 Dear student, here we will learn the characteristics function of multivariate normal distribution. What is the difference between the moment generating function and the characteristics function of multivariate? So the characteristics function is a function which completely determine a probability density function. But the moment generating function of every probability distribution may not exist. But the characteristics function of every probability density function do exist. Okay, what do we have here? Basically, the moment generating function cannot find every probability distribution. It cannot be determined. But the characteristics function of every probability density function can be determined. A little bit of mathematics of moment generating becomes heavy. We cannot properly solve it. That is why we use the characteristics function. So by definition of characteristic function of a random vector is defined as this. This is the psi of a t characteristic function. Or yeah, you have basically the notation of the characteristic function. So by definition which is equal to expected value of exponential iota t prime x. This is the by definition. So further is the expectation of the continuous distribution. The limit minus infinity to infinity up to so on limit minus infinity to infinity. Unit, unit Maripaskiya expectation expected exponential iota t prime x into f of x. This is the probability density. So let y equals to c prime x minus mu. So the value, find the value of x, x which is equals to t prime. We have the equality on that side. Then mu is equal to c y plus mu with c c prime which is equals to variance covariance matrix sigma and the y, y is distributed with standard multivariate normal with mean zero and variance identity as we learned before. So the f of y, this is the probability density function 2 pi minus p by 2 exponential of minus 1 by 2. So the characteristic function expected value of exponential iota t prime x. Now we have entered the value of x here. Further we have simplified it. Exponential you have multiplied with mu. Then we have multiplied this factor iota. This is multiplied with us. Now we know that this is the constant term. So we have bought the constant term by expectation because we apply the constant expectation of constant itself. Now this is the expected value of this one. So further we have to find the value of constant as it is. Basically we have to find the value of constant as it is. This is called the equation number 1. Now we have to simplify this. So this is the expected value of c prime ty. This is what we have to put further value of this. Look here you have t. Now we have to transform this in c prime t. So this is equal to c prime t which is equal to the expected value of this one. Now what we have here is the expectation we have opened. So we have the unit with the expectation open and its probability. Further f of y dy, f of y dy is standard normal. We have used its pdf and we have used the standard normal here. Then you know we have to simplify this term. This term is constant. We have taken it out of the integral. Then exponential of this variable and this which is equal to this. So now you have two exponentials multiplying. So when two exponentials are multiplying we have to take it in addition form. So this value you have first factor. We have got it first. Then what will happen here? Here it is 1 by 2 and here it is not 2. So twice time of this. Now next we have subtracted this factor add and in this exponential. So in exponential we have subtracted this portion add and. What we have added? The iota c prime t prime c prime t iota. Add and subtract. Now this is the negative term. This is positive term. Now we have taken the negative term first. You have this value with it. Further multiplying. 2 pi p by 2 as it is integral exponential. Now in this exponential look at this term I have taken first. Now you have iota iota iota square and the remaining function we have as it is. Here it is minus 1 by 2 minus minus plus. Then 1 by 2 as it is. So we have written this factor in exponential. Basically we have break the exponential. And the remaining part we have is part of another exponential. Now we are simplifying in this. Now this is the part of exponential and this also you have. Further how can I write this? Then i varies 1 to p yi square. This is the yi square minus. Because in the form of sum you have written it as further iota as it is. Here we are writing it as t i c i y i. And second part you have iota and iota iota square. c prime c prime c square t t t square. So we have written this term as iota t i c i whole square. Now further simplification. 2 pi minus p by 2 exponential iota square this value. Now you have p times integral. So p times integral is product term so we have done it one time. i varies 1 to p capital lambda limit minus infinity to infinity exponential of other part. So further simplification. Now you have opened this bracket. Here we have made this term further. This whole square is yi minus iota c t whole square. t i t i whole square. Now port i.e. we have lit this. This is equals to z i. This is t i and t i. So its derivative is dw i which is equals to dw y which is equals to d z i. Now here you have dy i which is equals to d z i. We have lit this factor for z i. So pi 2 pi minus p by 2 exponential of 1 by 2. This factor as it is exponential pi capital pi limit. In exponential you have term e power minus z square by 2. z square by 2 from here d z i. So we know that this value which is equals to 2 pi square root. So we have written 2 pi square root as 2 pi power 1 by 2. We have entered this. So the 2 pi limit its expected value will be 2 pi minus p by 2 exponential of this. Now t c prime which is equals to sigma. Further we will take it as sigma. This is equals to 2 pi power 1 by 2. How many times are we getting? P times. This is the product. How can we write this? 2 pi raise to power p by 2. This is the positive p by 2 and this is the negative p by 2. So this is remaining part. And do you know from where we have broken it? And we are finding its value. So this is the characteristic function of exponential. Where do you have this value? This is the equation 1. Now we have to enter this value in equation 1. In equation 1 we have entered into this c c prime which is equals to sigma. And what is the characteristic function of exponential? This is the iota t prime mu and this is the t prime sigma t. This is the characteristic function. And iota square which is equals to 1. Here you see that iota square is removed. The value of iota square is 1. Here we have characteristic function of multivariate normal distribution developed.