5

Suppose that Xt, Xuare iid N(U, 02) where U is unknown; but 02 is known: Here we have ~C <p < 0,0 < 0 < o and x = I. We would like to find the MLE for p...

Question

Suppose that Xt, Xuare iid N(U, 02) where U is unknown; but 02 is known: Here we have ~C <p < 0,0 < 0 < o and x = I. We would like to find the MLE for p Suppose that Xi, Xare iid N(A, 02) where both / and o2 are unknown. Here we have ~oo < A < o, 0 < 0 < o and X = I. We would like to find the MLE for p Suppose that Xt, Xaare iid Poisson(A) where 0 < 1 < 0 is the unknown parameter. Here X = {0, 1,2} and R+. We wish to find the MLE for 0_

Suppose that Xt, Xuare iid N(U, 02) where U is unknown; but 02 is known: Here we have ~C <p < 0,0 < 0 < o and x = I. We would like to find the MLE for p Suppose that Xi, Xare iid N(A, 02) where both / and o2 are unknown. Here we have ~oo < A < o, 0 < 0 < o and X = I. We would like to find the MLE for p Suppose that Xt, Xaare iid Poisson(A) where 0 < 1 < 0 is the unknown parameter. Here X = {0, 1,2, } and R+. We wish to find the MLE for 0_



Answers

Suppose $X_{1}, X_{2}, \ldots, X_{n}$ are iid with pdf $f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0<x<\infty$, zero elsewhere. Find the mle of $P(X \leq 2)$ and show that it is consistent.

Hi here our problem is to find that we have been given X one extra so on eggs in these are the random samples which follows personally distribution with probability of success be okay. And it is also given that piece restricted says that how less than equals to be less than it costs to what? No we have to find the Emily of B. Okay now first we first when we write the poverty mass function then we we will take the usual parametric space of B which is zero less than be less than one. Okay now the property mass function of this random variable is P. X equals two probability capital X equals two small X. Because to B to the power eggs 1 -2 to the power one minus X says that X equals to zero comma one. It takes only two values and zero less than be less than one. Okay now to find family of me we first construct the likelihood function which is united by L. B. Ex school equals two productive irons from one to end B. Exactly this is we do the parcel mentioning side 1 -4. to the power In- Submission Exile Islands. From one point in all cases. Okay, now maximizing this like look function is equivalent to maximizing the log likelihood function. Now we write log leg blue function as scribble the Kahlan excursion which is equals two submission exile lot B plus in minus the mission next day. No one minus and the family of the way first. And the first started derivative of this script log likelihood function And equated to zero. Okay. No first order derivative with respect to be script. L. P. X. Girl Equals to zero. That implies submission excited liberated by P. Plus in my estimation next day Only baited by one -P. And again -1 coach to syria so we can ride some issue. Makes minus someone should excite Into B equals two N. P minus pe submission. Except this is really empty. Okay now clearly Submission X. i. equals two MP. So we had Emily calls to submission excited by and the many of me Culture to submission. Excite islands from one doing for liberated by. And this is the value of P. When he is not restricted. See the maximum likelihood estimated of pes expert. Okay now Excite next of value zero and 1. Okay so expert definitely will lie between zero less than equals to expire less than equals two facebook users. Now we do you hide these interval in two intervals. Okay zero lesbian Eggs. two eggs were less than half and also half less than equals two. Expert less them. Just one. Now when if he is restricted half less than nickels to be less than equals to one. Then definitely we had Emily Takes two values. What are those two values when zero is done equals two X. Bar less than equals to have and another is sorry it will be only less than And another is half less than equals two. Expert. Less than industry. One when half less than the constraints for less than one. So definitely be Emily of people be expert. Mm. And when the expo lies between zero to have the definitely takes the value only. Have. Yeah. So what we get half when X body is less than half and expire when eggs, berries catered, then calls to This is in college too. Maximum of how comma. Thanks. So when we is restricted, Emily of, we had Emily of me wants to be had maxie mama. How coma explosion. It's one of the brands. Thank you.

