Problem 3.20

• $X_{i} \sim N(\mu, \sigma) \ (i = 1, \ldots, n)$
  • i.i.d.

\[\begin{eqnarray} T_{1} & := & \sum_{j=1}^{n} (X_{i} - \mu) \nonumber \\ T_{2} & := & \sum_{j=1}^{n} (X_{i} - \mu)^{2} \nonumber \end{eqnarray}\]

Then the followings hold:

• (1) $T_{1}$ and $T_{2}$ are independet,

• (2) $T_{1}$ is normal distribution $N(0, n\sigma^{2})$, that is, p.d.f. $f_{T_{1}}$ of $T_{1}$ is given by

\[f_{T_{1}}(t_{1}) = \frac{ 1 }{ (2 \pi n \sigma^{2})^{1/2} } \exp \left( \frac{1}{2 n \sigma^{2}} t^{2} \right)\]

• (3) $T_{2}$ is $\xi^{2}$ distribution with $n$ degree of freedom, that is, p.d.f. $f_{T_{2}}$ of $T_{2}$ is given by

\[f_{T_{1}}(t_{1}) = \frac{ 1 }{ (2 \pi n \sigma^{2})^{1/2} } \exp \left( \frac{1}{2 n \sigma^{2}} t^{2} \right)\]

proof

\[\begin{eqnarray} \bar{X} & := & \frac{1}{n} \sum_{j=1}^{n} X_{j} \sim \mathrm{N}(\mu, \sigma^{2}) \nonumber \\ S & := & \frac{1}{n} \sum_{j=1}^{n} (X_{j} - \bar{X})^{2} \nonumber \\ Y & := & \frac{ nS }{ \sigma^{2}(n - 1) } \sim \chi^{2}(n - 1) \nonumber \end{eqnarray}\] \[\begin{eqnarray} p_{\bar{X}, Y}(\bar{x}, y) & := & p_{\bar{X}}(\bar{x}) p_{Y}(y) \nonumber \\ & = & \frac{1}{(2\pi\sigma^{2})^{1/2}} \exp \left( - \frac{1}{2\sigma^{2}} (\bar{x} - \mu)^{2} \right) \frac{1}{\Gamma((n - 1)/2)} (\frac{1}{2})^{(n - 1)/2} y^{\frac{(n - 1) - 2}{2}} e^{-\frac{y}{2}} \nonumber \end{eqnarray}\] \[\begin{eqnarray} T_{1} & = & \sum_{j=1}^{n} X_{j} - n\mu \nonumber \\ & = & n \frac{1}{n} \sum_{j=1}^{n} X_{j} - n \mu \nonumber \\ & = & n \bar{X} - n \mu \nonumber \\ T_{2} & = & \sum_{j=1}^{n} X_{j}^{2} - 2\mu \sum_{j=1}^{n} X_{j} + n \mu^{2} \nonumber \\ & = & \sum_{j=1}^{n} X_{j}^{2} - 2\mu n \frac{1}{n} \sum_{j=1}^{n} X_{j} + n \mu^{2} \nonumber \\ & = & \sum_{j=1}^{n} X_{j}^{2} + 2\bar{X} \sum_{j=1}^{n} X_{j} - 2\bar{X} \sum_{j=1}^{n} X_{j} + n \bar{X}^{2} - n \bar{X}^{2} - 2\mu n \bar{X} + n \mu^{2} \nonumber \\ & = & \sum_{j=1}^{n} (X_{j} - \bar{X})^{2} + 2\bar{X} \sum_{j=1}^{n} X_{j} - n \bar{X}^{2} - 2\mu n \bar{X} + n \mu^{2} \nonumber \\ & = & n S + 2\bar{X}^{2} n - n \bar{X}^{2} - 2\mu n \bar{X} + n \mu^{2} \nonumber \\ & = & n S + \bar{X}^{2} n - 2\mu n \bar{X} + n \mu^{2} \end{eqnarray}\]

Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be

\[\begin{eqnarray} (g^{-1}(\bar{x}, y))_{1} & := & n \bar{x} - n \mu \nonumber \\ (g^{-1}(\bar{x}, y))_{2} & := & \sigma^{2}(n - 1) y + n \bar{x}^{2} - 2 n \mu \bar{x} + n \mu^{2} \nonumber \\ (g(t_{1}, t_{2}))_{1} & = & \frac{t_{1}}{n} + \mu \nonumber \\ (g(t_{1}, t_{2}))_{2} & = & \frac{ t_{2} - n \mu^{2} + 2 n\mu (\frac{t_{1}}{n} + \mu) - n \left( \frac{t_{1}}{n} + \mu \right)^{2} }{ \sigma^{2} (n - 1) } \nonumber \\ & = & \frac{ t_{2} - n \mu^{2} + 2 \mu t_{1} + 2 n\mu^{2} - \left( \frac{t_{1}^{2}}{n} + 2 \mu t_{1} + n \mu^{2} \right) }{ \sigma^{2} (n - 1) } \nonumber \\ & = & \frac{ t_{2} - 2 n \mu^{2} + 2 \mu t_{1} + 2n \mu^{2} - \frac{t_{1}^{2}}{n} - 2 \mu t_{1} }{ \sigma^{2} (n - 1) } \nonumber \\ & = & \frac{ t_{2} - \frac{t_{1}^{2}}{n} }{ \sigma^{2} (n - 1) } \nonumber \\ & = & \frac{ n t_{2} - t_{1}^{2} }{ n \sigma^{2} (n - 1) } \end{eqnarray}\]

