條件期望與重期望
條件期望的定義：
\(E(x|y)=\int_{-\infty}^{\infty}xf(x|y)dx\)（連續）
\(E(x|y)=\sum\limits_ix_i\rho(X=x_i|Y=y_i)\)（離散）
重期望的性質
\(1.E(E(g(x)|Y))=\int_{-\infty}^{\infty}E(E(g(x)|Y))f_{Y}(y)dy\)
=\(\int_{-\infty}^{\infty}[\int_{-\infty}^{\infty}g(x)f(x|y)dx]f_{Y}(y)dy\)
=\(\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(x)f(x|y)f_{Y}(y)dxdy\)
=\(\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(x)f(x|y)dxdy\)
=\(E(g(x))\)
\(2.E(h(y)g(x)|Y)\)