Hi here it is given that X one X two. So on XM follows a distribution in pdf f X F X data equals two. Twice six. Only one at birth. He does square. When zero is 10 X less than and zero. Other ways also integrated them. First we have to show that you have to find the Emily of theater which is today. Then we have to find C says that expectation of Sita had equals two. Yeah. 3rd we have to find the median of the distribution, median of the distribution, median of the distribution. Then we have to find that Emily of the median of the distribution and show that that Emily is a consistent estimator of that media. Okay now first problem we find that Amelia to to to find Emily of theater. We construct the likelihood function. L theater ex cult that is productive items from one to n F exiled. I wanted to that is the people in Product of ex Ii runs from one I think that whatever twice the conditions that zero less than excellent. Less than excellent. Less than theater. Zero less than extra. Less than theater and so on. zero less than equal to accept less than you don't sorry zito less than Nixon at least in question. We have to find the value of Peter says that the likelihood function is maximum. So as Kita is in the denominator so the minimum value of theater will make the likelihood function maximum. So we have to find the minimum value of Hello you get to Amelia. Yeah. Okay now from the flight condition we have Canada Get up then equals two x 1 from here. Similarly saying integration we have targeted tentacles two X 2. So on data better when it comes to accept so three days get at them all the real editions of the random center. So clearly theater is greater than equals to a maximum of all the real additions. Maximum mixing. Okay this is the maximum water statistic of the random. So definitely he died. Emily east Equals two Maxi Mama X. I. Where one less than equals to one less than equals twin. We write it as X. Order in there's two Y. N. So see the together then it goes to maximum of exiles to minimum fellow of theater is this maximum of eggs? A maximum order statistic. So first I have written that we have to find the minimum value. So um Emily of theater is maximum order statistic of X. I question why am I going forward to the second problem? Well to find the value of caesar's that expectation of safety diet Peter. Okay now peter had a college to where I am to find the expectation of to find the expectation of speed ahead. We need to know the distribution of this maximum water statistic expectation of wagon. Okay now C. X one X two X. And all follows with the F. F. Extra time. Now the oratory statistic Why are equals two x. Excellent art. The pdf of this with a G. X. R. Equals two. In factorial. Holy weighted by ar minus one. Factorial In two in- Artifact Real. Yeah if X. R. To the power AR -1 into one minus F. X. R. Exhorted art Already four in- are into F. X. Order. Hi where zero less than X. order are Less than equals two theatre. Now if we put the articles to end we'll get the pdf provided density function of the maximum water statistic. Thank you. So max when putting articles to win we have G. Y. N. You watch too twice end into my into the board. Do I send minus one? Only waited to get by grazing. Okay where zero less than Y. In less than equals to theater. This is the pdf of the maximum water statistic. Okay now you have to find the expectation value expected velo this wine so let Y capital Y. And it was to set this implies expectation of said so this is integration zero to theater zed into she said dessert. So this is integration zero to theater zen into bryce in The to the birth twice and -1 only reply the deputy poison disease. So this is Twice in only worried by twice and plus one indu Tinder. This is the expected value Brian. Now expectation of c. 3 to head equal strategic So that implies scene to expectation Peterhead that is twice and holy wanted by twice and plus one. The diagnostic theater. So value of C equals two twice end plus one who delivered it by twice. Okay how Went forward to the 3rd problem we have to find the media. Another distribution letters do not aim which is the median of the distribution. Now integration 02 a.m. F X. Dx request to have because the probability payments have indication 02 M twice sex or limited by theatre square D. X. Equals to have. So that in place mm square by Pieter square records to have this implies Amy college to hold this is the median of the distribution. The distribution has been given the period that from that period the distribution of the system million. Okay now with naked eyes. GT now from the previous one we have got that family of Canada. It is theoretical to Dwyane No parametric functions in the theatre equals two were limited by you too. This parametric function is a real and 1-1 function. So buy invariance property by invariance property of Emily we gain Emily of jupiter. Yes. Gt died. That means jeep Y in calls to Y unholy so Emily off to Hollywood to is Y. In Hollywood wine is the maximum statistic brian equals is the maximum order statistic of exile. One lesson equals two. I. List N equals to end. Okay now we have to find we have to show that this Emily y in Hollywood too is a consistent estimator of theatre Hollywood. So to show show that if it is a consistent estimated of this. So why in Hollywood where you will converges in probability to theatre Hollywood. If it converges to theatre Hollywood where you do then we say that the estimated Y and holy very Baruto is a consistent extinction of delivery. True. Okay. Now if it is a consistent estimator then limit Entrance to in three D probability, remarked why? And who liberated by Rupert minus theta only? Would it look more close? Less than excellent. He calls to what? So we started doing the finding the probability probability mud quien elaborate value to minus theater limited to place them absolute excellence is very small quantity which tends to zero that in place probability more dwyane minus theta. Not close less than roped you into. Excellent. See if theater change to zero to input data will also change to zero. It is a very small quantity so denoted by excellent staff. So sometimes to zero. Okay, so probability more dwyane minus theater. Less than excellent. Start we have to find this probability as entities to infinity. So probability three to minus. Excellent. Start less than y. And less than it applies. Excellent stuff. See why it is a maximums or a statistic and the value it takes from zero less than y and less than he calls to theater. Okay, so we'll partition it is probably too probably probability three to minus absolutely less than Y and less than theater plus probability To get up then equals two. A little less than equals two. I. N. Little lesbian equals to brian less than peter plus excellent stuff. So in the maximum value Y. And tax theater. So from the probability property of capitalist university in less than a close to less than 30 plus. Excellent start this probability will be zero because the fellow maximum fellow of wire needs thicker. So only he's lived with the ability 57. Uh huh. Less than Y in place them finger. Okay so we get integration peter minus epsilon start to Ryan is we ride G. Y. N. D. Way in. Okay so absolutely dominance absolute start to to to this is twice and why in the bar twice in -1 will give birth to what I said E why in So we get And they were twice in -30 to -12. Pretty poor twice and holding up a theater to the twice equals to one minus theater minus excellent star liberated by theater hold bracing Equals to one as in french too infinite. Okay because 3 to- Excellent star twice in tends to zero past intends to infinity since so clearly this problem tends to one has entered into infinity. So definitely Y N. Who liberated by Ruto is converges in probably two divided by two. Okay so this estimator is a consistent estimated of this. So the Emily of the median of the distribution. This is a consistent estimator of the immediate proof. Thank you.

