Question

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

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

Step-by-step solution

Given information

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

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

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 \(\sqrt {\frac{3}{2}} \).