Conditional Expectation of a variable given a sum of two random variables

Question

Suppose that:
$x_i = z_i + v_i,~~~~~ u_i = pv_i + e_i$
where $z_i, v_i, e_i$ are independent random variables ~$N(0,1)$.

I am trying to find:
$E[u_i | x_i] = E[pv_i + e_i | x_i]$
$~~~~~~~~~~~~~~= E[pv_i| x_i] + E[ e_i | x_i]$
$~~~~~~~~~~~~~~= E[pv_i| z_i + v_i]$
$~~~~~~~~~~~~~~= pE[v_i| z_i + v_i]$
I'm not sure where to go from here. How do I find the expected value given the sum of these two random variables?

Answer 1

The random variables $z_i$ and $v_i$ are i.i.d., so the vectors $(z_i,z_i+v_i)$ and $(v_i,z_i+v_i)$ have the same distribution. One can then check that $$E[z_i\mid z_i+v_i]=E[v_i\mid z_i+v_i]\tag{1}$$ almost surely. Now, by linearity, $$E[z_i\mid z_i+v_i]+E[v_i\mid z_i+v_i]=E[z_i+v_i\mid z_i+v_i]=z_i+v_i.\tag{2}$$ Combining $(1)$ and $(2)$ gives $E[v_i\mid z_i+v_i]=\frac12(z_i+v_i)$.