$\operatorname E(Y_i^2\mid X_i=x) $in regression

Question

I have a regression equation of the form \begin{equation} Y_i = f(X_i) + \varepsilon_i \end{equation}

I'm trying to figure out what $\operatorname E(Y_i^2\mid X_i=x)$ in this case. I know that \begin{equation} \operatorname{Var}(Y_i\mid X_i=x) = \operatorname E(Y_i^2\mid X_i=x)-(\operatorname E(Y_i\mid X_i=x))^2. \end{equation} and \begin{equation} \operatorname E(Y_i\mid X_i=x) = f(x) \end{equation} but I'm unsure how to figure out what $\operatorname E(Y_i^2\mid X_i=x)$ is.

Answer 1

\begin{align} \mathbb{E}[Y_i^2|X_i=x] &= Var(Y_i|X_i = x) + ( \mathbb{E}[Y_i|X_i = x] )^2 \\ &=Var(f(X) + \epsilon_i|X_i = x) + ( \mathbb{E}[f(X) + \epsilon_i|X_i=x] ) ^ 2 \\ & = Var(\epsilon_i|X) + ( f(x_i) + \mathbb{E}[\epsilon_i|X] ) ^ 2\\ & = \sigma^2 + f^2(x_i). \end{align}