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01 Maxinun Likolihood Estination 0f [4pts] Assume we observe dataset of occurrence counts D = {81.12. Iw } coming from N ii.d random variables distributed according...

Question

01 Maxinun Likolihood Estination 0f [4pts] Assume we observe dataset of occurrence counts D = {81.12. Iw } coming from N ii.d random variables distributed according to Pois(X 2:A). Derive the maximum likelihood estimate of the rate parameter To help guide you; consider the following steps:Write out the log-likelihood function logP(DIA)Take the derivative of the log-likelihood with respect to the parameterSet the derivative equal to zero and solve for A call this maximizing value ALE: Your answer

01 Maxinun Likolihood Estination 0f [4pts] Assume we observe dataset of occurrence counts D = {81.12. Iw } coming from N ii.d random variables distributed according to Pois(X 2:A). Derive the maximum likelihood estimate of the rate parameter To help guide you; consider the following steps: Write out the log-likelihood function logP(DIA) Take the derivative of the log-likelihood with respect to the parameter Set the derivative equal to zero and solve for A call this maximizing value ALE: Your answer should make intuitive sense given our interpretation of as the average number of occurrences_



Answers

Refer to Exercise $3.86 .$ The maximum likelihood estimator for $p$ is $1 / Y$ (note that $Y$ is the geometric random variable, not a particular value of it). Derive $E(1 / Y) .[\text { Hint: If }|r|<1$, $\left.\sum_{i=1}^{\infty} r^{i} / i=-\ln (1-r) .\right]$.

To find the maximum likelihood estimator of P, we must find the P value that maximizes half of P. To do this, we can assign of a to M and a valued and just pick a positive integer. I m equaling three. I don't say any equals four. Now we can grab this function in our calculator and the graph will look something like this, or it has a peek at the top. So we're like a bell curve. And if we find the maximum of this graph on our calculator here and find the P value in this case with assigning three and four p, value equals 0.75 How can we relate Eman end to find that p value? Well, if you look at 0.75 and how it relates to our assigned millions of m and N, you'll notice that in overt and is 3/4, which equals 0.75 No, Why does this make sense if you think about it, If you estimate m number of of successes, so three, which is our M successes in and number of trials which is four trials. Well, then, the long term proportion of success would be I'm over end or, in this example, 0.75 So, in the end, the maximum likelihood estimator of P to maximize P in the function F P is and liver end.

