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Chapter 8: Q5E (page 527)

Suppose that a random variable X has the normal distribution with mean 0 and unknown variance σ2> 0. Find the Fisher information I(σ2) in X. Note that in this exercise, the variance σ2 is regarded as the parameter, whereas in Exercise 4, the standard deviation σ is regarded as the parameter.

Short Answer

Expert verified

The fisher information is 2n/ σ2

Step by step solution

01

Given the information

It is given that X is a random variable that follows Normal distribution with known mean 0 and unknown variance >0. Therefore X1,….., Xn are iid Normal (µ=0, σ>0) .

02

Define the pdf


03

Define fisher information


04

Calculating fisher information for normal distribution

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Most popular questions from this chapter

For the conditions of Exercise 5, how large a random sample must be taken in order that\({{\bf{E}}_{\bf{p}}}\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - p}}{{\bf{|}}^{\bf{2}}}} \right) \le {\bf{0}}{\bf{.01}}\) whenp=0.2?

Question:Suppose that a random variable X has a normal distribution for which the mean μ is unknown (−∞ <μ< ∞) and the variance σ2 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 μ. Show that

\(\int_{{\bf{ - }}\infty }^\infty {{\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}\,\,{\bf{and}}\,\,\int_{{\bf{ - }}\infty }^\infty {{\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}{\bf{.}}\).

Suppose thatXhas the\({\chi ^{\bf{2}}}\)distribution with 200 degrees of freedom. Explain why the central limit theorem can be used to determine the approximate value of Pr(160<X<240)and find this approximate value.

Suppose thatX1, . . . , Xnform a random sample from the normal distribution with meanμand variance \({\sigma ^2}\). Find the distribution of

\(\frac{{n{{\left( {{{\bar X}_n} - \mu } \right)}^2}}}{{{\sigma ^2}}}\).

Consider the conditions of Exercise 10 again. Suppose also that it is found in a random sample of size n = 10 \(\overline {{{\bf{x}}_{\bf{n}}}} {\bf{ = 1}}\,\,{\bf{and}}\,\,{{\bf{s}}_{\bf{n}}}^{\bf{2}}{\bf{ = 8}}\) . Find the shortest possible interval so that the posterior probability \({\bf{\mu }}\) lies in the interval is 0.95.
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