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Chapter 8: Q 3E (page 484)

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.

Short Answer

Expert verified

The constant is\(\sqrt {\frac{3}{2}} \)such that the given form has a tdistribution.

Step by step solution

01

Given information

There are five random variables \({X_1}, \ldots ,{X_5}\). Each independent variable is identically distributed from a standard normal distribution.

02

Determine the joint distributions

Let us consider the joint distribution of \({X_1} + {X_2}\).

So, if\({X_i} \sim N\left( {0,1} \right)\;,\;\left( {i = 1,2,3,4,5} \right)\)then,

\(\begin{align}{X_1} + {X_2} \sim N\left( {0,2} \right)\;for\;\left( {i = 1,2} \right)\\ \Rightarrow \left( {\frac{{{X_1} + {X_2}}}{{\sqrt 2 }}} \right) \sim N\left( {0,1} \right)\end{align}\)

And \(X_i^2 \sim \chi _{\left( 1 \right)}^2\)

Similarly, \(X_3^2 + X_4^2 + X_5^2 \sim \chi _{\left( 3 \right)}^2\)

03

Determine the distribution

The t-distribution is,

\(\frac{{\frac{{{X_1} + {X_2}}}{{\sqrt 2 }}}}{{\sqrt {\frac{{\left( {X_3^2 + X_4^2 + X_5^2} \right)}}{3}} }} = \sqrt {\frac{3}{2}} \left( {\frac{{\left( {{X_1} + {X_2}} \right)}}{{{{\left( {X_3^2 + X_4^2 + X_5^2} \right)}^{\frac{1}{2}}}}}} \right) \sim {t_{\left( 3 \right)}}\)

Therefore, the value of c is \(\sqrt {\frac{3}{2}} \).

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Most popular questions from this chapter

Suppose that \({X_1},...,{X_n}\) form a random sample from the normal distribution with unknown mean 渭 (鈭掆垶 < 渭 < 鈭�) and known precision \(\tau \) . Suppose also that the prior distribution of 渭 is the normal distribution with mean \({\mu _0}\) and precision \({\lambda _0}\) . Show that the posterior distribution of 渭, given that \({X_i} = {x_i}\) (i = 1, . . . , n) is the normal distribution with mean

\(\frac{{{\lambda _0}{\mu _0} + n\tau {{\bar x}_n}}}{{{\lambda _0} + n\tau }}\)with precision \({\lambda _0} + n\tau \)

For the conditions of Exercise 2, how large a random sample must be taken in order that \({\bf{E}}\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - \theta |}}} \right) \le {\bf{0}}{\bf{.1}}\) for every possible value of胃?

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.

Suppose that a random sampleX1, . . . , Xnis to be taken from the uniform distribution on the interval (0, 胃) and that胃is unknown. How large must a random sample be taken in order\({\bf{P}}\left( {{\bf{|max}}\left\{ {{{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}} \right\}{\bf{ - \theta |}} \le {\bf{0}}{\bf{.10}}} \right) \ge {\bf{0}}{\bf{.95}}\) for all possible胃?

Suppose that\(X\) is a random variable for which the p.d.f. or the p.f. is\(f\left( {x|\theta } \right)\) where the value of the parameter \(\theta \) is unknown but must lie in an open interval \(\Omega \). Let \({I_0}\left( \theta \right)\) denote the Fisher information in \(X\) . Suppose now that the

parameter \(\theta \) is replaced by a new parameter \(\mu \), where\(\theta = \psi \left( \mu \right)\) and\(\psi \) is a differentiable function. Let \({I_1}\left( \mu \right)\)denote the Fisher information in X when the parameter is regarded as \(\mu \). Show thatShow that\({I_1}\left( \mu \right) = {\left( {{\psi ^{'}}\left( \mu \right)} \right)^{2}}{I_0}\left( {\psi \left( \mu \right)} \right)\)
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