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Let $Z_{1}$ and $Z_{2}$ be independent standard normal rvs and let$W_{1}=Z_{1} \quad W_{2}=\rho \cdot Z_{1}+\sqrt{1-\rho^{2}} Z_{2}$(a) By definition, $W_{1}$ has mean 0 and standard deviation $1 .$ Show that the same is true for $W$(b) Use the properties of covariance to show that Cov $\left(W_{1}, W_{2}\right)=\rho$ .(c) Show that $\operatorname{Corr}\left(W_{1}, W_{2}\right)=\rho$

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

Temple University

University of North Carolina at Chapel Hill

University of St. Thomas

Boston College

Lectures

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were given independent standard normal Random Berry Z one and Z two and we're giving W one is the one in the W two is Row Times z one plus squared of one minus row squared Z two. Where wrote is the correlation coefficient of Z one z two in part A were asked to show that he standard deviation of W two is one in the W two has a mean of zero. Well, first of all, we have the expected value of W two. Hey, linearity. This is going to be, by definition, expected value of Rosie one plus squared of one minus row squared times E to Yeah, this is equal to since Rosa Constant Road times expected value of Z one plus squared of one minus row squared times the expected value of Z two. Now we have that. They figured values of Z one z two are both zero. So this becomes row time zero plus the square root of one minus for a squared times zero. Because these are normal random variables, this is simply zero sweep Shanthi mean zero. And moreover, we have that the variance of w two again, this is the variance by definition of grow times e one plus the square root of one minus growth squared times z two And because Z one and Z two are independent, this is going to be rose squared times the variance of Z one plus square root of one minus row squared, which is simply one minus row squared. Okay, times variants Z to This is because Z one and Z two are independent, so they're co variance. Term cancels out, I guess. Rose Not really their co variance here. It's something else. Now we have the variance of Z one z two because he's their standard normal random variables is simply one. And once this is going to be row squared, times one plus one minus were squared times one which gives us one. And since the variances one follows that the standard deviation of Z two is also one. Next in part B were asked to show that the co variance of w one and W two is row using the properties of co variance Well, we had the Deco variants of w one and W two. By definition, this is the co variance of Z one and wrote times Z one plus squared of one minus row squared times z two and this is the same using. I guess you could call it a sort of partial linearity. You can write this as the co variance of Z one Rosie one, plus the co variance of Z one squared of one minus rose squared Z two, which can then be written again by a sort of partial homogeneity as row times the co variance of C one and C one plus the square root of one minus row squared times the co variance of Z one MZ too. And we have that co variants of a variable with itself is simply the variance. This is going to be wrote times the variance of Z one plus square root of one minus row squared times the co variance of Z one and Z two and we have that the variance of Z one. We already know that the standard deviation of Z one is one thing is just going to be one. And because Z one z two are independent, it follows that the co variance of Z one z two is zero. This is going to be row times one plus the square root of one minus row squared times zero, which is simply row. And so we have shown that row is the co variance of W one w two. Finally, in part C rest showed the correlation coefficient. W one and W two is also row. Well, we have by definition the correlation coefficient of W one and W two. This is he co variance of W one and W two over the standard deviation. If there be one times the standard deviation of W two, as we pointed out before both w one and W to have a staring deviation of one. And as we saw in part C, the co variance was simply Rose. This is going to be row over one times one which is simply row. And so not only is the co variant of W when w two row, but so is the correlation coefficient of W one and W two

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