I am just giving a brief idea, what is attribute inspection, when we talk about the qualitative data, that is attribute inspection, when we talk about the quantitative data, that is variable inspection. So, binomial data, it is actually as I said before, the binomial and Poisson. So, these are more used for the discrete distribution. There is another term that is connected to the binomial distribution, that can be the fraction defective or proportion defective, I can say here, fraction defective, ok, or we can call it proportion defective. So, fraction defective, I like to put some light on this as well. Like if we say, if the whole aggregate of a production has certain fraction defective, let me say 0.02 of fraction defective, ok. So, then that means 2 out of 100, 2 out of 100 pieces are defective, 2 out of 100 pieces are defective. Now, it is assumed that the production runs over a large number of days and that the aggregate production is large, the days production is a part of the aggregate production, that is the overall production that happens in a month. So, this is 2 out of 100 is just a small number. So, this is a sample size, out of 100, 2 are defective. I can say 2 defective and 98 not defective, 98 not defective. But, if I take the whole universe, the whole population in this case, now what happens, let me say if in a day 1000 units are produced, if they are produced for 1300 day, 2 percent means, 2 percent implies 20 pieces, 20 units, ok. Now, it can vary from like 20, if the 20 units are produced defective in a day, it can be 19 units some day, it can be 21 units some day, it can even vary from 18 to 22 or some 23 something. So, if sometimes it happens that if in on one day, if we have about, let me say 50 defectives, 50 defectives on a certain day. Now, this is alarming, 50 defectives in a certain day that means, only 20 has to be average, 20 is my mean, as I said the purpose of the probability distribution is to find the average and see that does our, do our average fit in the population or not, ok. Now, 20 is the average and the variability is there may be from 18 to 22. If some, if on some day I am having 50 defective that means, this is alarming some condition, in case some deterioration has occurred or some assignable causes are there, assignable causes which are to be traced, I will put it, find these, find these causes, what are the assignable causes. Otherwise, the random variation as we have discussed before, random variation is due to the non-assignable causes, the assignable causes, they can be systematic error, random error, this random error we are mostly dealing with, when we are talking about distributions or we are talking in the statistics in itself, we are talking of the random uncertainty. So, if causes are assignable for 20 to 50 defectives have happened, that might mean that there is some big cause or there is some reason, there is some reason because of this has happened, maybe some deterioration has happened to the machine, maybe some setup is a little changed or maybe the worker who is working is not skilled to that level, so we need to find that. So, I just need to touch the fraction of defectives with binomial distribution because these are always related. So, it is actually 2 out of 100 that is it can be I can say success and non-success, I can divide it in this way as well. So, next is log normal distribution. So, it is failure or durability projections, events whose outcomes tend to be skewed towards the extremity of the distribution. First, I will just put the PDF of this, PDF of log normal distribution is equal to 1 by sigma under 2 pi, this is actually log normal, so it should be derived from the normal distribution only. I will take normal distribution in the end because I like to explain that in detail. So, this is minus of natural log of x minus mu whole square by 2 sigma square. Now, what are the parameters here? Here x is my random variable, mu is mean, sigma is standard deviation, this is for the population. Now, we need to note here that we have constants here, e, pi are constants, e and pi are constants here. So, if you are performing some measurements that are finite and for instance, length, height, width, you are most likely going to end up with a data that resembles a log normal distribution. For the finite number, let me say I can have, if let me say this is my variable x and it varies from 0 to actually it varies from 0 to infinity, but I am taking 0 to 2.5 here. So, I can have different types of curves here, different types of the level of skewness depending upon the variation and mu. I will take mu as 0, mu 0, I will take value of sigma is equal to 1 first, it will be like this. Again, positively skewed and highly skewed. So, what happens actually? To give a better understanding on this, think of a calibrating gauge block. If you begin the calibration, you know the target length. You perform the measurements at a repeated point on a gauge block in such a way that majority of your