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Problem [10 points =5 + 5] Consider Brownin bridge {W() 0 <<} defined 45 Gnussian prOCCNS with Zero ICAD AIId cuvurinuce futction_ Cov [W(s), W(I min|s, t (0 ...

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Problem [10 points =5 + 5] Consider Brownin bridge {W() 0 <<} defined 45 Gnussian prOCCNS with Zero ICAD AIId cuvurinuce futction_ Cov [W(s), W(I min|s, t (0 < $.6<1) Evaluato covuriance, Cov [W() W(L ')]: where 0 < t< 1Derive the variance, Var [W( + W(T t)]: where << [Solution

Problem [10 points =5 + 5] Consider Brownin bridge {W() 0 <<} defined 45 Gnussian prOCCNS with Zero ICAD AIId cuvurinuce futction_ Cov [W(s), W(I min|s, t (0 < $.6<1) Evaluato covuriance, Cov [W() W(L ')]: where 0 < t< 1 Derive the variance, Var [W( + W(T t)]: where << [ Solution



Answers

For the hypothesis test $H_{0}: \mu=5$ against $H_{1}: \mu<5$ with variance unknown and $n=12,$ approximate the $P$ -value for each of the following test statistics. (a) $t_{0}=2.05$ (b) $t_{0}=-1.84$ (c) $t_{0}=0.4$

Hello. Welcome to this lesson in this lesson. We're looking for the p value. We have the alternative hypothesis us. The mean greater than 10. I mean you're dealing with. Yeah part two. So the P value with the probability that the bigger T. They were found from the table with a degree of freedom of and -1 is good to done. The T. Value the T. Statistic that's given to us. So Now 2.05 Given a degree of freedom of 14 this year. So 2.05 is between one 761 and 2.145. And that is zero point between 0.0 five and zero point 0 to 5. So looking at it this way Closer to 2.145. So we can have 0.0 five several itis There are 2.05. So between this so you can say P values opposed really their zero two for something because this public to three Cause that's very close to two 0.25. Right? Yes you would. The second part where you have They get to 1.8 for. So this one will take this one is less than it's around zero. We're dealing with the positive side. Okay we are dealing with the upper side. So this is around Gerald. So we have almost one. Okay that is On this figure. So almost one. Okay so the p value is almost one. Okay that's a negative. Okay like so let's go to the next part where we have t Just called 0.4. No the p value probability that t With a degree of freedom of 19 is created at the T. Statistic. So here we have the P value 0.4 is between zero and 0 0.6 92. So that is in between these two values. Just 0.5. Do we have 0.25? Mhm. But the value joke info is almost between it's almost the center of it. So we have the P value it as opposed merely They're all porn in three. Yes. All 0.33. It's not a approximate value. Was this cute toss? 0.25 was get into that. Okay. What? Yeah. Uh So yeah. Mhm. In between these two values. Yeah. All right. So France. What time? This is the end of the lesson.

Mm And this problem, random variable X is normally distributed with Mimi equals five and standard deviation sigma equals four. We wish to use a normal distribution table to find the following probabilities A through E that is we forced to find the relevant acts that satisfy these probabilities. So remember that for a standard normal distribution Aziz table maps Z scoring on the probabilities, that's probably easy. Greater than zero equals peanut implies that the area peanut purple is the area to the right, under the normal curve for score, is he? Not As an example, we have probably the greater than 0.5 because zero is the meat of the standard normal distribution, thus half of the total area is right. This example relates excellently to the to points we have here that we'll use to solve this problem, the symmetry of the normal curve and a total area under normal curve is one. So, to compute these probabilities, we need to compute them as the scores and then convert the Z score to our x variable value for x equals z, sigma plus meal. So the probability Z greater than 0.5 give zero equals zero. That's not easy. Not signal plus me equals five. Next, probably exported 6.95 gives equal 1.65. Thus X is 11.6 per. See we have a Z, not uh 0.36 or the problem is, you know, is less than zero, less than one. Where we converted our x equals nine to z. Score of one. This gives a 6.44 for D v solid follow a similar path. See we have TMZ score approaching infinity that his ex is not exist. The reason for this is that our x value of three correspondents t score negative 30.5. However there is no Xenon for which the probability that Z between negative 0.5 and 0.95 that is because 0.95 is around here notice how much area there is another standard normal that's leftover. Thus there can be no doubt that satisfies D finally any we look for negative X and X that satisfied from u minus five. Probably that's not the disease between negative zero and 9.99. This gives you an A plus or minus 2.56 So excess plus minus 2.56 times sigma equals plus or minus 10. 24. Notice how in party we did not add our meal because we subtracted mu originally

