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Exercise 2: An application of Ita's formula (3+4+5 Marks) Using the Ito $ forula with X, = W; and f(.1) = 2/2 prove the identity k WdM; Zw - 1 2 u By choosing...

Question

Exercise 2: An application of Ita's formula (3+4+5 Marks) Using the Ito $ forula with X, = W; and f(.1) = 2/2 prove the identity k WdM; Zw - 1 2 u By choosing appropriate functions Ito $ formula t0 show that W?dW %w- 6 Mds. +2u. ad WSaW; iw K wzds t 2 0 By taking expectation On hoth sicles ofthe (quatious in part (b). calculate E[WA] aud EWW,']. YOu ae USILg AHS fact from the lecture. state them and explain clearly why they can be applicxl this CaSC

Exercise 2: An application of Ita's formula (3+4+5 Marks) Using the Ito $ forula with X, = W; and f(.1) = 2/2 prove the identity k WdM; Zw - 1 2 u By choosing appropriate functions Ito $ formula t0 show that W?dW %w- 6 Mds. +2u. ad WSaW; iw K wzds t 2 0 By taking expectation On hoth sicles ofthe (quatious in part (b). calculate E[WA] aud EWW,']. YOu ae USILg AHS fact from the lecture. state them and explain clearly why they can be applicxl this CaSC



Answers

Prove the identity, assuming that F, ?, and G satisfy the hypotheses of the Divergence Theorem and that all necessary differentiability requirements for the functions f(x, y, z) and g(x, y, z) are met. $$ \iint_{\sigma} \operatorname{curl} \mathbf{F} \cdot \mathbf{n} d S=0[\text {Hint: See Exercise } 37, \text { Section } 15.1 .] $$

So what we're seeing in this problem here as we want to start with the greens first Identity, which states the double integral, um, of F del Square, G D A. Is equal to the in a rule of f of dell g dot and yes, minus the double integral of the Grady int of f dot the greedy int of g d a. So starting with that, we can rewrite the identity setting g equal to F. So now what we actually have is, um, an f right here. So we'll just use a different color. So have f and then with that in mind now, what we can write is that if f is harmonic, we know that this right here is gonna be equal to zero on D. So how we write this is that, um this right here will just be equal to zero. So now that we know that, um, we can show that if f of X Y equals zero than this right here is equal to zero as well. Um, and then this equation right here, the equation above all right here can now be reduced into the following since jersey gotta f we can just write that twice and then we know that's equal to zero. So with that, what we've shown is we can also use the property that a a equals the magnitude of a squared. So now what we end up having is that this right here is equal to zero. So if G is harmonic on D and f of X, Y equals zero on the boundary curve than this statements. True, So we've proven it.

The problem. We start. I mean, first on entity off beings Teoh Haiti's don't f it is body being waas closing f men Dean are envious My less no one indeed That Del Gene there meeting that the identity I mean the yeah, we're not gonna be And then this the Waas Roland Igel f enough not mbia's My last No one. That's about the No Yeah, If it is hot Morning, then what? Um no, no. You know, they f Dallas by f the a wasn't you? So if you know extra my wife was zero on the goal. See? Then they left the nds people zero Is it used to? No todo Della Donne being it wasn't You are using the property in our a was more a police fire me have don't video more Hold the equality

Now, here on this problem, we are told we have a random variable. Why with the moment generating function F. T. We're also told that we have you that's equal to a Y. Possibly. And we would like to find this moment generating function and then it's expected value. Mhm. Now, the moment generating function for you is equal the moment generating function of A. Y, possibly. Which means that this is the expected value of E. Mhm. To the A. Y possibly be team either the A wipo's B T. Use your properties of exponents here. This is equal to the expected value of E to the A. Y. T. Times E. To the Bt. Since the the Bt is just a constant, this means this is either the bt times the expected value of E to the A. Y. T. Yeah. Now either the I. T. Is just a moment generating function of why evaluated at 80. And so the moment generating function of you, as you call the E to the Bt. There's a moment generating function of why at 80. And that's what I wanted to show here. Now we want to use this to find the variance and the expected value for you. Mhm. Now the expected value of you is equal to the moment generating function of used derivative at zero first let's find a derivative here. Mm hmm. Using the product rule. This is equal to be eat of the Bt times. Innovating lost A. M prime of 80 each of the meeting. And so um U prime of zero. It will be easy to the zero mm zero plus a. M prime of zero. Even the zero each of the zero animals. There are both one. And so that term they're just becomes B in prime of zero is the expected value of why? And so this becomes a times mu because Mueller is the expected value of why. And so they expected value of you this B plus a mute. That's the expected value of you. Now for the variants We need the second derivative of you. So we take the derivative of the first room. We're gonna have to use the product rule twice here. And so it becomes B squared eat of the Bt mm of 80 Yeah plus a m prime of 80 times B E to the Bt boss. A squared mm double prime of 80. Either the Bt plus be either the Bt Hey, in prime of 18, that is the second derivative here of the moment generating function of you. Now we want this at zero. So this is B squared eat of the zero M of zero. Was a M prime of zero. Be either the zero. Yeah. Was it a squared mm double prime of zero. Either the zero plus B equals zero A. In premature E m m zero just one. And so we have B squared for that term. And prime of zero is mu That's just the expected value of Y. Time to be times one times A. That's because A B new in double prime of zero is the expected value. Why square this? We have plus a squared. You have I swear. Okay. And over here we have a B. M prime of zero. Such a B new because improvement zeros mute. This is our second derivative. And so this is the expected value of U squared can be expected by you. Square The variance of you is equal to the expected value of U squared minus the expected value of you squared. And so we just found that this first part here is a B squared plus to a be mu We have navy Munich maybe abuse will have those together plus a squared he of widespread. There were no subtract off the expected value are expected value of you was able to a mu because B square. Yeah. And so this becomes B squared was to a beam. You was a sward. You have widespread minus a squared U. Squared because to a new B I was in the spring. B squared minus B squared. Goes away to a be immune minus to a B mute goes away. So this is based where he of Y squared minus a squared music word which is a squared times E. Of widespread minus muse word. You have Y squared minus we just heard is they expect is the variance of why? And so this is a squared experience of woman. And so we showed that the variance of you is equal to a squared. That was a very handsome wife.


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