 Here in this lecture we are looking at the moment generating function of the multivariate normal distribution. In univariate we have seen the moment generating function. The moment is basically what you have. First moment about mean which is equals to zero. Second moment about mean which is equals to variance. So here we are generating a function. Moment generating function of multivariate normal. What do you have in this? By definition the moment generating function of a random vector is defined as this. This is the moment generating function. By definition we have this moment generating function. In univariate you know expected value of first moment we have to take. First order which is equals to zero. Second we have order which is equals to variance. So you better take a moment to make a function and according to that we generate its moments. fm of x t this is the moment generating function. By definition which is equals to expected value on average exponential t prime x. Where t is the every real number. If x follows the multivariate normal distribution with mean vector mu and variance covariance matrix sigma. Now the f of x this is the density function of the multivariate normal distribution. Here is the transformed linear transformation y which is equals to c inverse x minus mu. Further we know that tc prime which is equals to sigma. Now the expected value of y t inverse expected value of x minus mu which is equals to zero. Because we know the expected value of x which is equals to mu mu minus mu zero. Variance of x variance of y transformed variance of y c inverse expected value of x minus mu x minus mu prime. Covariance t prime c inverse transpose which is equals to identity. How the identity happened? That this value you have is equal to tc prime. Its t inverse t inverse prime finally cancel out and we have the answer identity. Because you know this is equals to sigma and sigma is equal to ct prime. So here if we take c transpose. So the answer we have variance of y identity. So the transformed variable which is follows the multivariate standard multivariate normal distribution. With mean vector zero and the variance covariance matrix identity. This is the f of x. In general this is the f of x. When we take transformation we have f of y cake. Transformed period. So the moment generating function expected value of exponential t prime x. Where x which is equals to this one. We have entered the value of x here. Then we have to simplify it. t prime multiplied t prime mu multiplied. Further we have this constant term. We have taken it from expected value. Then expected value applied to random variable. This is the random variable. Further what you have here. See what we have done here. Transpose applied. When we have applied transpose here. So c transpose t transpose. When I will open it. So t will come first and c will come later. So we have rearranged it here. E of t mu. This is the expected value of this. And this factor. Because it is transformed variable. So what we have here. C prime t. So this is the moment generating function of the transformed t prime t. This is the moment generating function of x. Now further we have to do this. This is the answer. This is the one. Now the expected value of e c prime t. Transpose into y. Integral minus infinity to this. Because you have expected value. So you have integral. This function. Into f of y. f of y. Previous time. Given this is f of z. This is f of y. Transformed. Now f of y. We have entered its value. We have entered the value of f of y. Then we are solving it further. This is constant. Now we have entered the value of integral. This is exponential. You know that two exponentials are multiplying. So we are writing it in one exponential. So first value we have taken. Minus 1 by 2. y prime y. You have taken this as lcm. Minus 2 is multiplied here. Minus 2 c prime t prime. Into y. Into y. Into y. Now further what we have done in this. Further we have subtracted. Into exponential. C prime t. Transpose. C prime t. Add and subtract. Into exponential we have subtracted this value. Add and subtract. From this we have subtraction. We have already written it. This factor as it is. It is coming. Exponential of this value. This is value. Minus. Plus. This factor came out to us. 1 by 2 as it is. Now the exponential of the remaining terms have come here. Now we have seen these remaining terms. We have done it here. y prime y is sum of yi squared. Minus 2 is its transpose. 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