Hello everyone. This is problem 9.82. So the park game, we need to find the sufficient statistic for data and they give us the pdf of uh we're just a pdf. So we're going to use the Factory Station theorem. So first step we need to get the likelihood, so the likelihood of data is equal to the product. Isaac will want to end of the pdf to one overstated, Multiplied by our so our it's just constant greater than zero and then Multiplied by Wiser Vie to the AR -1. Nor is the sky there because there's a lot or variables. A lot of ways, a lot of ways multiplying by E to the negative wives of I over a theta. Okay, now we need to simplify things so this is equal to so the are in the theater just constant. So we could say are divided by theater to to the end. And then we're going to do the product Isaac who want to end of Well, the wives of I right they're all different numbers of ways of life To the AR -1. And then as we've seen before we're going to do e mm to the negative one over data. So we take out the constant and then we're adding up this wise var. Okay. And yes so I think I forgot. Yes there's a little more in there. Okay so now we're good summation of why survive to the art. Okay now we need to look at two parts. So there's going to be a part that is just going to be made up of a ah statistic and parameters data. And there's going to be another one that is just made up of the um observation. So just the wise a lot. So let's write it down jean uh the summation of eyes. You couldn't want to end Uh voice of I two. The are so this is going to be the statistics and then the parameter is the data it's been now. So he's going to be good to one over theater to the end. E to the negative one over data summation of eyes. You could want to end the voice of I. To the R. Okay. And then we need to figure out which which was H. So end Each of Y. So one all the way to why? So then it's going to me it's made up of the wise of ice or to the end and multiplied by the product. Well I used to go to one to end before I survive To the AR -1. So as you can see this is just made up of them What survives and or two then it's just a constant self. And then well then elves ada so the likelihood is made up of two functions was made up of gov the summation of eyes. He could have want to end before I survived to the R. And data multiplied by H. For price of one. All the way to words when. So there's a theorem on the book and the theorem is 9.4. So theorem And this is just affect organization zero. There are 9.4. The submission of eyes you could want and okay some mission um of ways of lie to the R uh is a sufficient statistic poor data. Okay so this is part A. Now we're going to part B. Part B. We need to find the Emily of data. So we're already done some work um before but um let's just write it down. So everything is gonna be the same up to this point right here. So we could just start from there. Everything is the same there. Now we're gonna take the natural log of what's underline. If you do that you get the natural log both. Ells data is equal to end times the natural log of ar minus and times the natural log of data. Plus the natural log of the product with Isaac wouldn't want to end Before I survive to the AR -1 like this and then minus Data to the -1 summation advice you could want to end But why so I two r. Okay then up to this point we need to take the derivative now so then the partial derivative. The natural log of L. Data or the likelihood we respected data it's going to be negative and overstate. Uh huh. Plus data to the negative to summation of I is equal to 12 and before I survive To the ar which is equal to zero simplifying things. Oh yes and notice I said he'd go to zero and the theaters became theater has because on this step ah that's what you can do from here on we could write one over. Theater has squared let's play with the summation of Isaac. Who want to end of wine supply to the r. Is it going to end over there? A hat simplifying further. We can write the summation of eyes. You couldn't want to end of why I survived are all over and it is equal to data had and divided by an so there's a lot of algebra going on and then say it a hot is you go to the summation of eyes. You could want to end wiser lie to our over and this is part B. So yes this is the Emily uh Sarah and report see we want to find the envy horse theater. So we use um the sufficient statistic that we got before in part A. And we just call it you. So you is the sufficient statistic. We're just renaming it. Mhm. Okay. So now we have to do a transformation because the pdf that were given, it's not very common. So you want to rewrite it in a way that we can easily identify. So let's do that lit W equal towards survive overall. So we're just calling in W. And so because of this you could write down why is of i is equal to W. To the one overall. So I just isolated was why? Okay. So from here we're doing the transformation to rolling down. Well now it's just going to be in terms of W. So it's gonna be F. Of W. W. Is equal to f. Of uh W. To the one third. So it's gonna be are. And then with the squared of w. Well it's not the squared but it's just that the index are so we're going to multiply this by the derivative of. We will be the are through right cuba our our threat of W. With respect to W. Ah So this is a transformation and then simplifying. Well yeah one overseer multiply by our What's by by W. 2 1 over R. To the AR -1 E. To the negative W. Rosato. And then now we're going to do the derivatives. So multiply by one over R. W. One minus R. Uh huh. Or. Okay, simplifying. We'll get one over theater then E. To the negative W overstay to. And this only works if The W. is greater than zero. Okay. So they should look similar to another pdf and it is so why survive o to the R. Is distributed as an exponential random variable between data. Okay. Uh So in order to use the we're going to use the rail Blackwell down behind the M. B. So we need to see that data had is um by it. So let's first do the expected value of wiser. Vie to the arm what we know this isn't expected. Yeah it's the wise way to the Rs exponentially distributed. We're just going to get that its data right. Uh but let's just keep going. So now what about the expected value of the summation of ice? You could want to end of why survive door which is we're looking at well this is equal to end tires data so you can write down the data had Is he going to one over and Summation of Isaac and 1 to the end. Why did you lie to go or we'll this was just the Emily that we found before. So we know the data had is unbiased and a function of you. And how do we know it's somebody's well because if you take the expected value of this right here you just get data so you could just use ah this part right here but now you're just gonna do everybody and so you'll get the theater and then because of that so because it's an unbiased and a function of you we can say that. Uh huh. Is the and the U. A E four see you by row black wall the URL.

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

No. On the last problem we found that our of T. Mhm. Mhm. Which is equal to the natural log of the moment generating function is very useful for one. The expected value of a random variable is equal to the first derivative of are evaluated at zero and the variance is equal to the second derivative evaluated at zero. Now, here on this problem, we've been given a moment generating function for a person in variable is am to the five times each of the t minus one. That's what that tells us. That our ft is a natural on the the five times each of the t minus one. Okay, the natural beauty of power just cancels out. And so this is just E I'm sorry, this is just five times either the t minus one. That originally there will just cancel out. This is value of authority. How to find the expected value of the variance. Here we take our drip our prime. It's just five into the team. So the expected value of why, which is our prime of zero is five ft of the zero, which is five. And so are expected value is five. Okay, now are variants, is our double prime of zero. Now our prime is five years of the teachers who are double prime is also five ft of the T. So our double prime of zero is five ft of zero, which is still five. And so our variance is also five.


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