observations, majority of your measurements or measurement results will be centered around the actual length of the gauge block and some measurements will be larger than and a fewer would be lesser than that because you are limited with the let me say if this is the gauge block, I am just using this to measure length of some measurement, instrument that is manufactured. Let me say something is manufactured, I am just measuring because this is my length, I am limited with this length only. So, most of the observation would be close to this only, gauge block is this thing. So, we are limited by the length of the gauge block here. So, if we see the realistic situation, we cannot measure less than the length of the block. So, measurement results are finite or limited. So, we need to make sure that to consider log normal distribution, the data has to be or the performing measurements that are to be finite. So, it may prevent us from encountering measurement errors and miscalculate uncertainties. So, if sigma is equal to 1 and mu is equal to 0, it can take this shape. If the sigma, sigma is actually the shape parameter, I can put it here as well, sigma is the standard deviation, this is actually the shape parameter. By shape parameter, I mean because it is determining the shape. So, we can see that sigma is equal to 1 and mu is equal to 0, this is the shape that is coming. So, also the standard deviation of the log normal distribution is determined from the historical data. So, sometime we may be able to estimate it using current data. So, the shape parameter does not change the location or height of the graph, the location or height is not changed. What changes the location? So, mean is the location parameter here, mean is the location parameter. So, that determines the location of the where is our graph located. Also, we can use median. So, that may determine the how the graph shrinks or stretches that can be determined. So, for value of sigma is equal to 0.5 and mu is equal to 0, the skewness as we say that because what is sigma, sigma or what is standard deviation, standard deviation is actually spread, the spread more the is the value of standard deviation, more spread would be there. So, if this dispersion or spread, the value is reduced, the spread would be contracted. So, what we will see, we will have an away distribution like this here from 0 to 2.5 or I can even make one more curve. So, standard deviation is equal to 0.25 and mu is equal to 0, this curve would be further shrinked, it will be further more closer, but will have some like this because the area has always to be shrinked, but it would be lengthier now, it will be sorry taller now. So, this is sometimes known as kurtosis. Actually, kurtosis is something different than the center line is central tendency is exactly same and we have different heights of the curve, then kurtosis happen that we will also discuss. So, this is the log number. You can just see the approximate area, actually the actual area or the exact area below the red, blue and dark blue curves would be exactly same. The only thing is that because of the value of the sigma here, because the value of standard deviation, the curve is getting contracted here. Next comes the rectangular distributions. Processes in which a likely outcome falls in the range between a minimum value a and a maximum value b occurring with equal probability. In that case, rectangular distribution is used. So, sometimes what happens that we have some data or we have some observations, but we are not sure that what kind of distribution would it follow. We just know the minimum value is a ok, minimum value is equal to a and maximum value is equal to b. In between what is happening, we do not know, we do not have this great information about that. We can say that ok, there is nothing happening, there is a uniform distribution that is why it is called rectangular. It can be plotted like this. This is rectangular distribution. The minimum value here is a and the maximum value is b and pdf of this can be given by this is a and pdf is equal to it is 1 by b minus a ok, if x is between if x is between a and b ok, otherwise this pdf is 0 ok, I will put 0 otherwise. So, we just know the minimum value and the maximum value. If outliers are there, we will just take the pdf to be 0. If outliers means values greater than the maximum value or less than the minimum value, but if it is between a and b rectangular distribution can be used. Now, rolling a single die can be example of a discrete uniform distribution, but we are more concerned with the continuous distribution here. The processes in which a likely outcome falls in the range between minimum and maximum. So, the most common examples of this one is when you do not have any information as I just said and the day of the like you can see ok, day of the week of the hottest day of an year is about equally likely to be any of the 7 days. The or you can say the last digit of the security number of the you can say the cricket