 2DN, previous topic, we have generated, movement generating function Mathematically we have generated movement generating function We have got an idea that this is the movement generating function Now how we can use the role of movement generating function Now look at this, this is the example of the movement generating function Or we can say that, .Mgf, the movement generating function Provide a way to characterize a probability distribution by generating movement of a random variable. तीके, movement generating function बेसे के लिए क्या करीए? correctorize करीए probability distribution by generating movement of the random variable. In the case of the multivariate normal distribution, the MTF can be expressed using mean vector. You know that this is the movement generating function. Previously, अभी हम ने इसको, mathematically, derived here. Let consider an example of the movement generating function for a bivariate normal distribution. हम इसे bivariate normal distribution के अखर्ट करने लिए? This is the movement generating function. You know that this is the exponential t prime mu. This is the mean and plus 1 by 2. This is the total. This is the function, movement generating function. 1 by 2 t prime sigma t. अब यस मुम च़रेटिं को हम ने नमारिकली कैसे किया है? Suppose we have a bivariate normal distribution with mean mu1 mu2. Bivariate distribution का अप रेकोल करो Pdf क्या होता है? उसके अखर्टिं हम ने यस को सोल्फ करे लिए? With mean mu1 mu2. And variance is sigma1 square equals to 3 and sigma2 square equals to 4. Suppose we have done these values. And correlation coefficient rho which is equals to 0.5. 0.5 हम ने यस में कोरे लेशिन लिए? We want to find the movement generating function for this distribution. कोंसे दिस्टिबुशिन? Bivariate normal distribution के कोरे हम ने? Movement generating functions generate करे हैं? यह नी उदर से हम values, determine करे हैं? The movement generating function for a bivariate normal distribution can be expressed as m t1 t2, t1 t2 q, क्यों के bivariate normal distribution हम ने लिए? So this is the exponential mu1 t1 plus mu2 t2. कहाई आप के पास प्रीवियस? This is the t prime mu. तो mu1 t1 and mu2 t2 plus half. Look at this. This is half. तो half को हम ने क्या काई? 0.5. देन हम ने ये bivariate का आगे उसके अकोरे लिए? एस को अपन कर दिया है? तेगमा वान सके ये ती वान सके? प्लास सिगमा तुसके ती तुसके? प्लास तु रो? सिगमा वान सिगमा तु? ती वान ती तु? बाई वेरीएट का आप पीदेव ज़ा लिकोल करो? उसकी हम ने यापे उसके कोडिए? अप उसका मुमन ज़न्रेटिंग रिकोल कर लो? अम ने यापे वल्योज लिकी है? प्लास सिगमा तुसके ती तु? बाई वान सिगमा तु? तो हमारे पास नेएखस कि अगजा? तु ती तु? याने वान वान वान वालु सेंट की है? जो बसतति Нет लिकाom Work? आप ज़ो। बत्रास सिगमा तु? शवना वान नेएखस की है? अप सिगवा औप सिगमा तु? अप ज़ो Haven Mant?] आप था। शवना यापे च्ीन। शन्तौ कर लिजा अप कोना ने येखस की है व! यरू मारे पासवालfree products मैसाभोडय मैसाब Rail this which is equals to फोर, fourth value is entered या प्लाज तो रो appreciation which is equals to point five so this is total था थिस्बमा येडीश कि यानौ तो लिया मैसा बखातư� assumed this which is attached at zero तो तुऴ�ना कहीण ऑगी था after's तो तया man the exact line of which की मैसा चेज़ ब हैस था जो थो ठ॥ी हु now अदिश्के रूथ of four, T1 T2. तो after simplification, we have the values of this, अदिश्के कि बनाद चनरेटिंग चनरेटिंग खन्छन एक थी औई आप के पास यहां पे T2 के साथ 2, 1.5 T1 square यहां पे आप के पास नहीं एसको सुल्फ के आप, 3 multiplied by 0.5. So 1.5 point square, 0.5 multiplied by 42 square. So further, we have this result after simplification. This is the moment generating function for the given bivariate normal distribution. Okay, now you can recall how to solve this. This is the exponential. Now you have basically exponential of x. This is the e, e power x, basically. If I let x, e power x in general, which is equals to 1, plus x, plus x square over 2 factorial, plus up to so on. And if this is minus, then e raise to power minus x, which is equals to 1, minus x, plus x square over 2 factorial, plus minus up to so on. So we consider e raise to power x. Now e raise to power x, how we will open it? 1 plus x. Now this is the term of x. t1 plus 2t2, 1.5t1 square, this is x, 2t2 square plus 1.45t1t2. This is the x, plus x square over 2 factorial. So t1 plus 2t2 plus 1.5t1 square plus 2t2 square plus 1.45t1t2. e raise to the power 2 divided by 2 factorial. Now these are the coefficient of t1. Now what we need is coefficient of t1. What is the coefficient of t1? Plus up to so on, you are carrying it. So plus up to so on, we have carried it. We are doing it till here. Now here you see, what is most important? Now we need their coefficients. t1, t2, t1's coefficient. Where is the coefficient of t2 coming from? Coefficient of t1 square over 2 factorial. And coefficient of t2 square over 2 factorial. How do we determine this particular example? From moment generating, how do we determine this? So coefficient of t1. Now what I need is coefficient of t1. This is the t1. What is the coefficient of t1? 