Law of iterated expectations
Summary
- \(X,Y\) are two random variables. The law of iterated expectations states that
\[E\left( E\left( X \right) \right)=E\left( Y \right)\]
Example, discrete random variables
- \(f(x,y)\) is defined by the following table:
- \(E\left( X \right)\) is a random variable \(Z\) defined by \(f_Z\left( 0.2 \right)=0.5\) , \(f_Z\left( 0.8 \right)=0.5\) . The expected value of \(Z\) is
\[E\left( Z \right)=E\left( E\left( X \right) \right)=0.2⋅0.5+0.8⋅0.5=0.5\]
\[E\left( Y \right)=0⋅0.5+1⋅0.5=0.5\]
Example, continuous random variables
- \(f\left( x,y \right)=x+y\) for \(0≤x≤ 1\) and \(0≤y≤ 1\) . \(E\left( X \right)\) is then the random variable
\[Z= \frac{3X+2}{6X+3}\]
- and
\[E\left( Z \right)=E\left( E\left( X \right) \right)=\int_{5/9}^{2/3}{ z⋅ \frac{1}{18{\left( 2z-1 \right)}^3}dz }= \frac{7}{12}\]
\[E\left( Y \right)=\int_{0}^{1}{ y\left( y+ \frac{1}{2} \right)dx }= \frac{7}{12}\]