captain or of the various teams that would be always between 0 and 9. So, these are silly examples, but the thing is that the minimum and the maximum numbers are known there. So, in measurement uncertainty when its estimation is to be done, it often occurs when it is necessary to make choice between two alternatives of which one may be possible lead to somewhat overestimated uncertainty and the other one may be lead to underestimated uncertainty. In such situation it is usually reasonable to rather somewhat overestimate than underestimate uncertainty. I will put a note here it is reasonable to overestimate than to underestimate ok. If this is the case we can use the rectangular distribution ok. For instance if I using go and no go gauges or we are just measuring something and the upper tolerance is having higher limit or is higher. We can overestimate it a it is a permissible to have to overestimate, but underestimating is not allowed ok, not very much allowed. So, then rectangular distribution is preferred. There is another distribution like rectangular distribution that is triangular distribution. In triangular distribution also we do not have a much of data, but the thing is that there is one value that is the mode value. So, processes in which a likely outcome X falls between known lower bound A and upper bound B with a peak value mode this is used when population information is passed. We do not have much information about the population, but we have the lower value and higher value, but there is a middle value that is repeating multiple times that repeating maximum number of times ok. There is a mode in that case triangular distribution is there. So, it can be something like this. I can call it is value A, value D and a mode value. So, it is it takes its shape of a triangle that is why it is known as triangular distribution ok. I will make some other angle it is triangular distribution. So, PDF for this is PDF for triangular distribution is a different condition for that it is equal to 0 for X less than A ok. As we had in a rectangular distribution it is 2 into X minus A by B minus A into C minus A. So, this is B here ok. The mode is C minimum value A maximum value B upper bound is B lower bound is A ok. So, this is the PDF for probability distribution. This is for A is less than equal to X is less than C ok. Then this is equal to 2 into B minus X by B minus A into B minus C. You can see B is the upper bound. So, B minus X. So, X A was lower bound. So, there we had X minus A. In the second part and third part we have B minus X. This is for C is less than X is less than equal to B and again it is equal to 0 for X greater than B ok. I can put one more here. It is equal to actually 2 over B minus A for X exactly equal to C. So, this is PDF 1 2 3 4 and 5 cases this is PDF for triangular distribution. So, triangular distribution represents a non-maximum minimum and estimated central value. It is commonly referred to as lack of knowledge distribution when we do not have a large information or very good information about the population data. So, it is typically used where relationship between variable is known, but the data is scarce. So, data is small. For example, I can say if let me say if in any metrology lab I am using a thermo set controller that is constantly trying to achieve a target temperature set point ok. For example, I can put here there is a thermostat controller thermostat controller that is set to achieve a target set point. Most of the values will be just floating around this let me say this value is let me say 30 degrees 30 degree centigrade that is most of the values will be close to 30 degree centigrade. So, you know most of the time the thermostat will just give 30 degree centigrade like you say you can set thermostat controller you can say thermostat controller of your air conditioner. Air conditioner temperature is set to 18 degrees now here temperature is setting to a 22 degrees here. The most of the time the temperature would be 22 degrees. 22 degrees are central value. It can be 21 degrees or 23 degrees. So, values can vary, but the most of the time the value is 22. What is happening? The value is ok I can say thermostat controller if I take 30 degrees. So, 30 degrees it is it would the value would be like let me say if I take 10 values will be 30 degrees then 29, 30, 31, 32 ok then 28, then 30, then 30, then 30, 2, 3, 4, 5, 6, 7, 8, 9 then let me say first value was also 30. You can see the mode here mode value here 1, 2, 3, 4, 5, 30 is repeated 7, 6, 30 repeated 6 times and the value of A here is the minimum value 28. The value of B here is the maximum or the upper bound that is 32 and the mode value here is 30 ok. So, in this case triangular distribution can be used. So, when mode value is there sometimes mode value is not there sometimes or we can also use rectangular distribution in place of triangular distribution. Now, what determines the selection between triangular and rectangular I can put triangular versus rectangular