let-x_1-x_2-denote-the-variables-for-the-two-dimensional-data-in-exercise-1-find-a-new-variable-y_1-

Name: let x_1 x_2 denote the variables for the two dimensional data in exercise 1 find a new variable y_1
Uploaded: 2020-07-28
Duration: 1 min 13 s
Channel: Numerade
Description: Let $x_{1}, x_{2}$ denote the variables for the two-dimensional data in Exercise $1 .$ Find a new variable $y_{1}$ of the form $y_{1}=c_{1} x_{1}+c_{2} x_{2},$ with $c_{1}^{2}+c_{2}^{2}=1,$ such that $y_{1}$ has maximum possible variance over the given data. How much of the variance in the data is explained by $y_{1} ?$

Question

Let $x_{1}, x_{2}$ denote the variables for the two-dimensional data in Exercise $1 .$ Find a new variable $y_{1}$ of the form $y_{1}=c_{1} x_{1}+c_{2} x_{2},$ with $c_{1}^{2}+c_{2}^{2}=1,$ such that $y_{1}$ has maximum possible variance over the given data. How much of the variance in the data is explained by $y_{1} ?$

Jack Chen · Accepted Answer

according to a previous exercise, this new variable. Why one We will be 0.9 for 65 x y minus 0.3 to a to a steppin x two, Um, and this variable subject to the constant that&#x27;s on the coefficient in front, off X one next to our normalized. And, uh, why one has next to impossible variants over the given data. Um, so the various imitate eyes explained by why one, um, so the president do job flow? A very is explained by where, when we will be lambda one over lambda one plus lamb that you both this Agon values are, um, in the previous x ice. So the president you will be about 93.33%. So why one explains about 93.33% off Berries.

A M · Answer

so using the hint for this problem we have that variance of a experts be is equal to the expectation of a experts. Be minus the expectation of experts. Be inside Prince he&#x27;s squared, then close bracket. Now this is equal to the expectation of a X plus be minus eight times muse of exposed be and this quantity squared, which is equal to the expectation of now that we have to be here and they&#x27;ll be here, we can cancel them. So we have the expectation of a X minus a mule X squared, which is equal to the expectation off a squared times X minus mu X squared and this can be written us This could be redness, some of a square times x minus mu squared times pubic ce. And this is by definition, where ex lives in X. So now we can take out the A squared out of the summation because they escort is a constant with respect of X variable in Catholics. So we have a squared times a summation of X minus mu squared times p of X, where again excellent. The Knicks. So this is equal to a squared times Myriads of ex capitalists because Catholics is a random variable here and we&#x27;re taking the variance of that while in this definition, X is a realization of the random variable where we&#x27;re summing over the values of X dorky sex in capital X.

Jack Chen · Answer

Okay, so for this problem, part of one we know that X one up to eggs in Hi. In, Um I mean deviation for that means, um X one process x two up to eggs. Then we will be, ah zero vector which is and by 10 Victor Lemmens If we defied of extra top very close to one and by one vector with all with all inches are one We have a magics modification x one x two Exane multiplied by w equals 20 which we secretly into this condition. But this muse p transpose times um, this matrix multiplication because with zero. But by our definition, patrons post times on this metrics will be why one up to y in So basically change the the order off multiplication and that this line means why one upto y in How are also in a media in deviation form. So this is the proof for part one. And for part two, we know that there are coherence. Metrics for X well is defined in this way and the full for why the cover and men covers metrics s why he closed toe one over a minus one. Why times why transpose and by the definition of why, equals two p transposed eggs times teachers, post eggs and the amateurs supposed outside in a barrel off veterans posts. We have X transpose times p and that we we know that X choose post times extents extras post divided by impound and minus one equals two ways. So in the end we have Peter&#x27;s Post Times, Ace&#x27;s Times Peep, which proof the which proofs The second statement.

