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Consider the following "Xm)-(X (~)e)-( Nz Let discrete random variable with the probability function ply). Then the expected value of Y, E(Y), is defined to be...

Question

Consider the following "Xm)-(X (~)e)-( Nz Let discrete random variable with the probability function ply). Then the expected value of Y, E(Y), is defined to beE(Y) = Eyply):Derive directly the mean of hypergeometric random variableE(Y)(--)G3-) 2

Consider the following "Xm)-(X (~)e)-( Nz Let discrete random variable with the probability function ply). Then the expected value of Y, E(Y), is defined to be E(Y) = Eyply): Derive directly the mean of hypergeometric random variable E(Y) (--) G3-) 2



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 $E(1 / Y) .[\text { Hint: If }|r|<1$, $\left.\sum_{i=1}^{\infty} r^{i} / i=-\ln (1-r) .\right]$.

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.

Hello. This is problem 9.83. We're going to deport a so we have a uniformly distributed uh random variable um and it's made up of wise. So firstly we need to find the likelihood, we're trying to find the Emily of data. So the likelihood of data is equal to the product of eyes. You could want to end of one over tooth data plus one. And this is the important part right here. Those brackets. So it's gonna be from zero 12 0 is less than ricotta wise, java, Which is less than or equal to two times data plus one. So this is just given us our domain at least of just one why? But now it's all of the wise. Okay so we're trying to maximize this. So um so we'll do one right? Because just 12 then it's one and then the concepts too, Times Data Plus one. We're going to exponentially ate it so too then um but now uh we need to see well when does this work? So we could just write down, okay I put a one here. And so this only works when zero is less than or equal to wise of them. She lives in our equal to two Times Data Plus one. Um So then you must be thinking what's going on here? So you want uh the numerator to be the biggest and then the biggest that he could be is if you get the maximum of the wise which is this was then um But we're not done yet so it's not wiser than the biggest. We need to do a little algebra. Uh Because what we're trying to maximize uh data. So from here we could just write down the same thing one over two times data plus 1 to the end. And then what is this called indicator function? So if you were to um subtract one and divide by two on both sides, you get negative. Uh huh. Is less than or equal to worry. So then mine is one give other way too which is less than or equal to data. So this right here would be the biggest event. So now we could just write down the Emily. So Emily of data is theater. Uh huh. Which is equal to wise event. Maya is one all over to. Okay now we need to find the Emily of the variance. So her p we know that the variants. Why Is he grew one or 12 times two data plus one square by definition and the Emily uh variants. So why is so one of the toe stays the same? Is there for me too? Times whatever data is. So instead of the theater we write down the theater. Huh Too multiplied by wise event one is one divided by two plus one. We're going to square this simplifying. We'll get wiser men squared or 12


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