Mm And this problem, random variable X is normally distributed with Mimi equals five and standard deviation sigma equals four. We wish to use a normal distribution table to find the following probabilities A through E that is we forced to find the relevant acts that satisfy these probabilities. So remember that for a standard normal distribution Aziz table maps Z scoring on the probabilities, that's probably easy. Greater than zero equals peanut implies that the area peanut purple is the area to the right, under the normal curve for score, is he? Not As an example, we have probably the greater than 0.5 because zero is the meat of the standard normal distribution, thus half of the total area is right. This example relates excellently to the to points we have here that we'll use to solve this problem, the symmetry of the normal curve and a total area under normal curve is one. So, to compute these probabilities, we need to compute them as the scores and then convert the Z score to our x variable value for x equals z, sigma plus meal. So the probability Z greater than 0.5 give zero equals zero. That's not easy. Not signal plus me equals five. Next, probably exported 6.95 gives equal 1.65. Thus X is 11.6 per. See we have a Z, not uh 0.36 or the problem is, you know, is less than zero, less than one. Where we converted our x equals nine to z. Score of one. This gives a 6.44 for D v solid follow a similar path. See we have TMZ score approaching infinity that his ex is not exist. The reason for this is that our x value of three correspondents t score negative 30.5. However there is no Xenon for which the probability that Z between negative 0.5 and 0.95 that is because 0.95 is around here notice how much area there is another standard normal that's leftover. Thus there can be no doubt that satisfies D finally any we look for negative X and X that satisfied from u minus five. Probably that's not the disease between negative zero and 9.99. This gives you an A plus or minus 2.56 So excess plus minus 2.56 times sigma equals plus or minus 10. 24. Notice how in party we did not add our meal because we subtracted mu originally

So first I want to say there is a typo in this question. So where it says are is equal to X squared plus y squared. I want to say it should be r squared is equal to X squared plus y squared. So just kind of keep that in mind for when we're going through. If things seem to be a little bit off, I believe there was a typo, but they tells they want us to rewrite this in terms of just X, y and Z. So we want to get rid of all the tease. No first thing we could do is just go ahead and say each of the components equal to X y and Z So x Why and z So we'll have X is equal to 50 e to the negative t co sign T. And why is he going to 50 e to the negative? T silent e Only time you see signing co sign cropping up trying to get rid of it. It's not be a good idea to used with a gris. So by that I mean you want to use the fact that co sine squared of t plus sine squared of teeth is equal to one. So it's going and solve for side and co sign. So that's gonna give us X Her butt X co sign is equal to next times e to the t over 50 and then sign is gonna be the same. The only difference is is gonna be why instead, so we have these souls going and plug them in now. So if we square co sign, that's gonna give us X squared e to the to T over 2500. And then when we square sign that's going to give us why squared yea to the to t over 2500. And this is equal to what? Let's go ahead and multiply the 2500 over as well as divide by E to the to T. So doing that will give us Expert Plus y squared is equal to 2500 e to the to our negative to t. Now, this is supposed to be equal to if we think about in terms off unit circle R squared or even just a circle we know x squared plus y squared. Is he good? Our sport is the equation for a circle. So that's why I'm saying I'm pretty sure it should be equal to R Squared as opposed to our for what they say in the book. But now we could just go ahead and square each side because in our answer, they just have our so a square root, square root, square root, and that's going to give us Har is equal to the square root of X squared plus y squared and then 50 e to the negative t Now notice if we write out Z up here, though, So yeah, by minus. I don't know what happened to the five up here, but that there should be a five right there, minus five e to the negative. T if we were to divide all of these by 10 we get our over 10 is equal tubes of the square root of X squared plus y sward over 10. And this is gonna be five. Eat negative. So we could just go ahead and plug this in directly for it. It's on the stairs, I minus. So we said that this is supposed to be our over 10 and if we want to write this just in terms of ex and why this would be five minus X squared plus y squared square rooted, and that part is over 10. So this would be our equation Onley in terms of X and


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