 Soft-Runs, in the previous lecture, we have studied the data matrix, vector, matrices. Now, after the data matrix, next we have mean vector because we have seen the data matrix, we have seen a single column, that is vector and if we have collectively handled it, that is the matrix. I have to find the mean vector and variance covariance matrix of data. This is the example of the mean vector and variance covariance matrix. पूँज, now this is the supposition. तब रोग ती तब वाब स्तेत, with three variables. दीजार दी आप मुथन्तार्टीकार वेर्यबल्त, जिस को आपने खोथ से नाम दीए अगे एक से एक वाब थे। इस से एक वाब लिए लिए एक वेर्बल्त, एक चो एक वेर्बल्त, अगर मैं इसे कृल्लेक्टी लेज दू्गई थाडएस ती माद्प्रिк्ट कूल। तो यह तो जल्छ कूलाट्ए थी मैंने कित और वो अगर कुल्टतष मैं लेग दूगँ बूत के लगदर ँदेभाक क्रि इदिंट. the very important point is this कि अगर में लेएनीो, X1, X2, X3 and में कहीं लेएनिर एगार।, X1, X2, X3, it means के X1 किसीं न किसी variable को represent करता है, X2 मीं्स किसी variable को represent कर रहं, तो औं generally X1, X2, X3 कहते हैं, उसके behind ही रहोता है के अगर थी जी न किसी variable को represent करते हैं. और अधा जसो आप ज़ाप थी आप बच्टोऊग, एक वेरीबल gratuita ava , devo all observations, अखर टीवरेबल Road, may be, तो, to find the mean vector, we compute the average of each variable across all observations. क्या किया हमने, average find किया, each variable across all observations. अगेन में आपको यहापे, explain कर लिए हूँ, कि अगर में, single column लेए हूँ, that is called the vector, अगर में, collectively हमारे पास योजेगा, metric. अगर, single column कि आपके पास, dimensions कितने होँ, अगर मुझो वेक्टर की dimension देखनी है, तो, single column की basically कितने dimensions होँगें, we have, how many rows, we have 5 rows and 1 column. अगर, here is the 5 rows and 1 column, 5 rows and 1 column. अगर में लिए है, this is the vector, single man लिए है, अगर में भी से collectively metrics बनारे है, तो metrics की dimensions कितने होगे आपके पास, how many rows, we have 5 rows, and how many columns, we have 3 columns. तो यह आपके पास metrics की dimension होँगें कि, now further calculate the mean vector, a mean vector हमने इसका calculate करने है, now this is the mean of x1, x2, x3, आप में अगें variable, expenditure, expense, household income नहीं बोल रही हूँ, मैं क्या के लिए हूँ, the mean of x1 क्यों के हमने असको let कर लिए है, के expenditure कों x1 के रही है, mean of x2 and mean of x3, मैं के से क्यों के लिए है, sum of all values, divided by the number of values, तो हम ने सम कर लिए है, first का, and divide कर लिए है, 5 से, तो मैं, 1 आगया, mean 2, and the mean 3. दास दी मैं वैक्टर लेस दे, हमारे पस मैं वैक्टर कैसे लिए लिए है, वैक्टर लिए है, small x bar, वैक्टर लिए है, stand for mean vector, अर वैक्टर के लिए वेक्टर के लिए येस पे साइन आम ने नीजे डाला, दिस से ती मीं वैक्टर, खोलेक्टर लिए हमारे पस क्या होगे, this is the, 3.4 किस का है, x1 का, this is for x2, and this is for x3, उसे हम ने केसे लिए? This is the mean vector मुऽे असे पींगटर है.偏 इसके आप तोफी� trial एक अस link का है। हम ने औ конфpose शाँदन बेएक्टर Te Manty सेद सरोदन ड्वैरिए, पर नीजीग तोंगे लिए to formalize this. कैसे निकालते हैं, to calculate the variance, covariance, metric, we need to compute the variance of each variable and covariance between pair of variables. कैसे निकालते हैं, mean vector निकालते हैं? Variance, individual variable का निकलेगा? And covariance between pairs, covariance किसका, X1, X2 का? Pairs के बिट्विन हमने, covariance चेख करनें. First, we will find the variance of X1. Again मैं आपको के लिएगों के मैं X1, X2, X3 लेएगों X1 के बहांट कोईना कों वेरिएबल लेग. X2 के बहांट, वेरिएबल लेम होगा. तो बार बार नेम लेने से बेटेर हैं, हम उसकी नूटेचन ले लेतें, variance of X1. कैसे वेरिएँन्स फाण करते है? You know that, हार वालिओ कोसके मीन में से मैंनेस करके हमने स्कीर लेना है. तम वाई आई मैंनेस वाई बार होल स्कीर. तो फिरस्ट आपके पास क्या ता? This is the first value, 1.2 minus its mean, 3.4. Okay? 1.2 minus 3.4 mean, whole square. Second observation, minus mean, whole square. Third observation, minus mean, whole square, up to soon. Divided by... कितनी वालिओ स्ती? फाई वालिओ स्ती? But we wrote 4 here. तो वेरिएँस कैसे होता है? क्या हम उसको देवाट करते है? N minus 1 के साँ. Okay? N minus 1 के साँ देवाट करने के लिए? Total 5 observations, N minus 1 means 4. तो वेरिएँस अप एक स्वन आगया, हमारे पास 3.2 से. अप सेखन हम ने अपशवेशन लेनी है. अप सेखन हमारे पास वेरिएँबल है. X2, X2 आप के पास है 3 point 4 minus its mean. तो हमारे पास क्या होगया? 