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Let (KY) be independent standard normal N (0,1). Define U = a+6Y and V =e+dY for some &,b,€,d€ R(a) Determine QUV in terms of 0,6,0,d (b) Find all c...

Question

Let (KY) be independent standard normal N (0,1). Define U = a+6Y and V =e+dY for some &,b,€,d€ R(a) Determine QUV in terms of 0,6,0,d (b) Find all conditions such that U and V are independent:

Let (KY) be independent standard normal N (0,1). Define U = a+6Y and V =e+dY for some &,b,€,d€ R (a) Determine QUV in terms of 0,6,0,d (b) Find all conditions such that U and V are independent:



Answers

Let $X$ and $Y$ be independent rvs, with $X \sim N(0,1)$ and $Y \sim N(0,1)$
(a) Use convolution to show that $X+Y$ is also normal, and identify its mean and standard
deviation.
(b) Use the additive property of the normal distribution presented in this section to verify your
answer to part (a).

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

In this question. The given is you want and you too is given as U. One is equal to violent plus Y two and you two is equal to Y one minus Y two As violent and Y two are independent. Normal random variables with mean zero and variants. Sigma Oscar So what we have to find in this question we have to find the joint moment generating function of U. one and U. two. Where is the human and you do are also independent. Normal random variables with mean zero With me in zero and variants sigma square. So our answer becomes expectation of exponential T. one into violent, blessed by two plus T. Two. Why? When minus via to which is equals to expectation of exponential T. one Plus T. two 51 plus T one plus T two by two Which is equals two. M. Y. One Stephen plus Tito M. Y. Two T one plus T two as many as zero and variances sigma square. So exponential Sigma scare by two T one Plus T two. Old Scare. Exponential of sigma scare too Stephen minus T. To hold scare it equals to exponential sick Musker by to be even scarier exponential. six months whereby too T two whole skip This one is also a whole school. So a required answer is M you won T. One, I am you do T two. So since this is the joint MGF means mormon generating factors Where is the U. one and U. two are independent. So this is the required answer which we have to show. So thank you. So


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