Sonam Khatri · Answer

in this problem, we have toe derive the shortcut formula for the medians. A square is equal to submission X squared minus and experts square divided by n minus one. From the formula a square is equal. Do summation X minus eggs Bar square derided by end minus one By using the form less for summation that this by using the summation. Druids, this can be done s a square. Is it will do by using the formula for a minus Bi Break it square. This can be Rick Turness Summation off. Ex squid minus two times X My deployed by X pot Bless expired squid divided by end minus one. Yeah, we have used the formula for a minus Bi bread cakes. Grip. Now this is a well, too Now we will use the formula for summation which says that summation off X less on minus by. Is it well do Summation X bless or minus summation. But so this is a well toe summation. X squared minus So me. Shame two X expat. Bless summation experts. Quinn divided by end minus one. This is equal to now. We will use the summation rule which says that summation C X is equal. Do see summation X They&#x27;re C is a constant and also even use summation. See, they see is a cons JEC is it going to do and see? So mission X squared building me as it this minus so expert is a constant so we can take it outside from the summation Zain So this is being minus two eggs Bat summation X bless and that is a constant. So summation eggs bar square will be a summation Experts quit experts Quick summation sign will be removed It will be a expert squid divided by end minus one as it is now. This is a will do information X quipped will be as it this minus do expert No, We will use the formula off expert which says that expired Is it going to so Mission X divided But I am so using this we can write summation X s X But my team played back end plus an expert square bill remain and sit this divided by end minus one. Now this is a will do summation X squared. This will be minus toe and experts square and this is plus and experts square so minus two and experts quit bless and experts squared biblical minus and experts quit divided by end minus one as it is. So we have the right to short good formula for radiance that this s square is It will do summation X square minus and experts squared, divided by end minus one, which is about required formula.

Arulmozhi T · Answer

from exercise to We have the probability modern as ex it&#x27;ll 12 and be off access 0.2, 0.4 0.4 and US 1.2. We have to find the standard deviation. First we&#x27;ll find the radiant just Sigmar Square. And we noted that the weightings of random variable is expected value off the square deviation from the mean. So that is equal to summation X minus mu the whole square into P off X. Therefore, from the day boot, we have zero minus 1.2 just a mean whole square in tow. Zero point to bless one minus 1.2 the whole square into 0.4 place to minus 1.2 whole square in 20.4 It&#x27;s 1.44 It is it a point to this They&#x27;re appointed or four into 0.4, please. You will 0.64 in 20.4 this 0.288 less point 016 Bless 0.256 sequitur. Yeah, several 0.56 Sigma is equal to scrabbled Off Siddle Point, Phisix. This suitable 0.7 four Lanktree. No section B just given the probability. Morning excess 102 100 300 it 400 and Bill fix US 0.1, 0.2, 0.5 and 0.2. And muse equal to 280. WAY will first find the Radiance Sigma Square, which is summation X minus mu. The whole square into pure fakes. That is, 100 miners to 80. The whole square into little 0.1 place, 200 minus 2 80 the whole square inches. It&#x27;ll point to plus 300 minus two. Weighty the whole square into 0.5 plus 400 minus 2 80 the world square in 20.2. So we have 32,402 0.1 yes, 6400 in 20.2 this 400 in 20 point fight less. Oh, 14,400 in 20.2 3000 to 40 this 1000 to 80 plus 200 place yes, 8 2080 Yeah. News 7600 No. Sigma&#x27;s Square root off 7600 just 87.178 Is that a standard deviation?

