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(b) The moment generating function of random variable Z is defined as Wz (t) random variables then show that Yx-y (t) "x(t)y (t).E(etz-_ Now if X and Y are ind...

Question

(b) The moment generating function of random variable Z is defined as Wz (t) random variables then show that Yx-y (t) "x(t)y (t).E(etz-_ Now if X and Y are independentAlso, if Wx (t) (1 -p + pet)" then determine E(x) and Var(x) (Note: Try to compute these values without inferring what kind of random variable X is )

(b) The moment generating function of random variable Z is defined as Wz (t) random variables then show that Yx-y (t) "x(t)y (t). E(etz-_ Now if X and Y are independent Also, if Wx (t) (1 -p + pet)" then determine E(x) and Var(x) (Note: Try to compute these values without inferring what kind of random variable X is )



Answers

Consider two random processes $X(t)$ and $Y(t)$
(a) Show that if $X(t)$ and $Y(t)$ are uncorrelated random processes, then $E[X(t) Y(s)]=\mu_{X}(t) \mu_{Y}(s) .$
(b) Show that if $X(t)$ and $Y(t)$ are uncorrelated random processes and $X(t)$ has mean function
equal to zero, then $X(t)$ and $Y(t)$ are orthogonal.

So for this one, we want to show that the variance of these two independent, random, very loans extent why is equal to the expected value of X squared times the variance of why, plus the expected value of y squared times thieve arians X plus the variants of X times the variants of why so actually don't work with the right hand side of the equation here and get that too equal the left hand side. And so the first thing this just input the definition of variance each place it occurs. So we would get the expected value of X squared times and doesn't equal to the expected value of why squared minus the expected value of why squid. Okay, we're gonna add that to he expected value of why squared times. And then we put in the definition of the variance of X. So that's gonna be the expected value explain minus the expected value of X squared. And then we'll add that to these two variances. Multiply it together. So we get again the expected value of why squared minus the expected value of why squared. Multiply that against expected value. X squared minus the expected value of X square. Okay, I'm gonna move on to a new page. But when we do that multiplication, we end up getting the expected value of X square times he expected value of why it should be a little squid in there and then minus I mean the expected value. X squared times three expected value of why it's great. And can I was just the first expression. Multiply it out. We can zoom along their heads. Expected value off. Why squared times the expected value X squared minus the expected value. Weiss Where times the expected value. That X It's great. Okay, a few more expressions. So now we're doing The two variances were multiplied by each other. And the easiest way to do that is, of course, using our foil method. But whatever works for you should result in the same solutions. We get expected value of X squared times expected value. Why squared canine unit subtracts from that The expected value of X squared times expected value of why squared subtracting from that the expected value on X square in times the expected value of why square and last they will add the expected value of X Square times the expected value of why squared and you'll see that there's a few things that will cancel out. So we have. This was a value of y squared times. It's what the value of X squared minus the same thing here. Okay, this will have the expected value of X squared times they expected value of Y squared. And from that we can subtract that same expression here. There's one more we can eliminate. Yes, we have the expert's value of X squared times. You start value of why squared and we have two of those that cancel out. And so they were just left with just two expressions here. Okay, so if you write that out, this is equal to the expected value of exclaimed times the expected value of why spurting minus the expected value of why squared times the expected value of X squared. Okay, but this is equal to the expected value of X Y squared minus the expected value. That's why squared and so than that simply equal to the variants, despite a definition a variance of X y. Okay. And so then we've shown that this equation the right then equals the variants expression that we had him on the left. So that concludes the exercise

Here in this problem, we are doing more moment generating functions. We're told that why is rin variable with the moment generating function given my M. I. T. And we are told that we have another random variable W. And W is equal to A Y plus B. We want to show that the moment generating function of W. So I'll call that M W F T is equal to eat of the T. B. Time to him of 80. Or that innovates is related to that random variable. Why? Right. It's important remember that the moment generating function because the expected value of E to the T. Y. That's how you find a moment generating function of a function now. Mhm. As a result, if why has a moment generating function of them, see that tells us the expected value of E to the T. Y is a 14 FT. Now we're looking for that of W. So the moment generating function of W. There's E. Of E to the T. W. Now, since W. S. A. Y plus B. This means this is E. Of ear to the tear times A. Y plus B. Yeah. Which is the expected value of E. A. Y. T. Wasn't beauty. Now, this is equal to the expected value of E. To the X. Y. T. Times it'll be T. Using properties of exponents there. You're adding in the next moment when you break it apart. That's the same as multiplication. Mhm. Now, using the properties of expected value, this is the same as E. To the Bt times the expected value of E. So the A. Y. T. I have that eat at the Bt out front just like we wanted. But the expected value of E to the white is just the moment generating function of why? And 18. And so this is what we wanted to show. The moment generating function of W. Zero all the way to the Bt times the moment generating function of Why? 80

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.

