 Student, this is the Vishat distribution using R. When we have the sampling distribution of sample variance, in univariate case, we will move to the Kaisakir. When we have the sampling distribution of sample variance in multivariate case, we will move to the Vishat distribution. So this is the example of the Vishat distribution using R. Now here is the example, generate 100 random sample and find their mean matrix and using the below variance covariance matrix. Basically, what we have to do in this particular example, 100 random samples using the Vishat distribution generate, then find their mean and we have to find their mean matrix using the below variance covariance matrix. So let's move to the R, now this is the R studio, so here is the library mass, first of all, library mass install. So in the packages, here you have the packages, then install, in install we will write the library mass, mass is installed here, you have the method of the packages, in install we will install the library. Then we have P equals to 3, now P what, how many dimensions, 3 cross 3 you had, we have 3 dimensions, so P equals to 3. At scale, scale just, we have given its notation, this is the scale, scale how we are making the matrix which is given, matrix C, row Y we have entered the data, okay, row Y we are entering the data, where is the data, this is the data, 4101920216, this is the variance covariance matrix data. So 4101920216, how many rows, 3 rows generate because P equals to 3, 3 rows generate, 3 columns generate, 3 cross 3 we have the matrix, and what we have written next, by row, that is we arrange it row Y which is equals to true, that by row we have arranged it. So highlight this and then run, we have run the data, then df, df stand for degree of freedom, we have defined the parameters of degree of freedom, so we have supposed degree of freedom which is equals to 10, we have kept 10 degree of freedom here. Now this is the generate 100 random sample from the vishat distribution, now using the set.seat command, set.seat what will it do, random number samples which we are generating will remain as it is, that is it will not vary again when we generate random numbers. Now this is the command of the random samples using the vishat distribution, now look at this, this is the R vishat, R vishat means random vishat with sample 100, 100 sample we have to generate on these parameters, so what are the parameters, df, degree of freedom, degree of freedom which is equals to 10, we have let 10, and sigma, sigma we have defined, sigma we have said this is the scale, we have to generate on these particular values, okay we have used the set.seat command to generate samples, vishat samples, okay vishat samples we have generated, how many? 100, this is the 100, look at the various metrics you have, 100 times generated because we have said 100 samples, we have 100 samples using the vishat distribution with these values. Now what did he do next, in particular example what we have next is that the samples you have, you have to find further means, that means next what I am doing is mean underscore matrix, that means the samples I have here, these samples take out the mean matrix, take out the mean, so here it is written mean underscore matrix, you can write any value here instead of mean underscore matrix, I have just coded it, which is equals to call means, column wise means take out the mean, call means take out the samples, the samples generated here, column wise find the mean, so this is the command highlight this and then run, mean underscore matrix, command run, further highlight mean underscore matrix or we can use the command of print, print, parenthesis mean underscore matrix, then run, so this is the mean, now you have column wise mean that he has determined, first column wise mean, second, third, fourth, up to so on, how many means have generated us, total 100 means have been generated, column wise we have found the mean, so this is the example of the vishat distribution using the particular data that we have generated here, this is the example, 100 random samples have generated, vishat distribution and find the mean matrix, he has said to find the mean matrix, we have found the mean matrix using the below variance covariance matrix, so this is the example number one, so here the example number two, generate, this small data we have seen, generate five vishat random samples, just to understand the example, we have done five vishat random samples and find the mean matrix and we have determined the mean matrix using the below variance covariance matrix, now in particular example we have a degree of freedom which is defined by its parameters, with 10 degree of freedom I have generated this example, we have determined that to generate we have five vishat random samples and we have to find the mean matrix using this variance covariance matrix with 10 degree of freedom recording, again this is the R studio window, now look at this, this is the example N equals to five, five random numbers are generating me, with P equals to three, what is P, you have dimension with dimension three, this is the let X, we have let S here, equals to matrix, see what was the data you have, now this is the data, 10, 3, 4, 3, 5, 2, 4, 2, 6, row wise data we are entering, now row wise data we have entered, 10, 3, 4, 3, 5, 2, 4, 2, 6, how many rows generate, P, now P what is it, we have defined P, we have three, that is N row number of rows, how many will be P and P which is equals to three and Df, degree of freedom is given to us, if it is not given then we are letting it here, so degree of freedom which is equals to 10, I generate this data on 10 degree of freedom, so again this is the library matrix, now we are installing the library matrix and set dot seed command, okay set dot seed command, I have told you many times what does it show, and here is the X, X what are we saying, because we have called samples there, we are saying scale, so here we have let it X, what is under X, R which means random which is generating with N equals to N, because N we have defined earlier, N equals to five, Df, Df is defined, degree of freedom is also defined and Sigma equals to S, S where matrix is given according to this particular data, S is also defined, so I am running it, data ran, then print the result, here I have just X leak, X leak or we print X, so here is the result, five random samples have generated, because five is easily we have taken small data, one, two, three, four, five, meaning random samples generated using the Vishard distribution, now what I have to do is determine their mean, so here is the command call means, whose column means we have taken, this X which we have let, this is the X, so command I have a mean underscore matrix equal to call mean X, highlight, then run, again, mean underscore matrix, highlight, then run, so here is the mean, now column wise mean have come to us, now you have five random samples, you can check it manually, so manually you have five column wise mean coming, is it exact, yes, you have exact exact values, this is the mean for one sample for second sample for third, fourth and five, this is the example of the Vishard distribution, so in this particular example, here if we take this sample from the multivariate normal distribution, then the distribution of this sample mean will be the multivariate normal, but the distribution of this sample variance covariance will not be multivariate normal will be the Vishard distribution, okay, what do you have, if we check the mean distribution with the multivariate normal, then the mean distribution will follow the multivariate normal distribution, so this is the example of the Vishard distribution using R.