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Suppose that a statistic U has a probability density function that is positive over the interval a ≤ u ≤ b and suppose that the density depends on a parameter θ that can range over the interval α1 ≤ θ ≤ α2. Suppose also that g(u) is continuous for u in the interval [a, b]. If E[g(U) | θ] = 0 for all θ in the interval [α1, α2] implies that g(u) is identically zero, then the family of density functions { fU (u | θ), α1 ≤ θ ≤ α2} is said to be complete. (All statistics that we employed in Section 9.5 have complete families of density functions.) Suppose that U is a sufficient statistic for θ, and g1(U) and g2(U) are both unbiased estimators of θ. Show that, if the family of density functions for U is complete, g1(U) must equal g2(U), and thus there is a unique function of U that is an unbiased estimator of θ. Coupled with the Rao-Blackwell theorem, the property of completeness of fU (u | θ), along with the sufficiency of U, assures us that there is a unique minimum-variance unbiased estimator (UMVUE) of θ.
According to Fortune, solar power now accounts for only 1% of total energy produced.15 If this number was obtained based on a random sample of 8,000 electricity users, give a 95% con?dence interval for the proportion of users of solar energy.
The following table presents the total sample size and the percentage distribution of a grouped variable, "Hours of Shopping this Year." Calculate the mean and the variance
Carry out an appropriate text at a 5% level of significance to test george and Jerry's theory. Interpret the p-value you calculated in part A. Construct and interpret a 92% condidence interval for P.
Define f (r) to be the smallest integer n such that for all regular Markov chains with r states, the nth power of the transition matrix has all entries positive. It has been shown,14 that f (r) =r2 - 2r + 2.
Using the .05 level of significance: a) Is there a difference among types of gasoline? b) Is there a difference in the cars?
A random sample of 7 students selected from a large class revealed the following scores: midterm 70, 62, 57, 68, 89, 63, 82 and on the final 80, 79, 87, 88, 85, 79, 91. Is there evidence of an improvement on the final?
at a local university when scheduling classes the probability that there is no evening classes available on a certain
is it possible to run both test for the same experiment? if so how or what situation would warrant such
The annual real return Y on the Diversified International Fund has mean 11% and standard deviation 17%, and the correlation between X and Y is 0.6. The return on Linda's portfolio is R = 0.7X + 0.3Y. What are the mean and standard deviation of R?
Find the probability that the selected student is the oldest or an only and find the probability that the selected student is enrolled in the College of Arts & Science.
Human populations can be modeled using an exponential growth function. Use the Library for research or other resources to find the following:
A biologist has found that the average weight of 12 randomly selected adult mud turtles is 8.7 lb. with standard deviation 3.6 lbs. Construct a 99% confidence interval for the population mean weight of adult mud turtles.
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