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Chapter 8: Q16SE (page 529)
Suppose that a random variable X has the exponential distribution with mean胃, which is unknown(胃 >0). Find the Fisher information滨(胃)inX.
The fisher information for the random variable X is 1/胃2.
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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.
Question:For the conditions of Exercise 2, find an unbiased estimator of \({\left( {{\bf{E}}\left( {\bf{X}} \right)} \right)^{\bf{2}}}\). Hint: \({\left( {{\bf{E}}\left( {\bf{X}} \right)} \right)^{\bf{2}}}{\bf{ = E}}\left( {{{\bf{X}}^{\bf{2}}}} \right){\bf{ - Var}}\left( {\bf{X}} \right)\)
Suppose that \({X_1},...,{X_n}\) form a random sample from the Bernoulli distribution with parameter p. Let \({\bar X_n}\) be the sample average. Use the variance stabilizing transformation found in Exercise 5 of Section 6.5 to construct an approximate coefficient 纬 confidence interval for p
Question: Prove the limit formula Eq. (8.4.6).
Suppose that a random variableXhas the normal distributionwith mean渭and precision\(\tau \). Show that the random variable\({\bf{Y = aX + b}}\;\left( {{\bf{a}} \ne {\bf{0}}} \right)\)has the normal distribution with mean补渭+band precision\(\frac{\tau }{{{{\bf{a}}^{\bf{2}}}}}\).
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