 अढ़ाटय मैं, आदाडय दोंगा ठाटगितिड है। आदाटय ठाटगितिड है। तब आप आप सब वो आब वो विने सब विने अग़ादाचने में आप सब ढ़ाश्टी वह आप उग teni as the result of the experiment may follow a certain kind of pattern of distribution and this is what we want to call them ask them as a special random variables. So far we have introduced two kind of special random variables one is a uniform random variable these are all the continuous in nature. these are all the continuous in nature if you recall back we introduce the discrete random variables special random variables such as discrete uniform distribution binomial distribution Bernoulli trial negative binomial etc etc then we introduce a continuous type of special random variables so in the previous session we introduced uniform random variable we showed that it is useful for pseudo random is useful in random number generation and it uses the pseudo random number generator to generate any random pseudo random number that we wish to have from any distribution then we introduced a normal random variable we said that this random variable or this distribution was noticed actually it was noticed during the Galileo's time when he was very upset that under the same circumstances his students are bringing out different observations of stars and constellations but it took around 200 years for Gauss to come and say that this errors are normal to occur they are natural to occur and they have a certain kind of distribution and that distribution we called it a normal distribution or a Gaussian distribution we also said that the Gaussian error function is directly related to the normal random variable that is Gaussian error function actually shows the probability of a random variable y falling between minus x and plus x when y is distributed as a normal with a mean value 0 and a standard deviation or the variance as 1 over 2 or variance as a half now what we want to do this time is we want to introduce some of the distributions which are going to arise in future that is when we deal with assumption that certain data comes from a normal distribution and we do certain inference on the data we do the estimation of the data we do the confidence interval estimation of the data or we do the hypothesis testing all this we are going to do in the future when we do these things on the data we need certain distribution in order to estimate the probabilities or the estimate the uncertainties so this time we are going to in this session we are going to introduce three such distributions which are derived from the normal distribution they are i square distribution t distribution and f distribution so let us start as I described already in the inference for the to make the inference from the observations coming out of normal distribution we need a certain kind of distributions and these are the distribution we want to describe here first we go by chi square distribution let us say that we have n random variables which are independent and they are standard normal variants it means that they are all independently distributed and identically distributed as a normal distribution with mean 0 and variance sigma square in that case a random variable y defined as summation of xi square the square of all these random variables and summation of it is say 2 distributed as a chi square distribution with n degrees of freedom and is denoted as y distributed as chi square n n is a subscript in that case the expected value of y is n and variance of y is 2n and this is very interesting variance is exactly double the time of double the expectation value of y this distribution is useful to estimate the unknown variance sigma square of a normal distribution so let us consider that you have x 1, x 2, x 3, x n coming from a normal distribution with mean 0 but variance sigma square remember when we define the chi square we said sigma square is 1 here we are saying that there is a sigma square and we do not know the sigma square is unknown then we defined yi as another random variable which is xi divided by sigma and we defined a random variable w as a summation of yi square remember that now yi is distributed as I think we should write it down here that in this case yi is distributed as normal 0 1 and therefore w is distributed as chi square with n degrees of freedom and we know that expected value of w is n which is expected value of yi square so it comes to 1 over sigma square summation of xi square and therefore you can say that just make some changes expected value of same summation of sigma i square divided by 1 over n is sigma square and therefore expected value of w divided by n we come from here expected value of w divided by n is sigma square and this is the way if you have n random variable coming from a normal with 0 mean and sigma square and sigma square is unknown chi square distribution is useful to estimate the value of sigma square it gives you expected value of 1 over n summation xi square as a sigma square next we move on to students t distribution this distribution is also known as a students t distribution because the person I mean the t distribution in he took a takhalus he took the pen name of student and therefore it is called as students t distribution let x this be distributed as normal 0 1 and y be distributed as a chi square this distribution with n minus 1 degrees of freedom and these x and y are two independent random variables then you define a new random variable w as a ratio of x divided by square root of y divided by n then this follows a t distribution with the same n degrees of freedom and it is denoted as w distributed as small t n degrees of freedom sometimes n is also written as a subscript it is just a case of notation kindly note one thing that there is a square root in the bottom there is a square root in a while taking the ratio we have taken a square root you can imagine that if x is a result of an experiment it has a unit connected to it and y if you look at as a chi square distribution the unit of the random variables coming up for chi square distribution will have exactly the square of the unit of x please recall let us go back you have x random variables with some unit say l in that case the w which gives you chi square distribution has a l square unit and therefore in the case of student t distribution remember that it is unit less because you are dividing it by the square root and that is the reason you are dividing it by square root and therefore w is equal to x divided by square root of y by n easy to remember if you go by the units you know that t is not supposed to have any units and therefore t distribution it has to be divided by the square root it is so such a w follows a t distribution with n degrees of freedom if y by n as we discussed previously is an estimator of sigma square then sigma square is and if sigma square is not known w in a way represents the standardized normal variate in a way this is what i described in a different way if y by n suppose you assume that x is not the it is not a standard normal variate it has a variance sigma square which is unknown then what we are trying to say is that if y by n is an estimator of sigma then by taking x divided by square root of y by n you are kind of standardizing the variable you are standardizing the normal variate in order to get a distribution and that distribution is t so it is this distribution is useful to estimate an internal estimate of a normal variate or a normal mean value when the sigma square is not known we will go into detail in future when we come to the estimation parametric estimation the next distribution we would like to introduce is an f distribution let x or random variable be distributed as a chi square with n degrees of freedom and y be another random variable distributed also as chi square with m degrees of freedom these are two independent chi square distributed variates then the ratio w of x by its degrees of freedom divided by y by its degrees of freedom is said to be distributed as a f distribution with two degrees of freedom numerator degrees of freedom n and denominator degrees of freedom m f if we look at x by n and suppose x by n is also an estimator of a variance unknown variance sigma square of a normal variate as we derived in in the past if you recall we said that w by n when w is a chi square distribution with n degrees of freedom in certain circumstances w by n is an estimator of a variance sigma square unknown variance sigma square so here what it says is that f is a ratio of two independent chi square variates it is a two independent chi square variates and therefore if both the chi square variates are estimating the same say unknown sigma square then this becomes a test to see if the two variates are same or they are different this is what is going to come up when we learn analysis of variance this distribution is going to play a very important role it also plays important role to test the hypothesis that the two data sets have the same variance suppose you conduct you conduct an experiment or two independent experiment and you feel that the variation variance in the two experiments are very similar then to test such a hypothesis you need an f distribution so with this we introduce the three distributions derived from normal distribution please remember for normal distribution for we gave that it arises naturally in experimentation while this chi square t and f distribution do not necessarily arise naturally from performing any experiments but they arise they are useful for the future inference when we make on different kinds of normal data arising from a normal distribution and therefore we introduce the three distributions here chi square distribution as an estimator of unknown variance of a normal random variable t distribution which can be described a distribution of standardized normal variable when variance chi sigma square is unknown and f distribution as a ratio of two independent chi square variables thank you