distribution. So, before comparison I like to introduce the term uncertainty contributor uncertainty contributor. So, what is this? For instance I have a jar I get a jar from the market which has to measure 100 ml ok 100 ml jar. So, this is a kind of a gauge 100 ml if I fill the jar completely then 100 ml water or liquid should be there, but it might be marked there this 100 ml plus minus 100 ml or it is plus minus 5 ml. So, this plus minus 5 ml is my uncertainty contributor uncertainty contributor that is it is there the specifications are it is a no specification limit specific limits are there on my measuring instrument itself ok that this is the this can measure from 95 ml to 105 ml ok. This is uncertainty contributor and this uncertainty contributor when we talk about the actual uncertainty I will call it U i this is equal to uncertainty contributor which is actually capital U i this divided by k where k is my expansion factor ok. Then for normal this value of k may be the deviation standard deviation the more the spread is the let us say this is the value of U i here small u i. So, this k can be. So, in case of triangular distribution U i the value of U i that is generally used is equal to uncertainty contributor by under root of 6 and in case of rectangular it is U i by under root 3. So, based upon based upon this information if I use this data if I ok let me pick this jar only this jar 100 ml 100 ml this is plus minus 5 ml. So, I can call this 100 ml as my mean value ok or my center value and plus minus 5 can be the variation. So, if I try to draw a rectangular distribution ok I will try to draw the rectangular distribution and a triangular distribution 100 ml and plus minus 5 ml for the rectangular distribution I can have something like this. So, as I had from a to b I can even put from minus a to a and center as 0 ok. So, in this case in my specific case from a to b this value of a this is equal to 90 ok 5 ml is not a very good figure I will put 0.5 ok I will put 0.5 I will then I will put 99.5 I will put 100.5 as b and the center value is 100 here, but I am following rectangular distribution like this. So, what is my u i here? u i is equal to the uncertainty contributor that is 0.5 divided by under root 3 which is equal to 0.288 ok and in this case in triangular distribution I can have something like this 99.5 and 100.5 and most of the times it would be 100. So, it will be like this, but in this case a value of u i is equal to 0.5 by under root 6 which is equal to 0.204. So, if I see my uncertainty contributor here it is then reduced based upon the standard deviation or the expansion factor ok expansion factor this is expansion factor due to due to standard deviation or spread. So, I will pick the triangular distribution based upon the value of u i because this value is lower. If I need to follow the criteria or the tip that I just mentioned before that it is reasonable to overestimate if we need to just pick 100 ml, but we cannot afford less than 100 ml we can better choose rectangular distribution. If we cannot afford under estimation, but overestimation can be allowed ok for instance if we are like milk vendor is trying to measure using this 100 ml jar or if we are doing some experiments where we need 100 ml of water and we even consider evaporation might happen some evaporation might happen all the losses are there we cannot afford the underestimation here. Underestimation we cannot afford 99.5 or maybe if it is 5 ml is the uncertainty contributor we cannot afford 95 ml we can afford 105 it is acceptable 105 it is acceptable. So, in that case rectangular distribution is there ok in that case it will be selected. So, next is normal distribution most physical properties that are continuous or regular in time or space with variations due to random error follow the normal distribution the PDF for normal distribution is 1 by under root 2 pi sigma square. So, this sigma would can come out as well here e power minus x minus mu square over 2 sigma square and it is asymptotic distribution I will explain it asymptotic that it touch infinity at the extreme ends and it is symmetrical in distribution like this ok. So, this is my normal distribution. So, to get better understanding we can imagine to collect maybe 100 measurements of samples and create a histogram graph with these results like if I develop 100 histograms like this and so on ok. So, that we will see that the frequency if we plot the frequency we will see that it will follow normal distribution it can examples can be example it can be height weight then strength something like that ok this is our normal distribution. So, to recapitulate we did what are the probability density functions we discussed about a few discrete probability distributions then we discussed about a few continuous probability distributions we will discuss the log the normal distribution in detail and try to solve some problems with them and we will meet in next lecture where I will take this forward. Thank you.