Chapter 8: Q8E (page 527)
Suppose that X1,….., Xn form a random sample from the normal distribution with unknown mean µ and known variance σ2> 0 . Show thatX̄nis an efficient estimator of µ.
X̄n is the most efficient estimator of µ.
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Consider again the conditions of Exercise 12. Suppose also that in a random sample of size n = 8, it is found that \(\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{x}}_{\bf{i}}}{\bf{ = 16}}} \,\,{\bf{and}}\,\,\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{x}}_{\bf{i}}}^{\bf{2}}{\bf{ = 48}}} \) . Find the shortest possible interval such that the posterior probability that \({\bf{\mu }}\) lies inthe interval is 0.99.
Suppose that a point(X, Y )is to be chosen at random in thexy-plane, whereXandYare independent random variables, and each has the standard normal distribution. If a circle is drawn in thexy-plane with its center at the origin, what is the radius of the smallest circle that can be chosen for there to be a probability of 0.99 that the point(X, Y )will lie inside the circle?
By using the table of the t distribution given in the back of this book, determine the value of the integral
\(\int\limits_{ - \infty }^{2.5} {\frac{{dx}}{{{{\left( {12 + {x^2}} \right)}^2}}}} \)
Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the normal distribution with meanμand variance\({{\bf{\sigma }}^{\bf{2}}}\), and let\({{\bf{\hat \sigma }}^{\bf{2}}}\)denote the sample variance. Determine the smallest values ofnfor which the following relations are satisfied:
Question:Suppose that a specific population of individuals is composed of k different strata (k ≥ 2), and that for i = 1,...,k, the proportion of individuals in the total population who belong to stratum i is pi, where pi > 0 and\(\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{\bf{ = 1}}} \). We are interested in estimating the mean value μ of a particular characteristic among the total population. Among the individuals in stratum i, this characteristic has mean\({{\bf{\mu }}_{\bf{i}}}\)and variance\({\bf{\sigma }}_{\bf{i}}^{\bf{2}}\), where the value of\({{\bf{\mu }}_{\bf{i}}}\)is unknown and the value of\({\bf{\sigma }}_{\bf{i}}^{\bf{2}}\)is known. Suppose that a stratified sample is taken from the population as follows: From each stratum i, a random sample of ni individuals is taken, and the characteristic is measured for each individual. The samples from the k strata are taken independently of each other. Let\({{\bf{\bar X}}_{\bf{i}}}\)denote the average of the\({{\bf{n}}_{\bf{i}}}\)measurements in the sample from stratum i.
a. Show that\({\bf{\mu = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{{\bf{\mu }}_{\bf{i}}}} \), and show also that\({\bf{\hat \mu = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{p}}_{\bf{i}}}{{{\bf{\bar X}}}_{\bf{i}}}} \)is an unbiased estimator of μ.
b. Let\({\bf{n = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{n}}_{\bf{i}}}} \)denote the total number of observations in the k samples. For a fixed value of n, find the values for which the variance \({\bf{\hat \mu }}\)will be a minimum.
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