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Write down the principal components Y1 and Yz in terms of the Original variables X1and Xz [3]Write down the principal components P1 and Pz in terms of the Original ...

Question

Write down the principal components Y1 and Yz in terms of the Original variables X1and Xz [3]Write down the principal components P1 and Pz in terms of the Original variables Z1 and Zz [3]What percentage of the variation does Yi account for? [1]Find Cov(Y1 Y2) and Cov(Yi Xi)[3]Find Cov(P1 P-) and Cov(P1 Zi).[3]

Write down the principal components Y1 and Yz in terms of the Original variables X1and Xz [3] Write down the principal components P1 and Pz in terms of the Original variables Z1 and Zz [3] What percentage of the variation does Yi account for? [1] Find Cov(Y1 Y2) and Cov(Yi Xi) [3] Find Cov(P1 P-) and Cov(P1 Zi). [3]



Answers

Write a general formula to describe each variation. $$ z \text { varies directly with the sum of the cube of } x \text { and the square of } y ; \quad z=1 \text { when } x=2 \text { and } y=3 $$

Okay, so we have busy veering directly with the some of the cube of X and the square of why so that will look like Z is equal to the constant of proportionality times of this. Some of the cube of excess. So x cube and square of why? Why square plug in our values. One is equal to K times, so we have to. Cuba's eight three squared is nine, uh, 17. That's what that will equal solved for K and K is equal to 1/17 so the general formula will be easy is equal to 1/17 times x cubed plus y squared.

All right. So the first thing we're told in this problems that wide varies directly as acts in inversely as wise. What's that look like? Why Berries directly is acts that would mean a constant times acts and inversely Azzi. Rather, I said why, but so divided by Z that's what it looks like when it varies directly is X and inversely by the So we're given some values here were given that why is 2.37 we're trying to figure out k were given that X is pie and we're given that Z is the square root of two. So we have to solve this equation for K So I'm gonna start off by multiplying both sides by the square to and then off the viable sides by pie and that'll give me que so 2.37 times the square to all over high. Okay, not gonna be pretty, but that's our constant. So now we come back up here, right? Our equation. So we'll say why equals R K value, which is 2.37 square to over pie, um, Times X all over Z. So I think we can probably simplify this a little bit if we think about it. Has this, like 2.37 square to two acts over pie? Instead of dividing by Z, we could look at it as multiplying by Z, And then we can finish off the equation like this. 2.37 squared of two xz. Or maybe even like this, 2.37 x z, tens of square of two all over pie. Any of those would work. I kind of think this would be the most simplified way of writing instead of having a fraction in a fraction but

Right. So the first thing we're told in this problem is that why there's directly as accent inversely Aziz. So if it varies directly is acts, it would be K times accent inversely azi that would be divided by Z. We're trying to figure out OK, so if we multiply both sides this equation by se Z over X, then that would get k by itself. And I think that's easier to do now than to do later. So it be Z over. What rz times y over X equals K. All right, so let's use that equation. We have these values of why accident See that plug into that figure of constant variation. So Z is the square to to why is 2.37 and x is pie? So I guess 2.37 square to two over by. All right, so we're gonna plug that into the original equation here, So it's gonna be why equals are constant, which is 2.37 square to over pie times X all over Z. I guess I could probably center this a little bit better, So why equals all of that? I'm feeling like we could simplify that a little bit. Let's make it Ah, 2.37 square to two acts, um, over pie. And then instead of dividing by Z, we could multiply by one over Z. It's reciprocal, which would then simplify to this 2.37 squared of to X over, um pie Times Z. And that's what I'm gonna go with as the most simplified equation that we we can get. I guess technically, any of those green equations would work, but that was the most simplified instead of having a fraction in front.

First thing we're told in this equation that a is that a varies inversely and our started directly and inversely with something so very directly k times the thing and varies directly with which is the square root of em and then inversely with would be divided by the thing and varies inversely with which is the square of n or n squared. All right, so, uh, get k by itself, cause that's what we have to find. So if I multiplied by n squared over the square of em, that would cancel everything out over here and then multiplied by end squared over the square to em over here, we know that K is equal to a n squared over the square root of em. All right. And then we have all these values of a and Amon and that we can plug into that equation. So a is 5.47 aan is 1.625 and this squared of em, um, is irrational. I mean, so we're going to leave. It is the square to three. And that is what K would be. So I guess I mean, I'm gonna is doing air quotes right now, we could simplify this top part, but it's gonna be ugly. I'm typing 5.47 times 1.625 squared into my calculator, and I get a horrendous decimal that, I guess all right down. But it's one of 14 0.444 21875 over the square to three. I mean, technically, we shouldn't leave the square to three. And the denominator of the fraction. So I guess we'll fix that right now to its multiplied by the square to three over the square to three. And we'll get rid of that. And it's gonna be that K. So we're rationalizing the denominator, basically. Oh, boy. 14.444 21875 Um, all over away times the square to three. All over three. All right, that's k as ugly as it is. I stand by it. I've done that a couple times. So now we're gonna write our equation A equals K. Oh, boy, Why I keep saying that 14 times for or 14.44421875 Scored three over three, um, times the square to them. All over Z are no sorries IAM doing last from and squared, which could be simplified a little bit to this. About who? Here we go. Um, let's go 14 0.444 the last time in right does want to 1875 square root of three times and scored Obama's square 23 I'm we can combine those on. And then when we're dividing by N squared, it's kind of like the same as multiplying it by one over and squared so we could make it three and squared on the bottom. And I think that's technically the most simple. I could make this, um, make this equation. Yeah, I'm going to go with that. I don't know. You could probably get rid of all those decimals to if I wanted to, but we'll go with that. See what a teacher says about it


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