I have a doubt related to the MLE theory. Suppose we have a sample of i.i.d. real valued random variables $(Y_i, X_i)_{i=1}^n$ with $Y_i=X_i\beta+u_i$, $\beta\in B \subseteq \mathbb{R}$, $u_i|X_i\sim N(0,\sigma^2)$, $0<\sigma^2<\infty$. Then the sample log-likelihood function of $Y_i$ conditional on $X_i$ is $$ \frac{1}{n}\sum_{i=1}^n\log(f(Y_i|X_i;\beta, \sigma^2))=-\frac{n}{2}\log(2\pi)-\frac{n}{2}\log(\sigma^2)-\frac{\sum_{i=1}^n(Y_i-X_i\beta)^2}{2\sigma^2} $$
by the LLN we can write that $$ \frac{1}{n}\sum_{i=1}^n\log(f(Y_i|X_i;\beta, \sigma^2)) \rightarrow_p E(\log(f(Y_i|X_i;\beta, \sigma^2))) $$
Question: what is exactly $E(\log(f(Y_i|X_i;\beta, \sigma^2)))$? Is it computed using the distribution of $Y_i$ conditional on $X_i$ or the joint distribution of $Y_i,X_i$?