206 .. math:: E[g(Y)·X|Y] = g(Y)·E[X|Y]

   207 

   208 For invertible funtion :math:`h`:

   209 

   210 .. math:: E[X|h(Y)] = E[X|Y]

   211 

   212 Proof:

   213 

   214 .. math::

   215 

   216    E[X|Y=y] = E[X|h(Y)=h(y)]

   217 

   218 Law of Iterated Expectations

   219 ============================

   220 

   228 

   229 Generalisation of Law of Iterated Expectations:

   230 

   231 .. math:: E[E[X|Y,Z]|Y] = E[X|Y]

   232 

   233 Proof, for each :math:`y∈Y`:

   234 

   235 .. math::

   236 

   237    E[X|Y=y] = ∑_x\ x·p_{X|Y}(x|Y=y) = ∑_x\ x·p_{X,Y}(x,y)/p_Y(y)

   238 

   239    = ∑_x\ x·∑_z\ p_{X,Y,Z}(x,y,z)/p_Y(y)

   240 

   241    = ∑_x\ x·∑_z\ p_{X|Y,Z}(x|Y=y,Z=z)·p_{Y,Z}(y,z)/p_Y(y)

   242 

   243    = ∑_x\ x·∑_z\ p_{X|Y,Z}(x|Y=y,Z=z)·p_{Z|Y}(z|Y=y)

   244 

   245    = ∑_x\ ∑_z\ x·p_{X|Y,Z}(x|Y=y,Z=z)·p_{Z|Y}(z|Y=y)

   246 

   247    = ∑_z\ ∑_x\ x·p_{X|Y,Z}(x|Y=y,Z=z)·p_{Z|Y}(z|Y=y)

   248 

   249    = ∑_z\ p_{Z|Y}(z|Y=y)·∑_x\ x·p_{X|Y,Z}(x|Y=y,Z=z)

   250 

   251    = ∑_z\ p_{Z|Y}(z|Y=y)·E[X|Y,Z] = E[E[X|Y,Z]|Y=y]