Heena Haldankar · Answer

All right. So here, we&#x27;re supposed to prove, uh, the expression for cool villians. So over here we&#x27;re going to use the properties of Kobe Indians. So the first property that I&#x27;m going to use here is that obedience off X, with a constant, is always zero where c is any constant. Okay. And then the second property that I will be using here is that if I have obedience off E x comma D, then that is also equal. Does you know where a nd are constants and similarly cool? Aidan&#x27;s off any Constance a the We&#x27;ll also be completely zero. And the foot property that I&#x27;m abusing is that you have cool radiance off X comma by the negative, equal cooperating. So for a x comma vie. And it is also equal to a in the Koreans off X coma. Why? There is a constant here. So to start with, the first part of the relativists is work I&#x27;m going to consider. So they&#x27;ve given the letters as cool Billions off a X place be comma si vie les de. So to start with this and we&#x27;re forced, apply the property here. Where are it states that we can bear the germs off each variable in the first place, but the each available in the second place. So I&#x27;m going to write this as school billions off. We have a ex common. See why LSD? Plus I can say they have Koreans off B comma si vie les de further Nazi Web list Derek and Verity X, and we can split it further, so I will have cool billions off. This is ah e X comma. See why? Plus they have Kobe aliens off a X comma d then plus the same thing with the next Kobe shin stumps of your Corbin&#x27;s would be comma si vie. Plus I have court Gideon&#x27;s off be with thee. Now we know that every city we showed our constant and accent by are the main b enables. So then the terms that you will be zero I will just mention here of the terms which are becoming zero. So we will have Ah, this storm which will be zero. It is this time because I&#x27;m applying the second property here. Then I can see that this time, but also be zero because of the same second property. And, uh, we can write on this altar, a zero using the property. So now, out off The Fordham&#x27;s three terms are directly zero because of the properties, returnable So then I can directly, right. This is therefore equal to co variance off only ex common. See why? And we can write this as a sea into covert agents off X Y. And this is due to the property number four. So based on this became, say, this is equal to our chest. So we have proved that for that cove Aliens off, he explains, be with C y les de is actually equal toe a C intercooled agents off X y. So I&#x27;m putting this as equation number one. Now for the next part, we&#x27;re supposed to find the expression for correlation. So to start with the B part, we know that correlation off ex while will be given by Cove aliens off X Y and divided by standing division affects, understand interrogation or fired and putting this question number two. So now I&#x27;m gonna consider therefore co relation off the explosives being with sea by blessed days will be given by cool readings off a explains be received by LSD and this is divided by a standard deviation off a X place. Be into your standard division off, see by Blaise DE to find the standard deviations Sigma off the X Let&#x27;s be, we can say because of the property this is given by more off A into Sigma X and ah, Sigma FC by LSD we can drive this asked more off. See Percy my way. So, uh, this is because of the properties of standard deviation. And now I can therefore write down that sigma off explosives be interested more sea by LSD. It&#x27;s, um, equal, more off a interested mikes into more of seeing percent more wives, which I can write as more off a C for sig Monix and save my fight. Now we can say that since e and C have seeing signs so I can therefore write more off A C is equal to positive, easy, directly and business. So my expression for co relation off, he explained to me, and see why this deal will be equal. So the Kobe is Valley. We were already proved in the force part as a sea into covert agents off X y and the survey, divided by a sea into sig My extensive my wife&#x27;s We can see you can cancel the sea and what remains your risk? Obedience off ex ally upon standing ovation affects understand evolution of fire And this is nothing but a correlation off ex wife. So therefore, we&#x27;ve proved the second part also. Okay, now we&#x27;re going to the next part for the far tree. They&#x27;ve asked IFOR e N c out of different science thing Billy C b positive. So we see that if e and C have opposite signs so that gives more off cc will actually be negative a c Now, if this is negative a sea than therefore we say that correlation will be negative, right? And so in this case, we therefore can conclude that correlation off a explains be common sea by LSD will be actually below minus off course relation off X Y.

Problem

Repeat Exercise 7 for the data in Exercise 2

Problem 7 Easy Difficulty

Answer

$y _ { 1 } = 0.95 x _ { 1 } - 0.32 x _ { 2 }$
variance percentage: 93.33$\%$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Alayna H.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

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$y _ { 1 } = 0.95 x _ { 1 } - 0.32 x _ { 2 }$variance percentage: 93.33$\%$

Linear Algebra and Its Applications

Grace H.

Alayna H.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Alayna H.

Maria G.

Kristen K.

$y _ { 1 } = 0.95 x _ { 1 } - 0.32 x _ { 2 }$
variance percentage: 93.33$\%$

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications