Lecture 24 : Univariate Normal Distribution: Application in R Software

[Music] hello friend welcome to the course multivariate procedure with r so you can recall that in the last lecture we considered the univariate normal distribution and we had uh discussed different types of properties and one important aspect was that how are you going to compute different types of probabilities the rule which I had explained you in the last lecture to compute different types of probabilities in the normal distribution they are going to be useful when we are trying to uh consider different types of application like as test of hypothesis and other things so now we have understood that uh how one can manipulate different types of probabilities under the normal distribution what do I mean by manipulate manipulate doesn't mean we are going to do anything wrong but for example if I want to know the value of the CDF at minus a then it is going to be computed by 1 minus CDF at a so this type of manipulations and different types of rule uh we had understood and you can uh derive them very easily if you try to look at the curve and and they can be brought uh and understood very easily by understanding the rules of integration so now in this lecture we are going to show you the implementation of uh this uh normal distribution in the art software implementation mean we would like to see how we can generate the random numbers how we can uh compute different types of probabilities Etc in the our software so and for this thing you will not need any special package this I'm telling you because in the case of univariate you don't need any special type of package but when we were considering the multivariate then you will need a special package so let us begin our lecture and try to understand it right so you can see here that we had discussed the normal probability density function that uh random variable X is set to follow normal distribution with parameters mu and sigma square if its probability density function is given by this function which is 1 upon Sigma < tk2 Pi exponential of -1 upon 2x - mu upon Sigma s and we always write it in this way that X follows normal mu Sigma Square where mu is the mean and sigma square is the variance so obviously the sigma is going to be the standard deviation right okay so now in the r software there are various possibilities so we can compute the density distribution function quantile function as well as we can generate the random generation for the normal distribution and the two parameters the mean and variance they are specified by the parameters here me and mean that means it is something like here you can write down here mean is equal to Mu whatever is the value of mu which you want to specify and you have to remember one thing that this command does not use variance but it uses the standard deviation which is specified by the command SD so you will write down here SD equal to what would be the value for example here the value of Sigma right so many times you will see that in different problems you are given the variance so then you have to be careful that you use the square root of various there right right so now we will consider here four possible commands for example the First Command dorm dorm gives the density so X is here the data vector and then you have to specify here the value of mean and standard deviation similarly there is another function here pomm so pom gives us the distribution function so you have to assign here a value Q then mean standard deviation and there is one option here lower tail is equal to true well I will try to explain you all this uh things are here for example if you see here Q here is the vector of quiles right and similarly if you want to use here this term here Q so this qom is going to give you here the uh cotile function so here you have to specify here the p mean standard deviation SD and lower do tail is equal to true so you can see here this P here is the vector of probabilities right and then we have here another function here RN so RN generates the random numbers from this normal distribution so n is the number of random numbers which you want to generate and then mean is standard division that you have to specify that from which distribution you want to generate the uh random numbers right so you can see here any the number of observation mean is the vector of means SD is the standard deviation and lower do till if this is true this is the default actually then the probabilities are computed like this probability X less than equal to X which is something like your cumulative function and in case if you take lower dot tail is equal to false then this probability of X greater than x is computed you see in this normal distribution this part this part here is called as lower tail and this part here is called as upper tail right so if you want to compute here this probability this will be your here something if this is here some zero then if this will be your here probability of minus Z then this will become here probability of Zed so you want to specify whether you want to compute the probability on the left hand side of the normal distribution or on on the right hand side of the normal distribution and in case if the values of mean and variance are not as specified then mu that is mean is assumed to be zero and standard deviation is assumed to be one that they are the default values all right so now let me try to take here some very simple examples to explain you that how you can compute different types of probabilities and how they are useful so suppose an apple formal says the Apple in the boxes and the weight of the boxes vary and we and are assumed to be normally distributed with mean 20 and variance four that means mu is equal to 20 kg and the variance is 4 kilg Square you know that when the farmer is trying to pack the apples in the boxes means the apples have got different weights and you cannot ensure that every box had exactly they said 20 kilogram of apples sometime there will be some variation maybe some grams so this is reflected by the value Sigma square and the farmer wants to avoid customer being unsatisfied because the boxes are those are too low in weight customers will not be happy so therefore the farmer wants to know the probability that a box with a weight of less than 18 kg is sold that's a very genuine question because the farmer is trying to uh fill the the apples in the Box based on the approximate weight and if the weight is less than the far then the customers may not feel happy and and so the farmer wants to know its probability so if you try to translate this problem in the statistical term then I can say that farmer wants to know the probability that a box with a weight of less than 18 kg sold that means if I say that X is the weight of the box or the Apple then we want to compute the probability probability that X less than 18 right so this probability can be computed by the command pm and basically if you try to see when you try to compute that probability X less than 18 this is actually here f of x at X = to 18 this is the value of the CDF at X = to 18 right so then we try to give here the command here pom Q is equal to 18 then mean is equal to 20 and then SD is equal to squ root of 4 which is here the VAR variance and if you try to execute it on the r console this will come out to be 0.158 6553 so you can see here it is very easy to compute such probability right and if I want to show you hit it on the r console so that you can be confident you can see here it is like this I try to copy and paste this command over here and if I give it here in the r console it will give you here the same value so now means you can be confident that these values are very easy to obtain right and on the other hand if you want to use here the option here lower dot till is equal to True means you can put it here and then what will happen the same value will be obtained here right because lower do tail equal to True is the default right you can see here the screenshot that both the values are going to be the same now I try to modify this question yeah I just want to show you how to compute different types of probabilities now suppose the farmer wants to know the probability that a box with a weight of more than 22 kg is sold right so that means the farmer wants to compute the probability that probability X greater than 22 now this is the place where you have to use the rules which I explained you to compute different types of probabilities it is just like when I shown you that X probability X greater than 22 is less than 1 minus the area under the curve which is probability X less than equal to 22 right if you try to see on the curve this will be like this if somewhere it is here 22 suppose here and you want to compute here this probability right so this is going to be the 1 minus this remaining probability which is here written here like this this one so and this quantity here is actually F22 so if you try to see the trick the trick is this I want to convert the given probability in the form of this is CDF although in case if you try to use here the command lower do till correctly then also then you don't need to make this transformation so both the approaches can be used and uh they will give you the correct answer so now if you want to compute this probability that the first option is that you try to use the uh function here 1 minus P which is coming from here like this 1 minus pom and then you try to write down here qal to 20 which is coming here from this 22 and then you have to specify the mean is equal to 20 and SD is equal to root of 4 which is and do tril is equal to true this is the default even if you don't write there will be no problem and you get here this value or equivalently if you don't want to make it something like 1 - F22 then you can directly use this command to compute this probability that X get than 22 is p Norm Q is equal to 22 mean 20 SD uh squ root of 4 and after that you have to use the command here lower do tail is equal to false and if you try to use this command you will get the same values so now it depends on you whichever way you feel convenient and comfortable you can use it there is absolutely no issue right now I try to give about some idea about a different command this is a q Norm what this qor does it gives you the cotile function and it calculates the cotile which is defined by the smallest value of x such that FX is greater than or equal to P FX is the CDF of X it is here like this at any point a small X well you will see that when we try to conduct the test of hypothesis and and various other application this type of probability is is required to be computed right so suppose if you want to know the value of the 90th quantile or say 90% quantile suppose the the value here is Q then it is described by probability that X less than equal to Q is should be greater than or equal to 0.9 so this can be obtained by the command here Q Norm so you simply have to write down here p is equal to 0.9 which is coming from here and then you have to specify here the mean equal to 20 and D is equal to root of 4 right and if you try to execute this command here you will see here this is coming out to be 22. 