Hi. Yeah it is given that in random samples. So our next to accept each follows poison distribution is permitted peter and tita is restricted says that zero less than theater less than the cost. And then for that we have to find the Emilia theater. Okay now first we find the Emily of theater. In usual cases where theater has the usual parametric space where the usual parametric spaces belongs to 02 infinity. First we find the Emily of peter here then we'll go for the restricted case. Okay. No It's a random distribution. So the property mass function is p. x. equals two. It was they were minus theater. She did with the barracks hole. We were buying eggs factory And eggs 01 go. It will take accountably in finite values and he dies and get it then you okay now To find the family of the. We first construct the likelihood function which is given by L. 3 to eggs. Girl he calls to Productive islands from one twin B exciting column tender. The government must function and that is the R minus in theater. The deputy commissioner next day Cultivated by product of exile. Actually guidance from one doing okay are maximizing this is equivalent to maximizing the log likelihood function. Mhm. So log likelihood function we right questions cream tell linda. It's correct. That is minus in theater plus submission. Excite love it down minus lager productive except. Okay now to get the Emily of theater we first find the first order derivative of this log likelihood function with respect to theater and equity to zero. Okay now first order activity of of the script. L theater X skull equals to zero. That implies minus and plus submission. Excite fully verified theater calls to zero. That implies three calls to submission X. A delivery created islands from one to so from here we get to had Emily equals two summation ex II islands from one old vertical and he calls to X. This is the many of theater when theater is not restricted at zero less than theater less than infinity. Okay now here with the question it has been given that it is restricted. So according to the restricted peter if zero less than theater less than he calls to we have to find the Emilia Theatre. Okay so C. X. I can take the values from 012 and so on. Okay. Similarly expert the Hello eggs. Were these slides between 02 infinity. Right? That means zero leads the next part less than infinity. Okay now if he dies restricted. So the restricted Emily will be three to had Emily restricted. I do not restricted equals two. See when the value of X. Bar Is zero less than eggs. zero less than expert. Less than two. Less than equal to do. Sorry, another is two less than expert less than infinity. Okay, when zero less than X. Were less than it was to to see the parametric values also same then theater had Emily restricted will be same as the previous expert but when expert takes the value from two to infinity in that case peter is not the hello of Peter is not there. Right? So Emily of theater will be only to the nearest value of. Okay, so this implies eggs, but when experienced less than a question too And two when expert. Good, good thing. The this is meaning mama it was. But come on. Oh, so we can write the restricted. Emily of theater is equals two many mama eggs, but this is the answer. Thanks.

