Hi I am working on problem 6.8 from Sheldon Ross's Stochastic Processes book.
The given solution at the back of the book goes like:
Here I am confused on the red boxed part. How the assumption of X and Y's independency allows to go from $E[X_n|X_i + Y_i, i = 1,..,n-1]$ to $E[X_n|X_i, i = 1,..,n-1]$ in the inner expectation? I am having hard time to "prove" it systematically using properties of expectation.
For the problem, I am thinking my solution more like this:
$$E[X_n|X_i + Y_i, i = 1,..,n-1]$$ $$ = E[E[X_n|X_i + Y_i, i = 1,..,n-1]|Y_1,..., Y_{n-1}]$$ $$ = E[E[X_n|X_i, i = 1,..,n-1]|Y_1,..., Y_{n-1}]$$ $$ = E[X_{n-1}|Y_1,..., Y_{n-1}]$$ $$ = X_{n-1}$$
Could someone help me understand how to logically come to the rectangular boxed portion of the official solution?