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Consider two independent samples {}iI Nlur, 0}) and {yki_1 Nluy, 08) Show that x y is an unbiased and consistent estimator for Hx Hy:...

Question

Consider two independent samples {}iI Nlur, 0}) and {yki_1 Nluy, 08) Show that x y is an unbiased and consistent estimator for Hx Hy:

Consider two independent samples {}iI Nlur, 0}) and {yki_1 Nluy, 08) Show that x y is an unbiased and consistent estimator for Hx Hy:



Answers

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 $\operatorname{Corr}\left(W_{1}, W_{2}\right)=\rho$

Yeah. This problem will be given to probability densities. The first is G. Of X. It's 24 over X. To the fourth or X is greater than two. Yeah. And the second is H. Of Y. For each of wise to why with the Y being between zero and one now we are told that these are independent. And so this means that our joint distribution F of X. Y. Is the product of these two, says it's 24 over at the fourth times two. Why? Which is 48 X to the negative fourth. Why that is our joint distribution. And notice that acts is greater than two. And why is between zero and one. Know what we would like to find is the expected value of Z. Which is the expected value of X. Y. Mhm. No. This is going to be the double integral. That's why times or probability distribution sometimes 48. Excellent and so forth. Why? Yeah. No extras from two to infinity. And why it goes from 0 to 1. Yeah. And so this is april 2. 48 times the integral from 0 to 1 of why times Y. Is y squared do Y? And then times they entered all from two to infinity of X times X to the fourth which is excellent, negative third D. X. And so we just broke that integral into those two different parts of. Once we have this this just turns into evaluating each of these different intervals. So the first thing we have is that 48 out front. And so the 48 will just say his name. And then we have one third y cubed evaluated from y zero to one. Then we have negative one half X to the negative second evaluated from access to infinity. So this is 48 times for one third white cube evaluated from 0 to 1. We just plug in one, plug in zero and then subtract, giving us one third negative one half X the negative second evaluated from two to infinity. We're going to take the limit as we go to infinity and then plug into and subtract those two values. This is gonna give us 18 So now we have 48 times, one third times and eighth, which is to.

Come to this lesson and this lets we have two point estimated a rear checking whether or not they're on my ass estimate is for the population parameters that will find the variance of each parameter. Each estimated the first but we are looking at the expectation of this point is meeting what should be able to give us the population me. Yeah it virtually Azam by asked. Okay so let's see what let's see if that is she caught in that Huey chef the expectation of X one last X two along too. So we can split it into we can bring the half outside so that we have expectation of this one. The last expectation of X two then half half out Expectation of X one. The last expectation of X two. Okay. Okay. Mhm. So this would give us yeah for me Yeah. Mhm. Okay. And this will also give us this will also give us the population mean? So we have half Than two times the population and this becomes me. So the expectation of the point estimate has given us me which means it is on my ass. So it's just until I asked. Yeah. Yes. Okay let's check the second one. If The one is also I'm biased. Mhm. Yeah. Yeah. Yeah. Mhm. You're looking at the second one we have the expectation of the second estimated Which is excellent for last three times x She all over four. So expectation of this in the same we can bring the ones fourth out. Okay. And now this becomes from them before yeah Expectation of X one last three expectation of X two, This becomes one on 4. This is the population mean, than three times the population means, so this becomes one on four Times four times the population meat. And this becomes the population mean Yeah, it means the second point of tomato is also and I'm by asked estimate of the population mean this was so unbiased. The second path, we are looking at the variance of the two, so by professes to meet her, we have X one for all aspects to all of it too. So the to come out as a square, let's play both. So we have X one on 2 plug back. That's true, only truth. Yeah. Mhm. Yeah. Uh huh. So this is one on 4, the consent comes up as a square, Then the bar of X one becomes the experience Plus one on 4. The very mess. Okay, so this becomes two Time, is the variance all over four and this becomes squired. Uh two two goes into someone's, it goes into four two times. So we have the bar of the parameter as the variants all over two. Okay, that's the first one, Just go to the 2nd 1, you have the second Yeah stigmata which is where X one plus three. X. See all of the four. So with this one we can splits, you have bar that's a bar three or 4. That so this becomes one on 16, the variance for us, nine on 16 times very innocent. So this is equal to 10 times the variance all of US system, and that eventually becomes five with eight times the various. Yeah, so this is the bar of the 2nd Parameter, uh, estimate. Okay, some short time. This is the end of the lesson.


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