9.8. First We know that. Why one why two? All the way to. Oh yeah. Are distributed as a person random variable with me in lambda. And they're also I. D. Which means there independent and identically this rude for part A. We want to find Emily for longer. So for the first step we have this is a notation, its capital L. Of lander. So it's just going to be because we're trying to find the maximum likelihood or doing the likelihood right now and then inside the plant series is going to be the parameter which in this case is the land though and then it's going to be equal to the product. Um So it means that you're just multiplying a lot of things And you're going to go from I. is equal to one to end. And then after that we write down the probably distribution function. But this is going to be different. So first you'll do lambda and it's usually to the white. But now it's going to be to the wise of I why is it why? Because there's a lot of them and then multiplied by E. To a native lambda. And then we're going to divide it by wise of I. And Factorial. So that's the first step. Now we need to simplify things. So this is going to equal to the product from i. Is equal one to end. And then we could distribute the product on the factorial. So it's going to be one over wise of I factory. And then you're going to multiply. Whoa, Let's first right? Um with his work with the lambda. So you're gonna do lambda, right? And then you're adding up a lot of why? So, lambda to the summation. Well, the wise arrived From I is equal to 1 to end. And then we're going to distribute the product again and now to the alternative landau. But then since it's just a constant right into in Orlando, so it's just going to be e to the negative and lambda. All right. So this is the first step then the second step. We need to take the natural lovable science. So the natural log uh l of land, I would like to get her one is equal to the natural log. So over the first part of the natural log of the product of uh equal 1 to end of juan over wise I factorial like this. And then we're going to this room so that it's going to be plus the summation of I secret of one to end of ways of I. And then the natural log Atlanta. And then minus. And what? Okay. So what I have to say though is um you must be saying wait. So what happened right here? Right. So since we have we're using one of logarithms rules. So it will be natural love of land to the summation of Isaac didn't want to the and wiser I but since this right here is an exponent. Yes. Right here is an explanation then you could just bring it down to be here and then keep the natural level Philando. Okay, so there's that part afterwards we need to take the derivative of the natural love of the likelihood of lambda with respect to lander. Yeah. So gonna take the partial derivative or the derivative of natural law of the likelihood of lying there with respect to Miranda. So where you see a lambda uh that's the only part that's gonna be available, the rest is a constant. So for example this part it's just a constant. So the derivative of a constant respect to a variable is just we're just adorable. Constant is just zero. So we're going to focus on this part where yes the land us part and the part. So um we'll get summation Isaac All to 1 to the end of. Why is it Why? So this part is a constant. Uh But then the drift of natural log land respectful and you just bring the land down. Um The thing is that since we're trying to find the maximum likelihood instead of running land up we need to use a hat now. So we're going to write lambda hat. It's the same lander though. And then minus mm When I done though, after this we need a salary equals zero. And from there we need to solve for language. If you do that, you'll get an is equal to the summation of as you could want to end or why survive divided by land. Hat. And then solving for longer hat Yeah. Land uh huh is equal to the summation of uh Cipro one to end. The voice of I over and which is white boy. So this was part A. Okay, okay. For part B. When you find the expected value of land a hat, so expected value of land a hat is equal to the expected value. Why bar? Which is equal to So now instead of running white Bar, I'm going to use this right here, which is the same thing. So we're going to get this information, the eyes, you couldn't want to end. Oh the expected value of y survive all over. And so the expected value goes inside of the summation. And since the expected value of each, why is lambda each time? So and you're adding them and so you're adding lambda. End times you could just write down and land and we're going to do right by and so we end up getting just land now for the variants of lambda hat same thing, the variance of white bar which is equal to the variant ah The Summation of Isaac 01 to end who I survive over. And so since this and is a constant right here, we could just take it out. But when you have a uh the various of constant with them, let's say another variable. You get squared. So you just get one over and square and the rest summation. Both i is equal from one to end of the variance of wise. If I So the variance right, he wouldn't started the summation sign. So then we know that the variance of y survive. It's lambda. So the same thing we're adding and lambdas. So yeah, and times lambda over and square simplifying. We get lambda. All right. Now for part C. We want to make sure that mm. Why Bar is a consistent estimator of lambda? So there's different steps for this. Well let's just review a little bit more work done. So part C. From part B. Yeah, but the expected value of white bar is equal to land right seen this before. And we also know that the variance of why bar is equal to land the overhand. We notice, okay now cars, why bar is unbiased Orlando? So then you're saying, oh what is unbiased? Right, Some bias means that um the estimator that you have. So that that the expected value of the estimated that you that you got uh matches the mean of the random variable which in this case it does right here. That's unbiased. So we have them. Yeah, not just that. And we know that the variance. A white bar yeah, goes 20 S and approaches infinity. So we know this is true. So if you take the limit as n approaches to infinity of this part right here, you'll see that you'll get linda over infinity. So that will That would get close to zero. Okay, because of this. So because white bar is advice for lambda and the variance of white bar goes to zero as it approaches to infinity, it follows that white bond is a consistent estimator of laughter. No moving on to party. We want to find the Emily four. The probability the Y. is secret of zero which is equal to the native lambdas. There's a poor son running variable. Where Why is it with 0? Well, we could just use what we've got before. We are given linda has his secret away bro. So this is after all of the work that we've done. We know this so damn. Really? Uh probability Why secret is zero which is equal due to the negative lambda is just Mhm. So this is what happened. So when you're trying to find the Emily of a function that includes, for example, in this case we found the Emily for lambda, right? And we found that he was that he was just white bar, which we called Atlanta. Hat. So if so you know, function you're trying to find a middle of a function of this parameter letter lambda. Then the Emily is just going to be, for example, in this case is E to the negative of the estimated that we found which is a white bar.


