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(10 points) (10.3.4 from Probability and Statistical Inference) Suppose that Xi, Xn are iid Uniform(0,0) given that 0 where the parameter 0 is unknown. Suppose tha...

Question

(10 points) (10.3.4 from Probability and Statistical Inference) Suppose that Xi, Xn are iid Uniform(0,0) given that 0 where the parameter 0 is unknown. Suppose that has prior distribution Pareto(a , 8) , which has pdf _ given by h(0) Ba' 0-(5+1)1{a 0} where &, 8 are known positive numbers. Recall the sufficient statistic T' X(n) max{X,. X} which is the largest order statistic. (a) Show that the posterior distribution of 0 given x = {21,12, Tn turns out to be Pareto(a' max(t,

(10 points) (10.3.4 from Probability and Statistical Inference) Suppose that Xi, Xn are iid Uniform(0,0) given that 0 where the parameter 0 is unknown. Suppose that has prior distribution Pareto(a , 8) , which has pdf _ given by h(0) Ba' 0-(5+1)1{a 0} where &, 8 are known positive numbers. Recall the sufficient statistic T' X(n) max{X,. X} which is the largest order statistic. (a) Show that the posterior distribution of 0 given x = {21,12, Tn turns out to be Pareto(a' max(t,a), 8' = n + 8), where T = t = max{T1, Infa (b) What is the Bayes estimator for 0? (c) Suppose 10, Xn 1.5, a =4, 8 = 20. What is the posterior probability of Ho 0 < 22



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A PDF for a continuous random variable $X$ is given. Use the PDF to find (a) $P(X \geq 2),(b) E(X),$ and $(c)$ the CDF. $$f(x)=\left\{\begin{array}{ll}(8-x) / 32, & \text { if } 0 \leq x \leq 8 \\0, & \text { otherwise }\end{array}\right.$$

The function tea is a tea distributed random variable. And for part a we have tea with 5° of Freedom. And you want to use the table in the back of your textbook to find T sub 10 where the probability that T. Is greater than T sub 10 is going to be 100.10 So what that what is that asking you? Let's think about the shape of a T. Distribution. The shape of a T. Distribution is a bell shaped curve very similar to the standard normal curve. The average or the mean is zero. And it's found right at the center or the peak of that curve. So we're trying to do here is we're trying to find T sub 10, which means that we're gonna be over here and we want The area to the right to be a .10. So we can do this two different ways. We can use the table in the back of the book and we can use the applet that accompanies your book. So when you're using the table in the back of the book, what you're going to do is you're going to find the degrees of freedom in the right hand column And you'll find five and we're going to look underneath t Sub .10 because that's the area that we're talking about in that right tail. And when we find that it's going to be a 1.476. So what it's telling us is this value right here is 1.476 and there's 10% of the curve to its right. Now if I were to use the applet that accompanies the textbook you would see that very same curve. And at the top you'll see a location for you to enter the degrees of freedom. And at the bottom you'll see a box where you can enter an X. Value or you can enter a probability value. So we're going to enter the degrees of freedom of five And we're going to enter the probability value of .10. And we're going to let the applet Fill in that box. And when you let the applet fill in that box we're going to get that 1.476. And you'll see that the box from the applet and the value in your table do match in part B. Of this problem. You are asked to refer back to part a. And what Quanta will does T .10 correspond with T .10 corresponds with which Quanta will and to which percentile. So it's going to correspond to the .90 Quanta will Sometimes that is referred to the 90% Quantico. And in terms of percentile It's going to correspond to the 90th percentile. And why is that? Because when we're dealing with Quantum aisles or percentiles, we're talking about the area that is below. So in this particular problem the area that would be like be below the 1.476 would end up being 0.90 So therefore we are talking about the .90 Quanta will or the 90th%ile in part C. We're going to continue to use the applet And we're going to change our degrees of freedom from 30 to 60 to 120. So let's go to that applet and the first time we are going to fill in the degrees of freedom to be 30 the second time we're going to fill in the degrees of freedom to be 60. And the third time we're going to fill in the degrees of freedom to be 120. And in every one of these were trying to find the value for T Sub .10. So what we're doing here is we're putting .10 in for the probabilities and we're finding the corresponding X. Value or the boundary line. So when we put in uh huh 0.10 with the degrees of freedom of 30, you're going to get a 1.310. Keeping in mind that Zeros right at the center. So 1310. And you'll see the area shaded to the right When degrees of freedom were 60, You're going to see 1.296. So again zeros in the center and will be A little bit closer. 1296. And then when degrees of freedom were 120 you're gonna see 1.289. So again zeros in the center and it's gonna get even closer. 1.289. And we're gonna be shading to the right Yeah. Part D. In part D of this problem, It says when Z has a standard normal distribution, the probability of Z is greater than 1.2, Is equal to .10 and Z sub 10 Equals The 1.282. What property of the T. Distribution when you compare it to that? Standard normal distribution explains the fact that all the values obtained in the previous section, all these values right here were larger then the Z. Or the standard normal distribution. And the answer is going to be the fact that the standard normal curve is going to be more concentrated. The tales are smaller and the values in the bulk of the Bell are more concentrated. So because the T distribution has more variability you would expect their probabilities, sorry you would expect their X values to be greater in part E. Of this problem. What do you observe about the relative sizes of the values of T. Sub 10 for the t distributions of 30 60 and 1 20 degrees of freedom. What you should be seeing here is as the degrees of freedom was increasing the T distribution. Okay, was approaching the standard normal curve. Yeah. So what we're saying here is we kept the probability the same. The probability was always .10 and as the degrees of freedom we're going up yeah. Then the value T .10 was approaching the z distribution value Of 1.282. So let's take a look at it. So we started With a small degrees of freedom and we got a 1310. We increased the degrees of freedom. We got two a 1296. We increase the degrees of freedom. Again we got a 1.289. And as those degrees of freedom we're going up our area um Boundary line was getting closer to the 1.282. So if I were to look at it, the first boundary line was here Again, that was at 1.310. Then we had the next boundary line which was at 1296. Then we had the next boundary line Was at 1- 8 9. And then finally we got to that Z distribution. Yeah. And the Z distribution Boundary line for the 10% to the right was that 1.282. So as the degrees of freedom are going up, These values were approaching 1.282.

This problem totally have a binomial with an n value of 11th. We have we have 11 trials P is 0.75. Mhm. We want to find the probability that acts is greater than Wrinkle eight. Now setting up a binomial here, in general, the probability effects has 11 choose x Times .75 to the X Times .25 To the 11 -7. The probability of access greater than April eight is equal to the sum from Mexico's from 8 to 11 of this probability. So of 11 shoes at 6.75 to the X Times .25 To deal with one -X. Mhm. And then we're just going to evaluate this and so you part 89 10 and 11 11 and add them all together. And when we do As we have this .71 33 Okay, It's our probability .7133.


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