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Problem # 03 (LOpoints): Let Y,Y,,Y denote independent and identically distributed random variables from the following probability density function 2y02 1 fly| 0) 0...

Question

Problem # 03 (LOpoints): Let Y,Y,,Y denote independent and identically distributed random variables from the following probability density function 2y02 1 fly| 0) 0 <y< 0 otherwise (6 points) Find the method-of-moment estimator for 0. points) Find a sufficient statistic for 0.

Problem # 03 (LOpoints): Let Y,Y,,Y denote independent and identically distributed random variables from the following probability density function 2y02 1 fly| 0) 0 <y< 0 otherwise (6 points) Find the method-of-moment estimator for 0. points) Find a sufficient statistic for 0.



Answers

The joint density function of the random variables $X$ and $Y$ is $$ f(x, y)=\left\{\begin{array}{ll} 6 x, & 0<x<1,0<y<1-x, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Show that $X$ and $Y$ are not independent. (b) Find $P(X>0.3 \mid Y=0.5)$.

And here in this problem, we want to find that it required value so that the distribution is a discrete probability distribution. Now, the two factors that play into into making something to discrete probability distribution is first. Each of the probabilities must be greater than report zero, which in this problem is not, is not an issue. Because all the probabilities are greater than any closer. And the second is that the some of the probabilities must be wrong. One. And that is the point that we are going to make use of. Here is the sum of the probabilities. Must be one. Okay, so P of three plus p of four Hauspie of five. Possibility of serves must be for one mhm P three is 0.4. We don't know Pierre for we know the PF five is 50.1 and we know the PF searches point to So we're just going to solve this as no question. It was 0.7 SPF four is there for the one. And so it tells us that p f four is April 20.3 and that was our missing that mhm

Proper 14. We want to identify this probability dynasty function over the center. Then find the mean the variance and the standard division. Without integration we can rewrite F of X to be equal. Two divided Britain. Then simplify. It equals one divided by five. We can rewrite one divided by five to be one divided by five minus here. As we can write F of X As one divided by B minus A. Then it's only from distribution because B equals five and they equals you. Then you have X represents a uniform probability density function. Because it's a uniform distribution then you equals half, multiplied by a plus B equals half, multiplied by zero plus five Equals half, multiplied by five gifts 2.5. And for the variants it equals One divided by 12, multiplied by B -A. To the power of 12. Sorry. To the bar of two Equals one divided by 12. The boy boy B. It's just five. And this a which is zero sq Equals 25 divided by 12. And to get the standard division, we get the square root of the variants it equals, it's sigma Square root of 25, divided by 12 equals why, Divided by square root of 12? or in decimal mm Equals. 1.4.4 city needs to find an answer to our problem.

In this problem, it's required to find the value off New X and you Why? Let's start by New X. Your fax is integration off our boundaries from 0 to 1 for X and Y six exports three to play. Why the X do you want will be equal integration from 0 to 1 three over toe? Why do you, Roy, which is equal to 3/4 and for m y or mule boy? Thesis is mule Milton knew why so Integration from 0 to 1 seeks experts toe Why poor toe g x d y all these years? Thanks. So it will be integration from 0 to 1 two boy pose to do y which is equal to over three. So me, why is two or 3 a.m. u X is 3/4. Thank you.

So for this problem, the first thing that we're going to do is we need to know are actual equation that we're going to need. So we're doing a standard deviation and standard deviation can be denoted by this little funny guide and we're doing the standard deviation of X because this sub script says that we're doing it for X, so it is the square root of E. In the in parentheses you have X, and this is a capital X. That matters minus mu squared. And um so you need to know you first because for this problem we don't know what you is. And the way you find that is you do E to the X. M. X. And so what that looks like for this problem is U equals negative two times 0.3. Close three times two plus five times point for. And so what this is you're taking your exes and you're multiplying them by their function and then what you're doing is is at the end you are mhm, adding them together. And so, Let me fix this, this is .3 times 0.3. And these are all in parentheses, of course you're doing pimp dos. Um and so what your answer for this problem, and this is not the complete answer, 2.3. And so now, you know what mu is next, We're going to now put this all together. So we know that our formula for standard deviation is this. But so it's kind of write it again, the standard deviation of X is the square root of E. And the more times you write this, the more times you can kind of memorize this X minus ive X squared. And so what we're going to do is is what this turns into is is the square root E. And then in princes, you have x minus mu squared. And so then you can break this down even more. And if this looks funny, you can google on why this happens the proof for this. But you can put the summation of X times x minus mu squared times. Eh Fedex little eggs. Those are both little excess because that matters. A big X denotes a random variable. And so what that looks like, because I know that that looks like a handful. But when we put our numbers in there and remember you Is equal to 2.3. So we'll have negative too minus 2.3 squared times 0.3. Close and we can put curly brackets there, tended either. And then we're going to have more curly brackets and then we'll have The 3 -2.3 squared times 0.3 class, More curly brackets, five -2.3. That is not a 15, that's a a brace squared times 0.4 in curly brackets. And when you add these three numbers together this, you actually get 2.93, you go around To like 2.9 because you're only going to that figure in this problem. But if you want to be more precise, I would at least around like the 1000th place, especially if you're in a higher level class.


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