 Dear student, today we will going to learn the alternative method of movement generating functions. So last time we have developed the movement generating function. Now this slide is the alternative method of movement generating function. Alternative method is here we will use the standard normal variate. So y which is tends to normal distribution with mean zero and variant identity. So we have the probability distribution function of multivariate normal distribution is this. We know that this is the probability density function of the multivariate normal. Since sigma inverse equals to a, sigma inverse equals to a is a positive definite symmetric metric. This is the property of the positive definite symmetric metric. There exists a non-singular matrix C where C is the matrix of known values. So the C transpose AC which is equals to identity metric and C A we replaced with sigma inverse. C which is equals to identity. Sigma inverse value we know the sigma value of the sigma inverse. C we have taken on the side of equality. This is the C determinant. C prime inverse into C inverse. We have taken C prime into C inverse. This is called the equation number one. So let the y equals to C inverse x minus mu. This transformation we did in previous slides. So find the value of x where x which is equals to C y plus mu. Now why do we need x because we have x minus mu sigma inverse x. Here we have to enter its value. So using the standard normal variate with mean vector 0 and variance covariance metric sigma now we will get the probability density function which is equals to this one. Now we are finding the movement generating function of y as follows. A y because I have let y so y as follows. This is the by definition of the movement generating function m of y t which is equals to expected value of exponential of t prime y. Here the expected value exponential t prime y. Now we have written this in the sum up. So i varies 1 to p t i because we showed this in the i th variable because this term i varies 1 to p t i y i. So we know that the sum of this product terms then we have lambda. We have taken total lambda which is equals to i varies 1 to p. This is the product of expected value of exponential of t i y i. Since each component of y is distributed as the standard normal variate. We have y that every component is following the standard normal variate which is independent of each so movement generating function of y which is equal to pi i varies 1 to p capital pi i varies 1 to p expected value of exponential t i y i. So further exponential we have to write expected value as you know that the expected value of any random variable of x which is equals to sum x into its probability. This is the univariate case. So pi as it is expected value we have to continue this is the discrete for discrete and for continuous minus infinity to infinity exponential unit into its probability density function. So pi i varies 1 to p minus infinity to infinity exponential t i y i f of y i d i f of y i d i This is the f of y so f of y i d i value we have to enter. Further its simplification simplification how this factor you have constant constant we have to buy from integral and then expected value of t i y i and remaining part as it is. So further we have further simplification exponential now we have to see how we have opened its exponential here you have to see two exponentials when multiplied they come in addition form. So minus 1 by 2 first factor second factor we have written in first minus minus common minus 1 by 2 y i square this factor you have then minus because here you do not have 2 so twice time of t i y i just we have simplified it two exponential we have two exponential and multiplied so it will go in addition. Simplifying next in this we have simplified it i varies 1 to p this is the population pi or capital pi 1 over 2 pi under root limit minus infinity to infinity exponential now here we have done that ti square add and subtract here for complete whole square we have done ti square add and subtract here further now see this if I open this a square minus b square i.e. a square plus b square minus 2 a b so a square y i square ti plus ti square and minus 2 a b this one you have time ok we have just simplified it now again put the y i minus ti which is equals to w i that implies Y i variable w i variable so derivative here d y i which is equals to d w i now put this value this is the moment generating function i varies 1 to p 1 over 2 pi lambda e ti square integral minus infinity to infinity exponential minus 1 by 2 so this is equals to the w i and d w i d y i which is equals to d w i just put in here now for we know that this factor previous we have done this factor which is equals to 2 pi square root now its value i.e. this factor which is equals to 2 pi square root put the value of 2 pi square root 2 pi square root value we have entered the moment generating function of this this cancel out open now moment generating function of y which is equals to pi i varies 1 to p exponential raise to the power ti square over 2 now further how we have opened it you have written this product term in this term sum i varies 1 to p ti square by 2 and further how we can write it moment generating function of t exponential of t prime t by 2 so the moment generating function of x which is equals to expected value of this is by definition expected value of exponential t prime x which is equal to expected value of exponential t prime x which is equal to value of exponential t prime x. So, multiply these values expected value as it is t prime C y multiplied then t prime mu multiplied now this is the constant term expectation Now this is the constant further we have, we are writing it as T'C, now you know how we wrote it in transpose, we have written it as C'T transpose, so what will happen here, you see what will happen, this is the T' when we have opened the transpose, then C' and this prime, so we have written this factor further here, so this value as it is and this is which is equals to the moment generating of M of Y C'T, what will happen, this factor C'T, so further we are writing it as moment generating M of YT which is equals to exponential of T'T2, what have we done here, M of Y C'T, now M of Y C'T whose cake is this, exponential of 1 by 2 T'T whole prime this value and C'T, hence the M of X T exponential of T' mu as it is minus this factor's value we have entered, exponential of 1 by 2 C'T' into C'T, further we are multiplying it, we have constant M, constant M is exponential 1 by 2 T' now how we will write it, I have told you that T'T and what we have here is C'T and we know that this T, C' which is equals to sigma variance covariance, so enter this value, C, C' value entered and finally the moment generating function of X which is equals to exponential of T' mu and the T' sigma T, same result our moment generating function which we have already found is that it is also in the alternative method but in alternative methods we have mathematics which becomes a little simpler, there is no effect or difference in the results but we have mathematics which becomes simpler, now what is the difference we have here, T' coefficient which is equals to mu and T' coefficient which is equals to sigma square, so we have found the moment generating function with the alternative method.