All right. So we've got two independent and identically distributed function. Bannon processes A and B. Okay, we're gonna say the main function is view of tea or a Turk. A variance function. See off T of s if you want to use the order correlation function or are of tear ness. So that's why I sort of put etcetera at the end and what we're gonna do, because the fine exit t to be some A plus B. I'm going to find why to be the difference. A of T minus built. I got these two man in Fairview. Okay, sir. Part A, we gonna find the main functions off X and y. So the main function off X, it's just the expectant valley off Exit E, which really is just the expected value off 80 plus B f t are using the linearity expectations. This would be the expected value of a of T plus the expected value off be of tea. Okay, um, because these two are the same because there are identically distributed then. But some of these two things is just two times you off. That's the main function for X now. Similarly, you can compute the main function for why? Which if I jump to the second last step Oh, you get the expected value off a off T minus the expected value off B A T, which should give you sirrah. So the main function of X is just to beauty. And the main function of why is zero that is the main function of X. Why now, In the next part, part B, we're going to find the altar curve, various functions off X and what was a yacht basin? Boy. So the truth ever functions, I should probably point out we'll deal with the cross co parent suit. So the cross at the auto co variance function of X C sub x x of T and S by definition is the expected value off exit tee times X it s minus use up X as tee times muse up except s so the product off the main functions. Now, we already found from the previous question that new subjects of tea is to U of T. And so therefore, this product here will be four view of tea times. You've s come back to that soon. So we're going to use the definition off ex of tea to write the first term in different ways of God. Expected value of a of T plus beat off T times a off s plus B off s You got that? Minus for mew off T times New off s. All right, So we expand the brackets in the third line, So got expected value of a of T times A s plus a off t up. Be of s plus beat off t A off s plus B off T B s. You got a big drive. All that minus form you off t me off s okay, So let's sort of analyze this a little further. A and B were independent, so the expected value of these two terms ends up just being the products end up just being the product expected value of 80 plus the expected value. Or be of s okay, so I'll come back to that soon. So the first term here, um, read the expected value A of t times a mess, times a s, and I'm going to subtract this by, um, new off t times New off s. We're going to do that first. Furthermore, if I look at the last circle the first time I looked the last term, it will be the expected value off a bus or not Paper be off t times be off s and I'm going to subtract Issa's. Well, bye mu off t times. Mu off s every sort of created quick demarcation, so we know where we are. So I've got those. Thinks so I'll explain the madness. Um, off my methodology, sir. I got that. And now to deal with the last two terms. So we've gotten expect value of a of T, which is mu of tea. Expected value be of S is you've s and I have the same thing with third term. So I've got plus to that and what I what I need left over then is minus. So I've got minus one of these, minus another one to get to minus four. I just think it's attract too. Well, it was pretty obvious to see that these two terms will cancel out. So what's left over? What's left over is the first brackets bracket expression, which really is just the car variance function or the altar governance function. To be more precise of a, which is sea off T and s. And then the second bracket a term becomes D alter Co variants function off B. So all the left lift is two times the altar co variance function. See off T of s, That's the old ticker various function off X to get the altar convergence function of why we employed a similar approach. The only difference being in the second line Here you'll have minuses over there. And also the main function ends up being zero. Sorry. Fight. Skip to the second line, sir. I'm only left with the expected Valley off. Uh, what about what I have? So you're a of t minus B off t times a off s minus beat off s. So with that, what have I got? So you got the expected value off a tee times a s, so I'm going to do everything at once, sir, I'm gonna split the I'm going to use the linea aren t off expectations. So this minus the expected value off a of t times Be off s off. You in two lines. Look, that was with So you're minus. I'm gonna put a plus first the expected value off B of tee times be off s minus the expected value off B of tea times eight off s so it because A and B are independent. And moreover, they are identically distributed. So something you could do which is not incorrect is changed to be to a name. What changed me to a name, but imagine be as being identical distributed A. So what you have with these terms are just so you've got mu off tee times. Music s so because the independent you can split it up so you can expect if I have a tee time defect value be of s, and then you have that army of teacher would be a vest. This will also be mute off tee times view off s. Now we sort of go back to what we had before on this page. So we're basically in the situation where we have these four terms in the second last line. So the answer to the co various funk the altar convince functional why is just two times the auto co variant function seat off T. Let's so the to alter caverns functions of X and why Ah, the same. And finally, to get the cross co various functions, this will be the expected value off Expo ex off T minus so times Why of s minus the main function Observation your ex off tee times Mu. Why off s That's the cross co variance function. Now we know that the main function of why zero so this term doesn't exist. So all we're left with is the expected value off a off T plus B off T Time's a off s minus beat off s, um, so then using the linea arat e off expectations. You got a off t Time's a off s minus the expected value off a off T time speed of s minus. He expected value off. Be off tee times B. A s. So I'm writing in this order for a reason. Um, I'll put a plus here first, but I'll eventually rub it out, plus the expected Valley off be off tee times a off s. All right. Well, before the experience, what happens with this quality? This is just see off. T s likewise has experienced this quality as well before they seem what it is, but because there's a negative. Here this ends up being minus C off T anus. And if you Adam all up, what you get is zero. So what this tells us is that the well, the random processes X and Y are actually un correlated, and that should complete that question.


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