統計検定 1級 過去問 解答/解答例と解説
2014年11月30日 (日) 試験

$$\begin{align} P(Y_{11} = y_{11}) &= \frac{y_{1 \cdot}!}{y_{11}!~(y_{1 \cdot} - y_{11})!} \left(\frac{N p_{11}}{y_{1 \cdot}}\right)^{y_{11}}\left(\frac{N p_{12}}{y_{1 \cdot}}\right)^{y_{1 \cdot} - y_{11}} \\ &= \frac{y_{1 \cdot}!}{y_{11}!~y_{12}!} \left(\frac{N p_{11}}{y_{1 \cdot}}\right)^{y_{11}}\left(\frac{N p_{12}}{y_{1 \cdot}}\right)^{y_{1 \cdot} - y_{11}} \\ \end{align}$$

$$\begin{align} P(Y_{21} = y_{21}) &= \frac{y_{2 \cdot}!}{y_{21}!~y_{22}!} \left(\frac{N p_{21}}{y_{2 \cdot}}\right)^{y_{21}}\left(\frac{N p_{22}}{y_{2 \cdot}}\right)^{y_{2 \cdot} - y_{21}} \\ \end{align}$$

$$\begin{align} & P(Y_{11} = y_{11},~Y_{21} = y_{21}) \\ &~~~~= P(Y_{11} = y_{11})\,P(Y_{21} = y_{21}) \\ &~~~~= \frac{y_{1 \cdot}!~y_{2 \cdot}!}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\left(\frac{N p_{11}}{y_{1 \cdot}}\right)^{y_{11}}\left(\frac{N p_{12}}{y_{1 \cdot}}\right)^{y_{1 \cdot} - y_{11}}\left(\frac{N p_{21}}{y_{2 \cdot}}\right)^{y_{21}}\left(\frac{N p_{22}}{y_{2 \cdot}}\right)^{y_{2 \cdot} - y_{21}} \\ &~~~~\propto \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\left(\frac{N p_{11}}{y_{1 \cdot}}\right)^{y_{11}}\left(\frac{N p_{12}}{y_{1 \cdot}}\right)^{- y_{11}}\left(\frac{N p_{21}}{y_{2 \cdot}}\right)^{y_{21}}\left(\frac{N p_{22}}{y_{2 \cdot}}\right)^{- y_{21}} \\ &~~~~= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\left(\frac{N p_{11}}{y_{1 \cdot}}\right)^{y_{11}}\left(\frac{y_{1 \cdot}}{N p_{12}}\right)^{y_{11}}\left(\frac{N p_{21}}{y_{2 \cdot}}\right)^{y_{21}}\left(\frac{y_{2 \cdot}}{N p_{22}}\right)^{y_{21}} \\ &~~~~= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!} \left(\frac{p_{11}}{p_{12}}\right)^{y_{11}} \left(\frac{p_{21}}{p_{22}}\right)^{y_{21}} \\ &~~~~= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!} \left(\frac{p_{11}}{p_{12}}\right)^{y_{11}} \left(\frac{p_{21}}{p_{22}}\right)^{y_{\cdot 1} - y_{11}} \\ &~~~~= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!} \left(\frac{p_{11}p_{22}}{p_{12}p_{21}}\right)^{y_{11}} \left(\frac{p_{21}}{p_{22}}\right)^{y_{\cdot 1}} \\ &~~~~= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!} \left(e^{\delta}\right)^{y_{11}} \left(e^{\gamma}\right)^{y_{\cdot 1}} \\ &~~~~= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\exp[y_{\cdot 1}\gamma + y_{11}\delta] \\ \end{align}$$

$$\begin{equation} P(Y_{11} = y_{11},~Y_{21} = y_{21}) = \frac{C(\gamma, \delta)}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\exp[y_{\cdot 1}\gamma + y_{11}\delta] \end{equation}$$

$$\begin{equation} P(Y_{11} = y_{11},~Y_{12} = y_{12}) = \frac{C(\beta, \delta)}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\exp[y_{1 \cdot}\beta + y_{11}\delta] \end{equation}$$

$$\begin{align} &P(Y_{11} = y_{11},~Y_{12} = y_{12},~Y_{21} = y_{21},~Y_{22} = y_{22}) \\ &= \frac{N!}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}{p_{11}}^{y_{11}}{p_{12}}^{y_{12}}{p_{21}}^{y_{21}}{p_{22}}^{y_{22}} \\ &\propto \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}{p_{11}}^{y_{11}}{p_{12}}^{y_{12}}{p_{21}}^{y_{21}}{p_{22}}^{y_{22}} \\ &= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}{p_{11}}^{y_{11}}{p_{12}}^{y_{12}}{p_{21}}^{y_{21}}{p_{22}}^{N - y_{11} - y_{12} - y_{21}} \\ &\propto \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}{p_{11}}^{y_{11}}{p_{12}}^{y_{12}}{p_{21}}^{y_{21}}{p_{22}}^{- y_{11} - y_{12} - y_{21}} \\ &= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\left(\frac{p_{11}}{p_{22}}\right)^{y_{11}}\left(\frac{p_{12}}{p_{22}}\right)^{y_{12}}\left(\frac{p_{21}}{p_{22}}\right)^{y_{21}} \\ &= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\left(\frac{p_{11}}{p_{22}}\right)^{y_{11}}\left(\frac{p_{12}}{p_{22}}\right)^{y_{1\cdot} - y_{11}}\left(\frac{p_{21}}{p_{22}}\right)^{y_{\cdot 1} - y_{11}} \\ &= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\left(\frac{p_{11}p_{22}}{p_{12}p_{21}}\right)^{y_{11}}\left(\frac{p_{12}}{p_{22}}\right)^{y_{1\cdot}}\left(\frac{p_{21}}{p_{22}}\right)^{y_{\cdot 1}} \\ &= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\left(e^{\delta}\right)^{y_{11}}\left(e^{\beta}\right)^{y_{1\cdot}}\left(e^{\gamma}\right)^{y_{\cdot 1}} \\ &= \frac{1}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\exp\left[y_{1\cdot}\beta + y_{\cdot 1}\gamma + y_{11}\delta\right] \\ \end{align}$$

$$\begin{equation} P(Y_{11} = y_{11},~Y_{12} = y_{12},~Y_{21} = y_{21},~Y_{22} = y_{22}) = \frac{C(\beta, \gamma, \delta)}{y_{11}!~y_{12}!~y_{21}!~y_{22}!}\exp[y_{1 \cdot}\beta + y_{\cdot 1}\gamma + y_{11}\delta] \end{equation}$$