5631 right and in case if you just use here the command option here lower do T is equal to true this is the default then again both the values are going to be the same and that you can see here in the screenshot right so let me try to show you these values on the r console before I try to move forward right so in this case if you want to see that uh how I am Computing this quantity probability that X greater than equal to 22 this is here like this and if you want to use here the other command that uh not by subtracting from one but by using the option lower dot tail is equal to false then you can see here both are giving you the same value so this is about P Norm now similarly if you try to use here this Q Norm New Q Norm so you can see here that this is here like this so this is the giving you the value of the 90th conti and similarly if you try to use here the default part then it will be you here the same value that lower do tail is equal to True will give you again the same value here right so let me come back to our slide now I would like to give you the idea about this random number generation you see whenever we are trying to compute different types of simulation or we want to experiment with something where we want to have a data which is originating from the normal distribution so this R Norm function helps us in the generation of the random numbers from the normal distribution with a given mean and a given variance it is very helpful when we try to conduct different types of simulation studies monteo simulation etc for the command to generate the random number here is R Norm r n o RM and inside the parenthesis we write here the N which is the number of random numbers to be generated then the value of the mean and then the value of standard deviation right for example if you want to generate four random numbers from a normal distribution whose mean is 20 and variance is four so this can be obtained by the command here R Norm n is equal to 4 mean is equal to 20 and SD is equal to squ < TK of 4 so it will give you here this four values 1 2 3 and here four so these values have been generated from the normal distribution whose mean is 20 and variance is four the mean is 20 that is why you can see here most of the values are concentrated around 20 right and this is the screenshot but when I try to execute this command in the r console now then these values will not be obtained because these are random numbers every time you try to generate the set of random numbers they will be different unless and until you uh use a command like uh set seed right so that is why I wanted to handle it separately and that is why I had shown you the earlier commands now I will try try to show you here these commands on the r console so if you try to see here let me clear the screen and you can see here our nor will give you here these four values and in case if you try to repeat it you can see here that that this is going to give you different values and in case if you try to change here the number of observations you can see here now you're getting here 10 values and now if you try to suppose if I want to show you that what is the effect of variance suppose I try to make this variance High suppose I make it here 100 now if you try to compare the four values from this normal distribution you can see that although the mean is here 20 but the values are varying a lot 15 17 30 14 right and uh in case if you want to make this standard division to be very high I have made it suppose uh 1,000 that means the the SD square root of 1,000 you can see here it is giving you the value like 52 - 3.3 15 45 lots of varation is there among the values now if you try to make it very small suppose I now make it here say square root of 1 you can see here the values are very close to 20 so by conducting this type of this experiment you can very easily means identify and understand different statistical properties of different different probability density functions right so now let me come to an end to our uh this lecture and now you can see here in this uh short lecture we have understood that how are we going to generate the random numbers and how we can compute different types of probabilities using the art software be it the generation of random numbers computation of the CDF competition of the quantiles anything that is possible well it depends on the problem that what you want to solve that suppose if you want to compute some probability the probability can be computed as a difference of the cdfs now how to adjust those cdfs and how to take them for example in the last lecture we had discussed that if x is following normal mu Sigma Square so then you can always make a transformation like x - mu upon Sigma and then they will be following a normal 01 that is also possible otherwise in the r software now if you want to do it you don't need to do anything but you can simply use the uh same mean mu and uh Sigma directly over here you don't need any transformation also so this is how you can uh compute various types of probabilities without any problem and the problem of going into the tables and and choosing the value that is also also solved but my request to you all is here that now you try to take different types of examples try to create different uh say portions in the curve of normal distribution and try to compute various areas they are simply your probabilities for example if the areas on the left hand side areas on the right hand side areas in the center so how can you compute different different types of probabilities using the CDF because we have understood that probability that X lying between A and B is the same as FB minus fa f is here the CDF so now all those things you can verify here although I have explained you here only about the normal distribution but these types of uh activities can be done for all type of probability distributions and probability Mass function be it binomial be it potion be it geometric be it uh uh say gamma distribution but my objective in the course was to introduce you with the normal distribution because we are going to use them in the forthcoming lectures so my request to you is that you please try to take up some problems you can consult the books you can create your own problems but try to practice them on the art software so you try to practice it and I see you in the next lecture with by variate and multivariate normal distribution till then [Music] [Music] goodbye [Music]