Similar Solved Questions

5 answers
Considering the following reactions, you conclude that the stereochemistry of reactions #L and #2 is:Br H Hl Zn C-C_Br H;C CH; (Rxn #2)(Rxn #1)A) anti and syn, repectively syn and anti, repectively E anti and anti, repectively syn and syn, repectively none of the aboveBrz
Considering the following reactions, you conclude that the stereochemistry of reactions #L and #2 is: Br H Hl Zn C-C_Br H;C CH; (Rxn #2) (Rxn #1) A) anti and syn, repectively syn and anti, repectively E anti and anti, repectively syn and syn, repectively none of the above Brz...
5 answers
MF C1H12OzMW 164SC 73.1RH 74Hi111 J 1Lee] AteanttCopyright1994FPH2.51.5 Pr4_.0Proton NMA200 18016014012010020 PFMCarbon 43 NMA
MF C1H12Oz MW 164 SC 73.1 RH 74 Hi 1 1 1 J 1 Lee] Ateantt Copyright 1994 FPH 2.5 1.5 Pr4_.0 Proton NMA 200 180 160 140 120 100 20 PFM Carbon 43 NMA...
5 answers
Find the minimum sample size required to estimate the population proportion: Jse the given data to confidence level: 95%; from a prior study; p is estimated by the Margin of error: 0.07; decimal equivalent of 93%.
find the minimum sample size required to estimate the population proportion: Jse the given data to confidence level: 95%; from a prior study; p is estimated by the Margin of error: 0.07; decimal equivalent of 93%....
5 answers
HbI lo Ierul 04e n bnuldotuuzachbcaent cunonntr appaal Waumlyce 0nun' Wat ulnttnndy uuh Ceonter et NoAnndFAaJuy UudIHctnd Io Wunn (ncutalelacaTectFrunlaehotIclcidi WnFurdriedamoJn-Deorhu pucot nenDtjaEamnttmnmenennchueeenmmntFina e(49) . P(4) -[unand 10 mtoe doama plcot e5 naedod And PIALAT
HbI lo Ierul 04e n bnuldotuuzachbcaent cunonntr appaal Waumlyce 0nun' Wat ulnttnndy uuh Ceonter et No Annd FAa Juy Uud IHctnd Io Wunn (ncutalelaca Tect Frunlaehot Iclcidi Wn Furdrieda moJn- Deorhu pucot nenDtja Eamntt mnmenennchueeenmmnt Fina e(49) . P(4) -[unand 10 mtoe doama plcot e5 naedod A...
5 answers
Find the indicated term of the arithmetic sequence with first term, $a_{1},$ and common difference, $d$.Find $a_{70}$ when $a_{1}=-32, d=4$
Find the indicated term of the arithmetic sequence with first term, $a_{1},$ and common difference, $d$. Find $a_{70}$ when $a_{1}=-32, d=4$...
5 answers
MYBUFFSTATE 8 myBinghamton BASSIGN 7.S.UUorCheg[arget Pay and Be_ TYPultS]Let f(x) =Part 1 - f""(r)fis one-to-one , lind formula for (""(x): Assuming0 Pant 2 Functlon compositlonCheck your answer t0 pan using function composilion:( -1 f)(x)(f 0 f-1)(x)pana RangeFind the range 0f ( (-9,9) (-1,1)(-o.0) (-2,9) U (9,*) ~IU(-1,#)Additional HaterielWVatch ^OKosponsu
MYBUFFSTATE 8 myBinghamton BASSIGN 7.S.UUor Cheg [arget Pay and Be_ TYPultS] Let f(x) = Part 1 - f""(r) fis one-to-one , lind formula for (""(x): Assuming 0 Pant 2 Functlon compositlon Check your answer t0 pan using function composilion: ( -1 f)(x) (f 0 f-1)(x) pana Range Find...
