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Consider 3 continuous random variable X with the following probability density function: fbx) _ Jax+4b x? , F1sXs1, a and b are unknown parameters_Find point estima...

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Consider 3 continuous random variable X with the following probability density function: fbx) _ Jax+4b x? , F1sXs1, a and b are unknown parameters_Find point estimators of a and b based on a sample f size n.a = 1ZX;and 6 = 5 2n Ex} 8na =1ZXand 6 = 12* a= 3 2n Zx;and 6 = 52 2na # 1 Zx,and & = 526 = 12 andb= 4n 2 HnMoving #o the next questio prevents cHUNEFs to this answer

Consider 3 continuous random variable X with the following probability density function: fbx) _ Jax+4b x? , F1sXs1, a and b are unknown parameters_ Find point estimators of a and b based on a sample f size n. a = 1ZX;and 6 = 5 2n Ex} 8n a = 1ZXand 6 = 12* a= 3 2n Zx;and 6 = 52 2n a # 1 Zx,and & = 52 6 = 12 andb= 4n 2 Hn Moving #o the next questio prevents cHUNEFs to this answer



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Let $X$ be a continuous random variable with values between $A=1$ and $B=\infty,$ and with the density function $f(x)=4 x^{-5}.$
(a) Verify that $f(x)$ is a probability density function for $x \geq 1.$
(b) Find the corresponding cumulative distribution function $F(x).$
(c) Use $F(x)$ to compute $\operatorname{Pr}(1 \leq X \leq 2)$ and $\operatorname{Pr}(2 \leq X).$

In this question, very Kanda. If F X is bdf doesn't implies, then the integral on the f x the x from minus infinity to infinity every equal to one in discussion here were given the function f X go to two times X plus one bomb on those three for the X Quit equal to zero Now in the part, I will need to verify if the function f X here is a b, d f or not to verify. Is that be? Therefore not the FX here must be created than so far on X. And it's true here because always greater than zero here the second one with in the global two x plus one by minus three d x from studying from zero up to infinity getting coated the anti near. If there was any coach in the two on, then X plus one barrel managed to anybody in by managed to from zero to infinity. We see we can cancel that you hear the border infinity inside we got exactly equal to zero minus will be plus yeah, then you Buddhism. Inside we get equal to the, uh one power minus two so equal to the oneness. Well, from here this means. And then the F X is bdf. Now for the puppy we want Thio, find a humility distribution function F Captain the X by formula with any culture the integral starting from zero here after the x f t DT, they're going to cut you into problems that your ex thio x plus this one will be here. Yeah, the plus one by minus three The t here we get this one and then we get equal to the minus one of X plus one hour to from the Jew. Ah, x is me with a t here and then we get equal to it would deserve first. And then we get equal to the one and then minus one on the X plus one hour to for the X here in the interval from zero to infinity. And from here we can find a C Now, once you find a probability than the X in interval from 1 to 2. So it's a nickel to the F capital on the Tu minus F capital. No one ever, ever with two egos, you know, one minus 1/2 plus one hour to minus everyone equal to one minus one hour. One plus one about you. Do we get any go Jew? You don't have the one minus 1/9. So you go to the 8/9 minus doesn't get any coach in one man is, uh, one of the far, uh, it go to the three of a far and then we get Geico. Thio doesn't be 36 now, So him the ah, 30 to minus 27. So you go to the 5/36. That's gonna be the answer. Now, the next part. Want to find a probability that the exhale will be smaller, equal to three. Here we can incentives and smaller than infinity as well. And listen, technical to the F infinity minus f on the three have infinity in the bull. And so I'm going to cut, you know, one because yeah. And then, uh, minus 73. Equal to one minus 1/3. Plus one with you. So one counselor. Window one, some 100. Plus here. One other four square in coaches. 16. So that's gonna be the answer

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.

For the given exercise we have the probability density function we know is associated with a continuous random variable X. That has the form F of X. He calls A X must be a rats. Mhm. And this is where um one is less than or equal to X. That's awesome. You got to eat uh E. F. Of X is equal to two. We want to find the values of A. And B. So we know that uh E f X is going to be the expected value. So knowing that fx relax equals chill that are expected value. The values of a. and B is going to be 18 -6. E over negative E cubed plus three squared plus three E minus five. And B is going to equal 1 -18 -6. E over two times negative east cubed plus three E squared plus three E minus five, All Times Negative East or Eastward -1.

Everyone radio going to solve a problem. Number seven. Yeah. Therefore flex equals for express to minus five like less theological texts. Does the articles infinity expectation off Executes one to infinity Eggs in tow four. Access to minus fight the X. What do you think was for interval 1 2050 Existing minus four. These so which comes to be for limit. Be tensed in 51 Toby. Access to minus leap for the existing minus four. The X which comes to the fore limit be tends to infinity access to minus truly ordered by minus three. Want to be so which comes to be for limit Be tensed Infinity minus one day three B cube minus minus one by three into one cube which is sickles for by three. This is the a part of the question then the purpose variance off X equals. Want to infinity X squared daughter forward access to minus fight the x minus for by trader who square, which is it was full and Duke 1 2050. Access to minus three D x minus 16 by nine which will be equal to for Believe me, deal tends to infinity. Want Toby excess to minus three. D x Linus 16 by nine The difficult for limit Be intends to infinity access to minus two by minus two one toe deep minus 16 by nine. We just got the four Indo zero plus half Linus 16 by nine to buy nine. Thank you.


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