1. So you can write it as 1. Here also you have t1. But here what is the whole square of t1? So whole square will be t1 square. That is why I have coefficient of t1 in the first portion. So coefficient of t1 is equal to 1. Now for second, coefficient of t2. Where is the coefficient of t2? Where is the coefficient of t2? Here you have 2. You have 2. t2 is also here. But this will be the square of t2. Now next, the coefficient of t1 square over 2 factorial. Now how do we determine this? You need to check that t1 square over 2 factorial. Now look at this. This is the t1 square. And what do we need? Divided by 2 factorial. But here 2 factorial is not there. What will be 2 factorial? We are multiplying here 2 factorial. 2 factorial means 2. If I have divided here, then you will have to multiply here also. So what will happen? 2 into 1.5. We will solve this. 2 into 1.5. We have got 3. Into t1 square. Now look at this. t1 square over 2 factorial. Because it was a square term. Now look at this next term. t1 square over 2 factorial. So here you have t1. t1 square. Because you have the whole term. t1 square over 2 factorial is also here. t1 square over 2 factorial is also here. So what is the coefficient of t1? Which is equals to 1. t1 square over 2 factorial. So I am looking at their coefficients. t1 square over 2. What is their coefficient? 3 plus 1. 4. I have coefficient of t1 square. 4 is equal to 1. Now similarly. The coefficient of t2 square over 2 factorial. How do we determine this? Where is it? This is t2 square. What is the coefficient with this? 2. But we need its factorial. t2 square over 2 factorial. Again we have multiplied it with 2 factorial. And we have divided it with 2. And then we have multiplied it with 2. 2 into 2. 4. Here I have written 4t2 square over 2 factorial. So what is the coefficient of this particular term? 4. Plus. The term after plus. Here you have t2 square. So t2 square over 2 factorial. So we have it. Plus. 2 square. Here we do not have to divide it with 2. Because already t2 square over 2 factorial. But 2 square. 4. t2 square over 2 factorial. Okay. I have got the value of this second part. I have determined it from here. So coefficient of t2 square over 2 factorial. Here what I have got? 4 plus 4. Which is equals to 8. Got it? This is 8. Now this coefficient of t1 t2 t1 square over 2 factorial. t2 square over 2 factorial. Why are we finding it? Coefficient of t1. What are you showing? This is the mu1. Okay. This is the mu1. Mu1 is a particular example. 1. So we also have the same answer. 1. Coefficient of t1. t1's coefficient. 1 equals same result. Coefficient of t2. Which is equals to 2. So here you have mu2. Which is equals to 2. Got it? This is our result. Find it. First we have done. Then second we have mu1. Mu2 is also done. T1 square. Coefficient of t1 square over 2 factorial. Variance is equal to variance. If I say variance, you know that the variance of particular x. Suppose I am doing x here. So this is the by definition expected value of t square minus expected value of t whole square. So particular example I have coefficient of t1 square over 2 factorial 4. So this is my answer. This is the expected value of t square. Which is equals to coefficient of t1 square over 2. You have to find variance. So variance is equal to expected value of t1. 4. Minus expected value of t1. If I am doing this for t1. I have written general for t1. So expected value of t1. 1. Because this is the mu1. 1. 1 square which is equals to 1. So variance of t i.e. t1 if I am doing this for t1. So variance of t1 is equal to 4 minus 1 3. Recall what was previous. Sigma 1 square this is the variance which is equals to 3. Result we have 3. So next the variance of t2. Variance of t2. Which value you have? This is the 8. Minus. This is the coefficient of t2. Its whole square. Not 2. Whole square. So 8 minus 4. Which is equals to 4. So variance of t2. 4. Second variance was sigma 2 square. Which is equals to 4. So what you have? Moomin generating function we have easily generated its mean and variance. So in previous example if you remember I had told you that t1 these are the coefficient we have to use t2 coefficient given. Otherwise we will use their mean and variance. Now in particular example how we have generated it. In particular example we have done Moomin generating function. We have developed Moomin generating function mathematically. We have found its mean and variance. And in particular example we have seen given mean and variance how we have determined its mean and variance. So this is a very important example of movement generating. Basically what is movement generating? Expression. We generate its mean and variance. Mathematically and numerically. So this is the example of the movement generating function using the bivariate normal distribution we have determined.