Chapter 8: Q6 E (page 469)
For the conditions of Exercise 5, use the central limit theorem in Sec. 6.3 to find approximately the size of a random sample that must be taken so that \(P\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}} - {\bf{p|}}} \right) \ge 0.95\) whenp=0.2.
The needed sample size is \(n \ge 62\)
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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.
Question:Consider an infinite sequence of Bernoulli trials for which the parameter p is unknown (0 <p< 1), and suppose that sampling is continued until exactly k successes have been obtained, where k is a fixed integer (k โฅ 2). Let N denote the total number of trials that are needed to obtain the k successes. Show that the estimator (k โ 1)/(N โ 1) is an unbiased estimator of p.
Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the Poisson distribution with unknown mean ฮธ, and let
\({\bf{Y = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \).
a. Determine the value of a constant c such that the estimator\({{\bf{e}}^{{\bf{ - cY}}}}\)is an unbiased estimator of\({{\bf{e}}^{{\bf{ - \theta }}}}\).
b. Use the information inequality to obtain a lower bound for the variance of the unbiased estimator found in part (a).
Question:Suppose that a random variable X has a normal distribution for which the mean ฮผ is unknown (โโ <ฮผ< โ) and the variance ฯ2 is known. Let\({\bf{f}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)denote the p.d.f. of X, and let\({\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)and\({\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)denote the first and second partial derivatives with respect to ฮผ. Show that
\(\int_{{\bf{ - }}\infty }^\infty {{\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}\,\,{\bf{and}}\,\,\int_{{\bf{ - }}\infty }^\infty {{\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}{\bf{.}}\).
LetX1, . . . , Xnbe a random sample from the exponential distribution with parameterฮธ. Find the c.d.f. for the sampling distribution of the M.L.E. ofฮธ. (The M.L.E. itself was found in Exercise 7 in Sec. 7.5.)
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