stylistic:style_transfer, stylistic:tone
Decomposition

The conditional expectation E (X ∣ Y) {\displaystyle \operatorname {E} (X\mid Y)} of X {\displaystyle X} given Y {\displaystyle Y}, and the conditional variance Var (X ∣ Y) {\displaystyle \operatorname {Var} (X\mid Y)} may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation E (X ∣ Y = y) {\displaystyle \operatorname {E} (X\mid Y=y)} given the event Y = y. This quantity depends on the particular value y; it is a function g (y) = E (X ∣ Y = y) {\displaystyle g(y)=\operatorname {E} (X\mid Y=y)}. That same function evaluated at the random variable Y is the conditional expectation E (X ∣ Y) = g (Y). {\displaystyle \operatorname {E} (X\mid Y)=g(Y).}