Bias$-$Variance Tradeoff

$($a$)$ Derive the bias$-$variance decomposition for the squared error loss function. That is$,$ show that

for a model $f_S$ trained on a dataset $S$ to predict a target $y(x)$ for each $x,$

$E_S[E_{\text{o}\text{u}\text{t}}(f_S)]=E_x[Bias(x)+$Var$(x)]$

given the following definitions:

$F(x)=E_S[f_S(x)]$

$E_{\text{o}\text{u}\text{t}}(f_S)=E_x[(f_S(x)-y(x){)}^2]$

Bias$(x)=(F(x)-y(x){)}^2$

Var$(x)=E_S[(f_S(x)-F(x){)}^2]$