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Let $X(t)=A t+B,$ where $A$ and $B$ are independent random variables with $A \sim \operatorname{Unif}[0,6]$ and $B \sim$ Unifl $-10,10 ] .$(a) Describe the ensemble of $X(t)$ .(b) Determine the mean function of $X(t)$ .(c) Determine the autocovariance function of $X(t)$ .(d) Determine the autocorrelation function of $X(t) .$(e) Determine the variance function of $X(t)$ .

Intro Stats / AP Statistics

Chapter 7

Random Processes

Section 2

Properties of the Ensemble: Mean and Autocorrelation Functions

Probability Topics

Missouri State University

University of North Carolina at Chapel Hill

Idaho State University

Lectures

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09:06

Define $X(t)=A t+B,$ where…

14:13

Let $A(t)$ and $B(t)$ be i…

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Define a random process $X…

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A wide-sense stationary pr…

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Find (a) the mean of the d…

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01:52

04:58

Consider two random proces…

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Let $X(t)=A \cdot Y(t),$ w…

04:43

A subsample of a random se…

Okay, so this question will be divided up into five parts. So what we have is a random process ex of tea, uh, which will end up being a linear function specifically 80 plus b Uh, the coefficients A and B our uniform random variables themselves. So specifically, a is uniform in the interval, 0 to 6 and B is uniform Between modest 10 to 10 are fair minded birth A M B. Ah, continuous uniformed distributions. Just bear that in mind as we sort of progress off through his question. Okay, sir, we start with finding the ensemble off Exit E, which pretty much just beans describe it, so to speak. So not just describing a bit, sort of figuring out what possible, I guess. Sample functions you can obtain, sir. It's pretty obvious that exit T is linear. Moreover, the linear coefficients, selenia and constant coefficient coefficients to be more precise. Ah, uniformly distributed. It's really Maur at this point. Just describe what dysfunction in tails. So I'll probably leave it there because the next full pots become much more interesting. Her case and now to get the main function off the random verbal sir, to get change back the collar. So, um, you sub x off T by definition is just expected value off exit T, which ends up being the expected value off a T plus Be okay. Now T is a deterministic quantity. So if we use the Lini aridity of expectations, this ends up B tee times the expected value of a plus the expected Valley off be, um So if I go back to the first page, So if you look at these two things ah, the expected value off the random variables ends up just being the average valley off these two things. So if we go back to the main function, But this ends up being tee times three zero plus six, divided by two is three plus 10. Mona's standing fight about two ends up being zero and you get three tea. So that's the main function of X city, which we will use quite suit. Okay, so the next part we want to find the altar curve variance function of X t. I'm gonna go back to this. What I'm gonna do is I'm going to look at the altar correlation function because once we get the altar correlation function. Then we can get the water. Barbarians function just by subtracting the products off the main functions at T N s. More on that that suit. Okay, so the water co correlation function, which is denoted by our sub execs off s This by definition, is just expect. Valley off ex of tea times X off s. So if we substitute all the irrelevant quantities, uh, there are there, so should be a t plus be times a s plus, Hey, then what we can do is expand the brackets that the expected Valley off a squared T s plus a B T plus a B s. So what I can do is I can do that. So just factoring out tea out of a B, it just makes it easier later on to deal with the, um, deal with the product. And the other thing we need is b squared. A careful got to point something out at the start, which will become crucial here, and that is that I want to get the black color. Um A and B are independent. Okay, this is crucial for we going to do, um, in probably two steps time. So it's probably prudent for me to just get that it. All right, So what are we left with? So using live in the area of expectations so that the expected value off a squared times T s plus the expected value off a B Times T plus s plus the expected value off B squared. Okay. All right. So I'm gonna go back to the start again. So let me just recall a couple of things. I do it in here. So if X is uniformly distributed, say, between A and B, then the variants of X is really good. Just up beam. Honest, eh? All that squared, divided by 12. So I think this is in Sections three, chapter three off the book. In any case, it's always best to Google the continuous uniform distribution or searching on Wikipedia, which has sort of a list off what the variables are. All right, so once I am here. Okay. So I've got this. Um, I won't bother doing any extra supplication. So which is gonna use that quickly? So the expected value of a squared. So, by definition, the variance of a it's the expected value of a squared minus the expected value all squared. So this just ends up being the variance off, eh? Which is this? Minus the figures. I ve all squared. So just need to add on the expected value off, eh? Oh, squid. Okay, um then what we can do? He's suggesting now, because I am be our independence we can rewrite. This is the expected value of a times three expected value off B times T plus s and to finish off. I just need to write this in to start to expect about it be squared in the same way as I read the expected value of a squared, which is variants off B plus the expected value. Sorry Off, sir. I think it would be, um so I'm just trying. All right, The rigor. So expected value off be all square. All right, so let's cancel a few things out. So first of all expected value be was zero. So this term will not exist sick, and this term here will not exist. Okay, So what's next? What's next? Is that do compute eso variants off, eh? So remember a was uniform between Syria and six. So the parents of a is six minus zero or squared. Divided by 12 6 square. This 36 divided by 12 is three. On the various of B will be 10 minus minus. Austin, which is really just 10 plus 10. All that squared, divided by 12 eso 20 squared is 400 divided by 12. And if you really want to simplify this, it ends up being 100 over three. Okay, you've got those two qualities. So what is the altar correlation function? So the auto correlation function from what we've left off with. So this is going to be three, sir. Three squared is nine. Uh, this waas three and last of all this it is 100 over three. Hold it like that. All right, sir, Plugging all of daddy three plus nine is 12 served 12 times t times s plus the 100 over three terms on. That's pretty much it with the auto correlation function. So once we get this, um, getting the so getting the Walter Kerr variance function in Part three is very simple, sir. Us up where we are this time we see my bad c sub x x off t s. This is equal to the expected value off ex of T X s. By definition, I think this was also something you had to show in problem 12. And now we need to subtract. Um, use up ex off tee times, muse up, ex off s. Okay, Well, this is the first term here is purely derived from pot de. They're getting the altar correlation function. Um, so the altar correlation function waas if we call 12 times t times s plus 100 over three. Now, Musa Becks of tea is three t times that with three s users nine s t 19 s. And when you do this attraction, you end up with three t s plus ah, 100 over three. And that is our altar curve arians function. So we got one part left to figure out which is part E, which is D variance function, which we denote by Sigma squared sub X off T. And this is simply just the variance function evaluated at t and T. So from this formula here, we just substitute as equal to t and we end up with three t squared plus 103. And that pretty much concludes the question. The problem. Full team. Thank you

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