5 answers
Calculate the volume that 0.50 kg of copper occupies. Express the answer in L. Use unit conversion.
Calculate the volume that 0.50 kg of copper occupies. Express the answer in L. Use unit conversion....
1 answers
In Exercises $23-28,$ find the angle between the vectors. Use a calculator if necessary. $$\langle 1,1,-1\rangle, \quad(1,-2,-1)$$
In Exercises $23-28,$ find the angle between the vectors. Use a calculator if necessary. $$\langle 1,1,-1\rangle, \quad(1,-2,-1)$$...
5 answers
The lifetimes of group of 10 light bulbs are given below: 221,645,538,941,269,893,703,536,823 651. The standard deviation of the lifetimes is (to two decimal places)Select one: 622.0058155.11386884.00241.15
The lifetimes of group of 10 light bulbs are given below: 221,645,538,941,269,893,703,536,823 651. The standard deviation of the lifetimes is (to two decimal places) Select one: 622.00 58155.11 386884.00 241.15...
5 answers
(sal solution to77 2 x (t) = )*6) -1 -3and drop your files or click toS)
(s al solution to 77 2 x (t) = )*6) -1 -3 and drop your files or click to S)...
5 answers
TatataUeing Iha AyumoWcen Trestcm CnJ (Qunrhdek coolj'mottrt*9 neht trang € / 9 inchea chorterthan thg EnanuloIk Vonaet than tna lonatr 1ca Nra {na Ude Irnoths o( tLeneth elthc #henter ICo:Length 0f tha lcraet idlDant FenienlitArlletel20888
t atata Ueing Iha AyumoWcen Trestcm CnJ (Qunrhdek coolj' mottrt*9 neht trang € / 9 inchea chorterthan thg Enanulo Ik Vonaet than tna lonatr 1ca Nra {na Ude Irnoths o( t Leneth elthc #henter ICo: Length 0f tha lcraet idl Dant Fenienlit Arlletel 20 888...
5 answers
Find the exact value_ if any, of the following composite function. Do not use calculator:10x sinsinSelect the correct choice below and, if necessary, fill in the answer box within your choice.0A 10x sin sin (Simplify your answer: Type an exact answer; using I as needed: Use integers or fractions for any numbers in the expression:) 0 B. It is not defined:
Find the exact value_ if any, of the following composite function. Do not use calculator: 10x sin sin Select the correct choice below and, if necessary, fill in the answer box within your choice. 0A 10x sin sin (Simplify your answer: Type an exact answer; using I as needed: Use integers or fractions...
5 answers
The number of electrons that can fit into the 5322d subshell is [sub]
The number of electrons that can fit into the 5322d subshell is [sub]...
5 answers
6s8 +4s2 6s 10 Find the inverse Laplace transform of F(s) 292(82 + 2s + 3)
6s8 +4s2 6s 10 Find the inverse Laplace transform of F(s) 292(82 + 2s + 3)...

-- 0.019474--