Then we have

\[\begin{eqnarray} g^{-1}(\bar{X}, Y) & = & (T_{1}, T_{2}) \nonumber \\ g(T_{1}, T_{2}) & = & (\bar{X}, Y) . \nonumber \end{eqnarray}\]

Hence

\[\begin{eqnarray} p_{T_{1}, T_{2}}(t_{1}, t_{2}) = p_{\bar{X}, Y}(g(t_{1}, t_{2})) |J_{g}(t_{1}, t_{2})| \nonumber \end{eqnarray}\]

where \(J_{g}(t_{1}, t_{2})\) is Jacobian matrix of $g$ at \((t_{1}, t_{2})\).

\[\begin{eqnarray} \frac{\partial g_{1}}{\partial t_{1}} & = & \frac{1}{n} \nonumber \\ \frac{\partial g_{1}}{\partial t_{2}} & = & 0 \nonumber \\ \frac{\partial g_{2}}{\partial t_{1}} & = & \frac{-2 t_{1}}{\sigma^{2} (n - 1) n} \nonumber \\ \frac{\partial g_{2}}{\partial t_{2}} & = & \frac{1}{\sigma^{2} (n - 1)} \nonumber \\ J_{g}(t_{1}, t_{2}) & = & \left( \begin{array}{cc} \frac{1}{n} & 0 \\ \frac{-2 t_{1}}{\sigma^{2} (n - 1) n} & \frac{1}{\sigma^{2} (n - 1)} \end{array} \right) \nonumber \\ |J_{g}(t_{1}, t_{2})| & = & \frac{ 1 }{ \sigma^{2}(n - 1)n } \nonumber \end{eqnarray}\]

Therefore we can explicitly write the joint p.d.f. as follows:

\[\begin{eqnarray} p_{T_{1}, T_{2}}(t_{1}, t_{2}) & = & p_{\bar{X}}( \frac{t_{1}}{n} + \mu ) p_{Y} \left( \frac{ n t_{2} - t_{1}^{2} }{ n \sigma^{2} (n - 1) } \right) \frac{ 1 }{ \sigma^{2}(n - 1)n } \nonumber \\ & = & \frac{1}{(2\pi\sigma^{2})^{1/2}} \exp \left( - \frac{1}{2\sigma^{2}} (\frac{t_{1}}{n})^{2} \right) \frac{1}{\Gamma((n - 1)/2)} (\frac{1}{2})^{(n - 1)/2} \left( \frac{ n t_{2} - t_{1}^{2} }{ n \sigma^{2} (n - 1) } \right)^{\frac{n - 3}{2}} \exp \left( -\frac{ \frac{ n t_{2} - t_{1}^{2} }{ n \sigma^{2} (n - 1) } }{ 2 } \right) \frac{ 1 }{ \sigma^{2}(n - 1)n } \nonumber \\ & = & \frac{ 1 }{ (2\pi\sigma^{2}n^{2})^{1/2} \sigma^{2}(n - 1) } \exp \left( - \frac{1}{2\sigma^{2}n^{2}} t_{1}^{2} - \frac{ n t_{2} - t_{1}^{2} }{ 2 n \sigma^{2} (n - 1) } \right) \frac{1}{\Gamma((n - 1)/2)} (\frac{1}{2})^{(n - 1)/2} \left( \frac{ n t_{2} - t_{1}^{2} }{ n \sigma^{2} (n - 1) } \right)^{\frac{n - 3}{2}} \nonumber \\ & = & \frac{ 1 }{ (2\pi\sigma^{2}n^{2})^{1/2} \sigma^{2}(n - 1) 2^{(n - 1)/2} \Gamma(\frac{n - 1}{2}) (n \sigma^{2} (n - 1))^{\frac{n - 3}{2}} } \exp \left( \frac{ -(n - 1) t_{1}^{2} + t_{1}^{2}n }{ 2\sigma^{2}n^{2} (n - 1) } - \frac{ t_{2} }{ 2 \sigma^{2} (n - 1) } \right) \left( n t_{2} - t_{1}^{2} \right)^{\frac{n - 3}{2}} \nonumber \\ & = & K \exp \left( \frac{ t_{1}^{2} }{ 2\sigma^{2}n^{2} (n - 1) } - \frac{ t_{2} }{ 2 \sigma^{2} (n - 1) } \right) \left( n t_{2} - t_{1}^{2} \right)^{\frac{n - 3}{2}} \nonumber \\ & = & K \exp \left( \frac{ - \left( t_{1}n\mu(n - 1) - t_{1} \right)^{2} - t_{1}^{2} n^{2} \mu^{2} (n - 1)^{2} ( + (n - 1) 2t_{1}n\mu + (n - 1) n^{2}\mu^{2} ) }{ 2\sigma^{2}n^{2} (n - 1) } - \frac{ t_{2} }{ 2 \sigma^{2} (n - 1) } \right) \left( n t_{2} - t_{1}^{2} \right)^{\frac{n - 3}{2}} \nonumber \end{eqnarray}\]

$\Box$