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Chapter 8: Q 9E (page 472)

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

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

Short Answer

Expert verified

\(\frac{{n{{\left( {{{\bar X}_n} - \mu } \right)}^2}}}{{{\sigma ^2}}}\) has \({\chi ^2}\) distribution with degrees of freedom 1.

Step by step solution

01

Given information

\({X_1},{X_2},...,{X_n}\)be a random sample from \(N\left( {\mu ,{\sigma ^2}} \right)\).

02

Determine the distribution

Since, \({X_1},{X_2},...,{X_n} \sim N\left( {\mu ,{\sigma ^2}} \right)\)

\({\bar X_n} \sim N\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)\)

Hence, \(\frac{{\left( {{{\bar X}_n} - \mu } \right)}}{{\frac{\sigma }{{\sqrt n }}}}\) has a standard normal distribution.

And \(\frac{{{{\left( {{{\bar X}_n} - \mu } \right)}^2}}}{{\frac{{{\sigma ^2}}}{n}}}\) has \({\chi ^2}\)distribution with degrees of freedom 1.

Therefore, \(\frac{{n{{\left( {{{\bar X}_n} - \mu } \right)}^2}}}{{{\sigma ^2}}}\) has \({\chi ^2}\)distribution with degrees of freedom 1.

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Most popular questions from this chapter

In the situation of Exercise 9, suppose that a prior distribution is used for胃with p.d.f.尉(胃)=0.1 exp(鈭�0.1胃)for胃 >0. (This is the exponential distribution with parameter 0.1.)

  1. Prove that the posterior p.d.f. of胃given the data observed in Exercise 9 is

\({\bf{\xi }}\left( {{\bf{\theta }}\left| {\bf{x}} \right.} \right){\bf{ = }}\left\{ \begin{align}{l}{\bf{4}}{\bf{.122exp}}\left( {{\bf{ - 0}}{\bf{.1\theta }}} \right)\;{\bf{if}}\;{\bf{4}}{\bf{.8 < \theta < 5}}{\bf{.2}}\\{\bf{0}}\;{\bf{otherwise}}\end{align} \right.\)

  1. Calculate the posterior probability \(\left| {{\bf{\theta - }}{{{\bf{\bar X}}}_{\bf{2}}}} \right|{\bf{ < 0}}{\bf{.1}}\), which \({{\bf{\bar X}}_{\bf{2}}}\)is the observed average of the datavalues.
  2. Calculate the posterior probability that胃is in the confidence interval found in part (a) of Exercise 9.
  3. Can you explain why the answer to part (b) is so close to the answer to part (e) of Exercise 9? Hint:Compare the posterior p.d.f. in part (a) to the function in Eq. (8.5.15).

Suppose that \({X_1},...,{X_n}\) form a random sample from the normal distribution with known mean 渭 and unknown precision \(\tau \left( {\tau > 0} \right)\). Suppose also that the prior distribution of \(\tau \) is the gamma distribution with parameters\({\alpha _0}\,\,\,{\rm{and}}\,\,\,\,{\beta _0}\left( {{\alpha _0} > 0\,\,\,{\rm{and}}\,\,\,{\beta _0} > 0} \right)\) . Show that the posterior distribution of \(\tau \) given that \({X_i} = {x_i}\) (i = 1, . . . , n) is the gamma distribution with parameters \({\alpha _0} + \frac{n}{2}\,\,\,\,{\rm{and}}\,\,\,\,\,{\beta _0} + \frac{1}{2}\sum\limits_{i = 1}^N {{{\left( {{x_i} - \mu } \right)}^2}} \).

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?

Suppose that\({X_1}...{X_n}\)form a random sample from the normal distribution with mean 0 and unknown standard deviation\(\sigma > 0\). Find the lower bound specified by the information inequality for the variance of any unbiased estimator of\(\log \sigma \).

Suppose that the five random variables \({{\bf{X}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{X}}_{\bf{5}}}\) are i.i.d. and that each has the standard normal distribution. Determine a constantcsuch that the random variable

\(\frac{{{\bf{c}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}} \right)}}{{{{\left( {{\bf{X}}_{\bf{3}}^{\bf{2}}{\bf{ + X}}_{\bf{4}}^{\bf{2}}{\bf{ + X}}_{\bf{5}}^{\bf{2}}} \right)}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}\)

will have atdistribution.
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