3 point 4 minus its mean, 5 point 6. हार अपशवेशन को हम ने असके में में से मैने तोर देवाट करती है. 5 point 6, whole square divided by 4. 06. Now, we have also found the mean vector. We have also found the variance. Then, we have to do co-variances between each pair of variables. Now, we have to check the co-variances between x1, x2, between x1, x3 and the last between x2, x3 co-variances. Now, this is the co-variance between x1 and the x2. Now, we have to give the co-variance between x1 and x2. Now, the variable 1x1 is 1.2 and it is mean 3.4. And the next variable is 3.4 minus it is mean 5.6. Now, look at this 1.2 minus 3.4, 3.4 minus 5.6. So, this is the co-variance between the x1 and the x2. Similarly, we will take all the values. And the final result is the minus 1.16. Co-variance value can exist. But the variance value will not be negative because variance is in care form. Now, next is the co-variance between x1 and the x3. Now, the 1 is 1.2 and the third x3 is 1.2 minus 3.4, 5.6 minus 7.8. We have to check the co-variance between 1 and 3. So, 1.2 minus it is mean, third variable minus it is mean. Again, we did not have the observation. And the final result of the co-variance between x1 and x2 is the 2.475. And the last is the co-variance between the x2 and the x3. between x2 and x3, we have to check the co-variance. And that is x2 is 3.4 minus it is mean 5.6 and x3 is 5.6 minus it is mean 7.8. Now, look at this. Similarly, we have generated all the results in the same way. Divided by 4, q4 because we have to take n minus 1 terms. And the final result is the 1.1011. Now, we have got the mean vector. We have also found the variance and then co-variance. Now, we have the variance-co-variance matrix which we call sigma. This is the variance-co-variance matrix sigma. What dimension was it? Because we have taken 3 cross 3, we have selected it. So, variance-co-variance matrix we have got sigma. What is the first diagonal element we have? These are the variances. Now, we have the previous variances. Variance of 3.26, 3.26 and 3.06. So, what is written in the diagonal? Off-diagonal. Off-diagonal, what are the values we have? That is the co-variance. Whose co-variance is this? Between x1 and x2. And whose co-variance is this? Between x1 and x3. So, what have we got? Variances in diagonal. Co-variance in off-diagonal. And whose co-variance is this? Between x2 and x3. Now, we have got the co-variance. This is the minus 1.16, 2.475, 1.10. So, we have adjusted all the values here. So, this is the numerical application of the mean vector and the variance-co-variance matrix. Whose we have found? Again, x1 expenditure, x2 expense, x3 household income. And students, again I am repeating you that we will say x1, x2, x3 again and again. We do not have to recall the variable names every time. We just recall its notations. So, x1 means that there must be some variable behind it. x2 means that there is a variable behind it. So, what have we seen? We have checked the mean vector of this data. We have checked the variances. We have checked the co-variance. And we have adjusted it. Variance to variance matrix. And this is the mean vector x bar. Now, what is x bar? It is transpose. If I remove it, then I will take the transpose. This is the 3.4. It has been converted to the rows column. 5.6 and the last 7.8. So, we have the value of x. The value of the mean vector will come according to the given data. So, this is the example of the mean vector and the variance-co-variance matrix.