期待値・分散を条件付き期待値・条件付き分散で表す

期待値を条件付き期待値で表す

\(\displaystyle E_Y[Y]=\int_y y f_Y(y)dy\)

\(\displaystyle = \int_y y \left(\int_x f(x,y)dx\right)dy\)

\(\displaystyle = \int_y y \left(\int_x f_{Y|X}(y|x)f_X(x)dx\right)dy\)

\(\displaystyle = \int_x \left(\int_y yf_{Y|X}(y|x)dy\right)f_X(x)dx\)

\(\displaystyle = \int_x E_{Y|X}[Y|X]f_X(x)dx\)

\(\displaystyle = E_X\left[ E_{Y|X}[Y|X]\right] \quad \cdots (1)\)

分散を条件付き期待値と条件付き分散で表す

\(V_Y[Y]=E_Y[Y^2]-E_Y[Y]^2\)

\(=E_X\left[E_{Y|X}[Y^2|X]\right]\)\(-E_X\left[E_{Y|X}[Y|X]\right]^2 \quad \)\((1)\)を利用

\(=E_X\left[V_{Y|X}[Y|X]+E_{Y|X}[Y|X]^2\right]\)\(-E_X\left[E_{Y|X}[Y|X]\right]^2 \)

\(=E_X\left[V_{Y|X}[Y|X]\right]\)\(+E_X\left[E_{Y|X}[Y|X]^2\right]\)\(-E_X\left[E_{Y|X}[Y|X]\right]^2 \)

\(=E_X\left[V_{Y|X}[Y|X]\right]\)\(+V_X\left[E_{Y|X}[Y|X]\right]\)