 Hi everyone, it's MJ and we're looking at question 8 from the September 2017 paper now This question is dealing with the properties of Estimators and we're gonna see that part one is probably the hardest part and then the rest is fairly straightforward Okay, so the question starts with saying the following the two random variables x1 and x2 are independent from each other and follow a Uniform distribution where our parameter is greater than zero and We have our estimator and we're saying that it is equal to 3z and we're saying that Z is equal to a maximum of x1 and x2 Now we need to show that the probability density function of z is given by the following and This is I mean the examiners are recognizing that this is a difficult question Not only are they telling us what the answer should be so we'll know if we get it right or if we get it wrong So yeah, not only they're giving us the answer, but they're also giving us a clue They say by first deriving its cumulative distribution function So with those two clues, we should be able to give quite a good attempt at it Yeah, it would have been a much harder question. They just said show the probability density function of z But now we know what to aim for and we have our clue which is to do the cumulative distribution first Okay, so let's do that. So cumulative distribution denoted by the big fz and This is just equal to the probability that the z is less than little z Which as we've been told is equal to the probability max x1 and x2 is less than z Which is going to equal the probability that x1 is less than z and x2 is less than z and we know this because our x1 and our x2 are Independent and identically distributed you do get a mark for saying that so don't leave that out in the exam Okay, we also know that this is coming from the uniform distribution and the continuous one So we want to keep that in mind when we now come here where we have the following We have Cumulative distribution z here is going to be equal to the probability x1 less than z squared and What that's going to do is if we look in the book of formulas we can see what the cumulative distribution is of z of the uniform distribution and actually let's just get it out. Let me just Write it out. Let me find it in the little book. Yeah, we go. So I'm looking at page 13 of the formulas where it will say that f of x equals x minus a over b minus a but now we're in a situation where B is equal to our parameter and a is equal to the negative of that parameter and Instead of using x. We're using z. So and we're squaring all of it So we're going to get a situation where we have z minus the negative parameters So we're going to be plusing that parameter Subtracting that just that we're going to get to and then we have it squared Okay, cool, but we're not out of the woods yet We're not out of the woods yet We know that the answer or the question is asking us to find the probability density function And now we have just done the cumulative distribution function So how do we get to the next step? Well, we know that we need to take the derivative we know that probability distribution function is equal to the derivative of our cumulative distribution Which is going to be equal To the following and this is where You do need to be quite sharp with your maths. You need to know how to handle the sort of derivative and you'll get the following as your answer and What is nice? We can now check that with what the question told us and we can see oh look at that We did get the same answer. So we know we're going to get the four marks Remember you had to have said that in order to get to this step over here So, yeah, that is four marks. That is the hard part of the question I mean what we're going to see now the rest is fairly straightforward We want to show that the expected value of z is equal to a third of the parameter So what we do here? I mean expected value of z the general formula is You know z Z DZ so this is going to be equal negative theta Positive theta We have our Z then we have I guess what's nice the fact that they gave it to us over here Is that even if you bombed out in this question? You can now use this for the second part and you're not just gonna lose all your marks But you know you do need to This was a very mathematical question because now what we're doing is we did We did the other derivative in the previous question now we're having to find the integral of this Which also some people find a little bit of a challenge, especially if they are Rusty on their maths, but you should get to the following plus z squared over for their theta negative theta and then you plug those values in and Ta-da Gonna play plugging those into the Z values and you are going to get your theta over 3 Once again, they are being nice in the sense that they're telling us what the answer is because I guess this is going to be Leading on to the next question where we have to use that result So now they're asking us to find the bias Yeah, that's what the next question saying it says Derived the bias of this estimator Okay, and what we know the we know the formula of the bias because we have been studying and You know, that's the expected value of whatever the function is minus a little parameter so if we look here, we have the following we have expected value of 3z minus theta, which is same saying 3 expected sorry z minus theta Which is equal to 3 times That we're getting that value from over there Minus theta and we're getting a bias of 0. Okay, which is sometimes a common answer, which is quite nice So yeah, we've done that now says derive an expression for the mean squared error Okay, in terms of the unknown parameter Okay, and once again, this is going to be a bit of a tricky question What we do know we do know is because the bias is equal to 0 We can start writing this out. Yeah, we know that the mean square error is equal to you know the variance of That plus the bias squared So function, but we know that this is zero so we just need to find the variance of ZX of well variance of our little parameter guy who in this case is the variance of 3z Which is equal to 9 Variants z reason why we do that constant. We take it out and we can square it Okay, but this is where the fun starts. We know variance is equal to Equal here. I must try to check following it on. It's the expected value of the z squared minus Expected squared z Okay, and this is What we need to do what we do need to do is Find out what this term is. Yeah, we know what this one is because we just is that It's gonna be squared over nine But what we do need to do is Find this one. Let's go expect the value of Z squared that is going to be equal To our integral Z squared times our probability distribution function There d z you do the math and you get the following z power four eight theta squared plus z cubed over six theta by our two bounds and That is going to give us theta squared over three okay Which means we can now Return to this and we see we have the following nine We have pizza squared over three minus Peter squared divided by nine. Where am I getting that from? Where am I getting that from? I'm getting that from this value here squared okay because that is easy and we're looking at easy it squared okay, so you have this here and Ta-da our mean square error. It's going to be equal to two theta squared Which is actually quite quite large that is. Yeah, that is quite large Anyway that is Those two questions. They're done what we now move on to is gonna do basically the same thing again They've just told us that we now have another estimator where it is equal to two Z Remember Z is still going to stay the maximum of X1 and X2 Different estimator. Oh, yeah, once again there it says okay show that the bias now is equal To that we have a little bit of a negative bias so once again we have Bias of our second estimator is Going to be the expected value of this estimator minus that parameter And that's going to be equal to the expected value of two Z minus theta which is equal to Expected value of Z minus theta and we know that two Z is Equal to two theta over three minus theta and we have Our answer of negative theta over three Which is what they'd given us yay, you know why that was two marks that maybe should be worth one mark And then what we know now is that we have to use a more complicated formula for the MSc because our bias isn't zero So here our MSc is going to be equal to the second estimator It's going to be equal to the variance of the second estimator plus the bias squared and This is going to give us a situation of The variance of two Z plus our bias which is Sigma squared over nine Okay, which is equal to four times the variance of Z That's sigma squared divided by nine and We have figured out what the variance is From the earlier questions So what we can simply now do is use that to Oh, yeah, we have the expected value of the Z chair That is our variance So we just use that over here this time now Where we will have or maybe write it in Or sigma squared over three minus sigma squared over nine plus sigma squared over nine and What we're going to see is that this all comes down to an answer of Sigma squared Okay, cool and There we are done Final question now asks us to comment on how good the two estimators are based on your answers in part 3 and 4 so because asking us to look at just 3 and 4 Don't make the mistake of trying to work out consistency or one of the other properties of the estimator use what you have and What you basically need to talk about in the sense that if we had to compare estimator one and Estimated to we'll see estimated one is unbiased Where this one is biased? But this one has a large MSc and this has a smaller MSc unbiased is desirable smaller MSc is desirable so each have got a desirable trait well each of them also has an undesirable trait and I think you have to state that you're going to get your two marks for this question and There we are done It was a bit of a tricky question in the sense that they were combining a lot of the probability and cumulative distribution function mathematics Along with the properties of the estimator But if you've got any questions, please feel free to ask and you guys need to brush up on your maths in order to Do these questions keep well. Cheers