Expectation of an Expectation

Question

I need to solve two exercises:

1. Calculate V[u_i|x_i] using E[y_i|x_i] and u_i = y_i - E[y_i].
2. Calculate E[$y^2$|x_i].

Information given for the exercise:
E[$u^2$|x_i] = V[y_i|x_i]
E[$u^2$|x_i] = V[y_i|x_i] because E[u_i|x_i]

There are also a values given for E[y_i] = 2λ_i, V[y_i] = 2$λ^2$ and λ_i = (e$^{xβ}$)/2

What I know:
V[y_i|x_i] = V[$y^2$|x_i] = E[$y^2$|x_i] - $(E[y|x])^2$

It would be easy to set the value for E[y_i] into u and square it. Afterward set the solution into E[$u^2$|x_i] and V[y_i|x_i] would be solved.
Obviously I wouldn't use E[y_i|x_i] in this case, so I tried it by rearranging terms, but often end at a point where they cancel each other out and I am back at the beginning.

So my question is: How do I properly derive V[u_i|x_i] and E[$y^2$|x]. Especially important for me is how to link it to E[y_i|x_i].

Answer 1

In the part where you write "what I know" it does not make sense to say $V(y|x)=V(y^2|x)$.

1. $V(u|x)=V(y-E(y)|x)=V(y|x)$ (because E(y) is a constant). This in turn is $E(y^2|x)-\left[E(y|x)\right]^2$.

2. Also, $E(y^2|x)=V(y|x)+\left[E(y|x)\right]^2$

But, I am not clear how all these conditional expectations and variances tie into the particular values. One way to do it is for (2), $\frac{e^{2x\beta}}{2}+e^{2x\beta}=\frac{3}{2}e^{2x\beta}$. And for (1), we can go directly from $V(y|x)=\frac{e^{2x\beta}}{2}$.