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Let X, Xz, Xjz be 12 independent random variables, each having the same geometric PMF e6)={(2) '(4) x=l,2,3, otherwise mc 'he Cxact ana an approximate val...

Question

Let X, Xz, Xjz be 12 independent random variables, each having the same geometric PMF e6)={(2) '(4) x=l,2,3, otherwise mc 'he Cxact ana an approximate value of Pl 20sEx,s30 CamScanner

Let X, Xz, Xjz be 12 independent random variables, each having the same geometric PMF e6)={(2) '(4) x=l,2,3, otherwise mc 'he Cxact ana an approximate value of Pl 20sEx,s30 CamScanner



Answers

Let the pmf $p(x)$ be positive on and only on the nonnegative integers. Given that $p(x)=(4 / x) p(x-1), x=1,2,3, \ldots$, find the formula for $p(x)$. Hint: Note that $p(1)=4 p(0), p(2)=\left(4^{2} / 2 !\right) p(0)$, and so on. That is, find each $p(x)$ in terms of $p(0)$ and then determine $p(0)$ from $$ 1=p(0)+p(1)+p(2)+\cdots $$

Yeah. Hello this problem. 9.97 were given the what is attributed as a geometry uh distribution with parameter P. Okay, so we're trying to find the method of moments estimator. So in order to do that we just set the population moment equal to the sample. Well, so the population moment, it's just one of the P. Because this is the mean of geometric random variable in the sample moment. It's just white bar. And so from here on we can solve for P. We're just going to put a little hat so he had is equal to one or a white loss. Yeah. So this is the message uh moments estimator. Sure. This is part of a for poor B. We need to find the Emily for P. So the likelihood of P is equal to the product. Well as you couldn't want to and of p times 11 sp to the Y. Survive minus one Which is equal to p. to the end 1 -2. To the summation. I secretly want to end the voice of I minus. And you know, we're going to take the natural law. So the natural log of L. P. Is equal to end times the natural log of P. Plus the summation of Isaac didn't want to end a wiser I minus And Times of Natural Law one -P. Now we're going to take the partial derivative natural law of L. P. With respect to pee. So it's equal to end overpay hat minus assassination of eyes. You gotta want to end the voice of my minus and Over 1 1 is P. had which is equal to your wall. We said at a four year 0 Is not equal to zero. Okay and then from here we're trying to solve for P. I. So and overpay had is equal to the summation of Isaac would want to end a voice of I minus and over one minus P hat simplifying well, yeah and minus P hat multiplied by N. Which is able to pee hat what supplied by the summation of Isaac will want to end. Why survive minus P hat multiplied by N. Soapy had is equal to end over the summation of Isaac. Little want to end always. Why? And this is just he could have worn over. Why bar. So this is the M. Always of P.

Problem. Then we have the probability mass function for the random variable X, which is defined as one half to the bar of the absolute value of X when x equals -1 -2 -3. and so we want to find the probability mass function for the random variable Y, which is defined as X to the power of four. For the stated values of X. You can find that why? And X is want one. Each value of X here corresponds to a unique value, avoid. But we notice that to get X as a function of boy, we should get the negative branch of why? It was about one divided by four. Of course this negative sign and not make any difference because we have here an absolute but we should know that we take the negative branch because we have here the negative values of X As long as X&Y is want one. Then we can find B. Why sequence Why? By substituting by the value of X Year By this exhibition. Then it's 1/2 to the power of the absolute of minus Y. Should be small. White To the power of one divided by four. And the values of Y. We substitute here by each value of X. We substitute, then swamp and two. It was about four 16, then It's 81 and so on. These are the only possible values that we can substitute in our function and we can rewrite it as half. It is a bar of why one quarter? Because the absolute function doesn't make any difference with this. Find the sign. And we knew that Why here is always supposed.

The problem is we have the probability mass function for the random variable X, which equals half to the war of X. When X equals 123 the infinity and zero elsewhere. We want to find the probability mass function for the random variable Y where y equals execute for the stated values of X. We can find that Why and exes want one and we can get X as a function of boy. Then we can rewrite this to be X equals cubic root voice. And to find the probability mass function of boy. We just replace X where they give a good boy. Then the probability of boy to be equal wide equals uh huh. To the power of go back through. Why? When Y equals the substitute here by X equals 123 When Y. Equals one it 27 and so on. And of course it's zero else. Where And this is the final answer of our problem.

Yeah. This problem will be given to probability densities. The first is G. Of X. It's 24 over X. To the fourth or X is greater than two. Yeah. And the second is H. Of Y. For each of wise to why with the Y being between zero and one now we are told that these are independent. And so this means that our joint distribution F of X. Y. Is the product of these two, says it's 24 over at the fourth times two. Why? Which is 48 X to the negative fourth. Why that is our joint distribution. And notice that acts is greater than two. And why is between zero and one. Know what we would like to find is the expected value of Z. Which is the expected value of X. Y. Mhm. No. This is going to be the double integral. That's why times or probability distribution sometimes 48. Excellent and so forth. Why? Yeah. No extras from two to infinity. And why it goes from 0 to 1. Yeah. And so this is april 2. 48 times the integral from 0 to 1 of why times Y. Is y squared do Y? And then times they entered all from two to infinity of X times X to the fourth which is excellent, negative third D. X. And so we just broke that integral into those two different parts of. Once we have this this just turns into evaluating each of these different intervals. So the first thing we have is that 48 out front. And so the 48 will just say his name. And then we have one third y cubed evaluated from y zero to one. Then we have negative one half X to the negative second evaluated from access to infinity. So this is 48 times for one third white cube evaluated from 0 to 1. We just plug in one, plug in zero and then subtract, giving us one third negative one half X the negative second evaluated from two to infinity. We're going to take the limit as we go to infinity and then plug into and subtract those two values. This is gonna give us 18 So now we have 48 times, one third times and eighth, which is to.


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